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Seminars (RGM)

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Event When Speaker Title Presentation Material
RGMW01 12th January 2015
10:00 to 11:00
JP Miller Gaussian Free Field 1
RGMW01 12th January 2015
11:30 to 12:30
Random Planar Maps 1
A map is a gluing of a finite number of polygons, forming a connected orientable topological surface. It can be interpreted as assigning this surface a discrete geometry, and the theoretical physics literature in the 80-90’s argued that random maps are an appropriate discrete model for the theory of 2-dimensional quantum gravity, which involves ill-defined integrals over all metrics on a given surface. The idea is to replace these integrals by finite sums, for instance over all triangulation of the sphere with a large number of faces, hoping that such triangulations approximate a limiting “continuum random surface”.

In the recent years, much progress has been made in the mathematical understanding of the latter problem. In particular, it is now known that many natural models of random planar maps, for which the faces degrees remain small, admit a universal scaling limit, the Brownian map.

Other models, favorizing large faces, also admit a one-parameter family of scaling limits, called stable maps. The latter are believed to describe the asymptotic geometry of random maps carrying statistical physics models, as has now been established in some important cases (including the so-called rigid O(n) model on quadrangulations).

This mini-course will review the main aspects of these themes.

RGMW01 13th January 2015
13:30 to 14:30
Schramm-Loewner Evolution 1
RGMW01 13th January 2015
15:00 to 16:00
Random Planar Maps 2
A map is a gluing of a finite number of polygons, forming a connected orientable topological surface. It can be interpreted as assigning this surface a discrete geometry, and the theoretical physics literature in the 80-90’s argued that random maps are an appropriate discrete model for the theory of 2-dimensional quantum gravity, which involves ill-defined integrals over all metrics on a given surface. The idea is to replace these integrals by finite sums, for instance over all triangulation of the sphere with a large number of faces, hoping that such triangulations approximate a limiting “continuum random surface”.

In the recent years, much progress has been made in the mathematical understanding of the latter problem. In particular, it is now known that many natural models of random planar maps, for which the faces degrees remain small, admit a universal scaling limit, the Brownian map.

Other models, favorizing large faces, also admit a one-parameter family of scaling limits, called stable maps. The latter are believed to describe the asymptotic geometry of random maps carrying statistical physics models, as has now been established in some important cases (including the so-called rigid O(n) model on quadrangulations).

This mini-course will review the main aspects of these themes.

RGMW01 13th January 2015
16:00 to 17:00
JP Miller Gaussian Free Field 2
RGMW01 14th January 2015
09:00 to 10:00
Random Planar Maps 3
A map is a gluing of a finite number of polygons, forming a connected orientable topological surface. It can be interpreted as assigning this surface a discrete geometry, and the theoretical physics literature in the 80-90’s argued that random maps are an appropriate discrete model for the theory of 2-dimensional quantum gravity, which involves ill-defined integrals over all metrics on a given surface. The idea is to replace these integrals by finite sums, for instance over all triangulation of the sphere with a large number of faces, hoping that such triangulations approximate a limiting “continuum random surface”.

In the recent years, much progress has been made in the mathematical understanding of the latter problem. In particular, it is now known that many natural models of random planar maps, for which the faces degrees remain small, admit a universal scaling limit, the Brownian map.

Other models, favorizing large faces, also admit a one-parameter family of scaling limits, called stable maps. The latter are believed to describe the asymptotic geometry of random maps carrying statistical physics models, as has now been established in some important cases (including the so-called rigid O(n) model on quadrangulations).

This mini-course will review the main aspects of these themes.

RGMW01 14th January 2015
10:00 to 11:00
Schramm-Loewner Evolution 2
RGMW01 15th January 2015
13:30 to 14:30
Schramm-Loewner Evolution 3
RGMW01 15th January 2015
15:00 to 16:00
JP Miller Gaussian Free Field 3
RGMW01 15th January 2015
16:00 to 17:00
Random Planar Maps 4
A map is a gluing of a finite number of polygons, forming a connected orientable topological surface. It can be interpreted as assigning this surface a discrete geometry, and the theoretical physics literature in the 80-90’s argued that random maps are an appropriate discrete model for the theory of 2-dimensional quantum gravity, which involves ill-defined integrals over all metrics on a given surface. The idea is to replace these integrals by finite sums, for instance over all triangulation of the sphere with a large number of faces, hoping that such triangulations approximate a limiting “continuum random surface”.

In the recent years, much progress has been made in the mathematical understanding of the latter problem. In particular, it is now known that many natural models of random planar maps, for which the faces degrees remain small, admit a universal scaling limit, the Brownian map.

Other models, favorizing large faces, also admit a one-parameter family of scaling limits, called stable maps. The latter are believed to describe the asymptotic geometry of random maps carrying statistical physics models, as has now been established in some important cases (including the so-called rigid O(n) model on quadrangulations).

This mini-course will review the main aspects of these themes.

RGMW01 16th January 2015
09:00 to 10:00
Discrete Lattice Models 1
RGMW01 16th January 2015
10:00 to 11:00
Discrete Lattice Models 2
RGMW01 19th January 2015
10:00 to 11:00
Gaussian Multiplicative Chaos 1
RGMW01 19th January 2015
11:30 to 12:30
JP Miller Gaussian Free Field 4
RGMW01 19th January 2015
13:30 to 14:30
Gaussian Multiplicative Chaos 2
RGMW01 19th January 2015
15:00 to 16:00
Discrete Lattice Models 3
RGMW01 20th January 2015
09:00 to 10:00
Schramm-Loewner Evolution 4
RGMW01 20th January 2015
10:00 to 11:00
Gaussian Multiplicative Chaos 3
RGMW01 20th January 2015
13:30 to 14:30
Gaussian Multiplicative Chaos 4
RGMW01 20th January 2015
15:00 to 16:00
JP Miller Gaussian Free Field 5
RGMW01 21st January 2015
09:00 to 10:00
Discrete Lattice Models 4
RGMW01 21st January 2015
10:00 to 11:00
Schramm-Loewner Evolution 5
RGMW01 21st January 2015
11:30 to 12:30
Random Planar Maps 5
A map is a gluing of a finite number of polygons, forming a connected orientable topological surface. It can be interpreted as assigning this surface a discrete geometry, and the theoretical physics literature in the 80-90’s argued that random maps are an appropriate discrete model for the theory of 2-dimensional quantum gravity, which involves ill-defined integrals over all metrics on a given surface. The idea is to replace these integrals by finite sums, for instance over all triangulation of the sphere with a large number of faces, hoping that such triangulations approximate a limiting “continuum random surface”.

In the recent years, much progress has been made in the mathematical understanding of the latter problem. In particular, it is now known that many natural models of random planar maps, for which the faces degrees remain small, admit a universal scaling limit, the Brownian map.

Other models, favorizing large faces, also admit a one-parameter family of scaling limits, called stable maps. The latter are believed to describe the asymptotic geometry of random maps carrying statistical physics models, as has now been established in some important cases (including the so-called rigid O(n) model on quadrangulations).

This mini-course will review the main aspects of these themes.

RGMW01 21st January 2015
13:30 to 14:30
Gaussian Multiplicative Chaos 5
RGMW01 21st January 2015
15:00 to 16:00
J Miller Gaussian Multiplicative Chaos 6
RGMW01 22nd January 2015
09:00 to 10:00
Schramm-Loewner Evolution 6
RGMW01 22nd January 2015
10:00 to 11:00
Random Planar Maps 6
A map is a gluing of a finite number of polygons, forming a connected orientable topological surface. It can be interpreted as assigning this surface a discrete geometry, and the theoretical physics literature in the 80-90’s argued that random maps are an appropriate discrete model for the theory of 2-dimensional quantum gravity, which involves ill-defined integrals over all metrics on a given surface. The idea is to replace these integrals by finite sums, for instance over all triangulation of the sphere with a large number of faces, hoping that such triangulations approximate a limiting “continuum random surface”.

In the recent years, much progress has been made in the mathematical understanding of the latter problem. In particular, it is now known that many natural models of random planar maps, for which the faces degrees remain small, admit a universal scaling limit, the Brownian map.

Other models, favorizing large faces, also admit a one-parameter family of scaling limits, called stable maps. The latter are believed to describe the asymptotic geometry of random maps carrying statistical physics models, as has now been established in some important cases (including the so-called rigid O(n) model on quadrangulations).

This mini-course will review the main aspects of these themes.

RGMW01 22nd January 2015
11:30 to 12:30
Gaussian Multiplicative Chaos 6
RGMW01 22nd January 2015
15:00 to 16:00
Discrete Lattice Models 5
RGMW01 23rd January 2015
09:00 to 10:00
JP Miller Gaussian Free Field 7
RGMW01 23rd January 2015
10:00 to 11:00
Gaussian Multiplicative Chaos 7
RGMW01 23rd January 2015
11:30 to 12:30
Discrete Lattice Models 6
RGMW01 23rd January 2015
13:30 to 14:30
Schramm-Loewner Evolution 7
RGMW01 23rd January 2015
15:00 to 16:00
Random Planar Maps 7
A map is a gluing of a finite number of polygons, forming a connected orientable topological surface. It can be interpreted as assigning this surface a discrete geometry, and the theoretical physics literature in the 80-90’s argued that random maps are an appropriate discrete model for the theory of 2-dimensional quantum gravity, which involves ill-defined integrals over all metrics on a given surface. The idea is to replace these integrals by finite sums, for instance over all triangulation of the sphere with a large number of faces, hoping that such triangulations approximate a limiting “continuum random surface”.

In the recent years, much progress has been made in the mathematical understanding of the latter problem. In particular, it is now known that many natural models of random planar maps, for which the faces degrees remain small, admit a universal scaling limit, the Brownian map.

Other models, favorizing large faces, also admit a one-parameter family of scaling limits, called stable maps. The latter are believed to describe the asymptotic geometry of random maps carrying statistical physics models, as has now been established in some important cases (including the so-called rigid O(n) model on quadrangulations).

This mini-course will review the main aspects of these themes.

RGMW02 26th January 2015
10:00 to 11:00
W Werner Renormalization via merging trees
Co-authors: Stephane Benoist (Columbia Univ.), Laure Dumaz (Univ. of Cambridge)

We present a rather simple setup in which one can formulate some renormalization group ideas, conjectures as well as some results in a concrete manner: For FK-type random cluster models, one relates the scaling limit of a (slight perturbation of the) critical model in terms of a stationary measure of a Markov process acting on a space of weighted graphs. This setup is not restricted to two dimensions (but one expects to be able to prove results only in 2d at this point).

RGMW02 26th January 2015
11:30 to 12:30
The extremal process in nested conformal loops
By analogy with the Liouville measures constructed by Duplantier and Sheffield in the case of the Gaussian Free Field, we construct a random measure on the unit disc related to a collection of nested conformal loops. Then, we study the extremal process associated to points in the disc with high conformal radius. We show that it gives a decorated Poisson point process, as can be expected from the analogy with branching Brownian motion.
RGMW02 26th January 2015
13:30 to 14:30
Ising Model, Conformal Field Theory, etc
Co-authors: Stéphane Benoist (Columbia), Dmitry Chelkak (ETHZ), Hugo Duminil-Copin (University of Geneva), Konstantin Izyurov (University of Helsinki), Kalle Kytölä (Aalto University), Stanislav Smirnov (University of Geneva), Fredrik Viklund (KTH)

I will give an overview of recent results on the two-dimensional Ising model scaling limit and then explain some results in progress to make some probabilistic sense of Conformal Field Theory ideas.

RGMW02 26th January 2015
15:00 to 16:00
Aperiodic hierarchical conformal tilings: random at the ends?
Co-author: Phil Bowers (Florida State Univ.)

Conformal tilings represent a new chapter in the theory of aperiodic hierarchical tilings, whose most famous example is the Penrose tiling of 'kites' and 'darts'. We move away from tiles with individually rigid euclidean shapes to tiles that are conformally regular and get their rigidity from the global pattern. I will introduce the structure for individual conformal tilings and illustrate with several examples, including the conformal Penrose, snowcube, and pinwheel tilings. At first these might seem quite concrete, but there is profound ambiguity in the long range structure --- indeed, any finite patch can be completed to uncountably many global conformal tilings. In other words, hierarchical tiling families display a type of randomness in their ends.

RGMW02 27th January 2015
09:00 to 10:00
Parafermionic observables and order of the phase transition in planar random-cluster models
Co-authors: Vincent Tassion (Université de Genève), Vladas Sidoravicius (IMPA)

This talk will be devoted to the study of the parafermionic observable for planar random-cluster models. We will present a few applications of such observables, including the determination of the order of the phase transition for cluster-weights q between 1 and 4, as well as the computation of the critical value.

RGMW02 27th January 2015
10:00 to 11:00
Liouville Quantum Gravity on the Riemann sphere
Co-authors: David (), Kupiainen (), Vargas ()

In this talk, I will explain how to rigorously construct Liouville quantum field theory on the Riemann sphere and precise conjectures relating this object to the scaling limit of random planar maps conformally embedded onto the Riemann sphere.

RGMW02 27th January 2015
11:30 to 12:30
tba
Schramm Loewner Evolution (SLE), Conformal Loop Ensemble (CLE), and Gaussian Free Field (GFF) are three important planar objects that arise from Statistical physics and quantum field theory. In the first part of this talk, I will introduce these three objects. In the second part, I will explain the construction of a conformally invariant growing mechanism on CLE4. In the last part, I will discuss two couplings between GFF and CLE4.
RGMW02 27th January 2015
13:30 to 14:30
A conformally invariant metric on CLE(4)
Co-authors: Scott Sheffield (MIT), Hao Wu (MIT)

Werner and Wu introduced a conformally invariant way of exploring the loops in a CLE$_4$ $\Gamma$ in a simply connected domain. Using the relationship between CLE$_4$ and the Gaussian free field, we show that the dynamics of this exploration process are a deterministic function of the CLE$_4$ loops, and we use this fact to construct a conformally invariant metric on $\Gamma$ for which a ball of radius $t$ coincides with the set of loops explored up to time $t$ by the exploration process. It is conjectured that this metric space is related in the $\epsilon \to 0$ limit to the contact graph metric on CLE$_{4+\epsilon}$ as well as the contact graph metric on $\epsilon$-neighborhoods of CLE$_4$ loops.

RGMW02 27th January 2015
15:00 to 16:00
Conformal invariance of boundary touching loops of FK Ising model
Co-author: Stanislav Smirnov (University of Geneva and St. Petersburg State University)

I will present a result showing the full conformal invariance of Fortuin-Kasteleyn representation of Ising model (FK Ising model) at criticality. The collection of all the interfaces, which in a planar model are closed loops, in the FK Ising model at criticality defined on a lattice approximation of a planar domain is shown to converge to a conformally invariant scaling limit as the mesh size is decreased. More specifically, the scaling limit can be described using a branching SLE(?,?-6) with ?=16/3, a variant of Oded Schramm's SLE curves. We consider the exploration tree of the loop collection and the main step of the proof is to find a discrete holomorphic observable which is a martingale for the branch of the exploration tree.

This is a joint work with Stanislav Smirnov (University of Geneva and St. Petersburg State University)

RGMW02 28th January 2015
09:00 to 10:00
SLE correlations and singular vectors
We discuss relations between SLE correlations, in particular the (higher order) differential equations they satisfy, and highest weight representations of the Virasoro algebra.
RGMW02 28th January 2015
10:00 to 11:00
Almost sure multifractal spectrum of SLE
Co-authors: Jason Miller (Massachusetts Institute of Technology), Xin Sun (Massachusetts Institute of Technology)

Suppose that $\eta$ is an SLE$_\kappa$ in a smoothly bounded simply connected domain $D \subset \mathbb C$ and that $\phi$ is a conformal map from the unit disk $\mathbb D$ to a connected component of $D \setminus \eta([0,t])$ for some $t>0$. The multifractal spectrum of $\eta$ is the function $(-1,1) \rightarrow [0,\infty)$ which, for each $s \in (-1,1)$, gives the Hausdorff dimension of the set of points $x \in \partial \mathbb D$ such that $|\phi'( (1-\epsilon) x)| = \epsilon^{-s+o(1)}$ as $\epsilon \rightarrow 0$. I will present a rigorous computation of the a.s. multifractal spectrum of SLE (joint with J. Miller and X. Sun), which confirms a prediction due to Duplantier. The proof makes use of various couplings of SLE with the Gaussian free field. As a corollary, we also confirm a conjecture of Beliaev and Smirnov for the a.s. bulk integral means spectrum of SLE.

RGMW02 28th January 2015
11:30 to 12:30
From the critical Ising model to spanning trees
We will show an explicit mapping between the squared partition function of the critical Ising model defined on isoradial graphs and the partition function of critical spanning trees, thus proving a strong relation between two classical models of statistical mechanics.
RGMW02 28th January 2015
13:30 to 14:30
GFF with SLE and KPZ
This talk concerns three objects that remain in fashion: the Gaussian free field (GFF), the Schramm-Loewner Evolution (SLE) and the Liouville measure. We first discuss a way to derive the exponential moments of the winding for the chordal SLE curves and then show how this enters into determining the quantum fractal dimension for the SLE curves in their flow line coupling with the GFF. It comes out that the usual KPZ relation does not hold for the SLE flow lines of the GFF, yet it is not clear whether one should be actually surprised.
RGMW02 28th January 2015
15:00 to 16:00
Generalized Multifractality of Whole-Plane SLE
Co-authors: H Ho (Orléans University), B Le (Orléans University), M Zinsmeister (Orléans University)

We introduce a generalized notion of integral means spectrum for unbounded conformal maps, depending on two moments, giving access to logarithmic coefficients. We study this (average) generalized integral means spectrum for unbounded whole-plane SLE. The usual SLE multifractal spectrum, predicted by the author in 2000 and proved by Beliaev and Smirnov in 2005 and Gwynne, Miller and Sun in 2014, crosses over to a novel spectrum along phase transition lines in the plane of moment orders. A conjecture is proposed for the universal generalized multifractal spectrum, which is proved for a certain range of moments.

RGMW02 29th January 2015
09:00 to 10:00
Liouville quantum gravity as a mating of trees
Co-authors: Bertrand Duplantier (CEA/Saclay), Scott Sheffield (Massachusetts Institute of Technology)

There is a simple way to “glue together” a coupled pair of continuum random trees to produce a topological sphere. The sphere comes equipped with a measure and a space-filling curve (which describes the “interface” between the trees). We present an explicit and canonical way to embed the sphere into the Riemann sphere. In this embedding, the measure is Liouville quantum gravity with parameter gamma in (0,2), and the curve is space-filling version of SLE with kappa=16/gamma^2. Based on joint work with Bertrand Duplantier and Scott Sheffield.

RGMW02 29th January 2015
10:00 to 11:00
Geodesics in Brownian surfaces
In this talk, we introduce a class of random metric spaces called Brownian surfaces, which generalize the famous Brownian map to the case of topologies more complicated than that of the sphere. More precisely, these random surfaces arise as the scaling limit of random maps on a given surface with a boundary. We will review the known results about these rather wild random metric spaces and we will particularly focus on the geodesics starting from a uniformly chosen random point. This allow to characterize some subsets of interest in terms of geodesics and, in particular, in terms of pairs of geodesics aiming at the same point and whose concatenation forms a loop not homotopic to 0.

Our results generalize in particular the properties shown by Le Gall on geodesics in the Brownian map, although our approach is completely different.

RGMW02 29th January 2015
11:30 to 12:30
V Beffara Drawing maps
I will discuss various natural ways of drawing maps, and some of the mathematics behind a few nice pictures.
RGMW02 29th January 2015
13:30 to 14:30
Some scaling limit results for critical Fortuin-Kastelyn random planar map model
Co-authors: Ewain Gwynne (MIT), Cheng Mao (MIT)

Sheffield (2011) introduced a discrete inventory accumulation model which encodes a random planar map decorated by a collection of loops sampled from the critical Fortuin-Kasteleyn (FK) model and showed that a certain two-dimensional random walk associated with an in finite-volume version of the model converges in the scaling limit to a correlated planar Brownian motion. We improve on this scaling limit result by showing that the times corresponding to FK loops (or "flexible orders") in the discrete model converge to the \pi/2 cone times of the Brownian motion. Our result can be used to obtain convergence of interesting functionals of the FK loops including their lengths and areas toward the corresponding "quantum" functionals of the loops of a conformal loop ensemble on a Liouville quantum gravity surface, hence provides a solution to the whole plane version of a question by Duplantier, Miller, and Sheffield (2014).

RGMW02 30th January 2015
09:00 to 10:00
D Chelkak Scaling limits of critical Ising correlation functions in planar domains
In this talk, we plan to summarize rigorous results on scaling limits of critical Ising correlation functions in bounded planar domains obtained during last five years. Those include fermionic, energy density and spin correlations (joint results with C. Hongler and K. Izyurov) as well as recent convergence results for a lattice version of the stress-energy tensor (joint results with A. Glazman and S. Smirnov).
RGMW02 30th January 2015
10:00 to 11:00
Renormalization Approach to the 2-Dimensional Uniform Spanning Tree
We show how to rigorously complete the program outlined in Wendelin Werner's talk in the special case of two-dimensional spanning trees, building on some ideas and results about the scaling limits of loop-erased random walks and their lengths.
RGMW02 30th January 2015
11:30 to 12:30
The Z-invariant massive Laplacian on isoradial graphs
Co-authors: Béatrice de Tilière (LPMA, UPMC), Kilian Raschel (LMPT, University of Tours)

Isoradial graphs form an interesting subset of planar graphs to study critical integrable models: the geometric properties of their embedding are related to the Yang-Baxter equation and allows one to develop a discrete theory of complex analysis.

After having reviewed some results about critical models on those graphs, we will define a massive Laplacian on isoradial graphs with integrability properties.

This massive Laplacian can be used to study off-criticality models from statistical mechanics on these infinite non-periodic graphs (e.g. spanning forests), for which local correlations are obtained, and phase transition as the mass vanishes can be studied analytically.

RGMW02 30th January 2015
13:30 to 14:30
G Pete On near-critical SLE(6) and on the tail in Cardy's formula
First, I give a simple but tricky proof that the Loewner driving process of the near-critical SLE(6) curve is a sub-martingale. Then I explain a conjectural exact form of this driving process. This is from joint work with Christophe Garban and Oded Schramm. Then, using a very different method, I will prove that the probability of having a left-right crossing in a square in \lambda-near-critical percolation, as \lambda\to-\infty, is about \exp(-|\lambda|^{4/3}).
RGM 2nd February 2015
16:00 to 17:00
Rothschild Distinguished Visiting Fellow Lecture: Random maps and random 2-dimensional geometries
A map is a graph embedded into a 2-dimensional surface, considered up to homeomorphisms. In a way, such an object endows the surface with a discrete metric, so that a map taken at random is a natural candidate for a notion of "discrete random metric surface". More precisely, it is expected (and proved in an evergrowing number of cases) that upon re-scaling the distances in an appropriate fashion, a large random map converges to a random metric space that is homeomorphic to the surface one started with.

This is reminiscent of the well-known convergence of random walks to Brownian motion. Similarly to the latter, the random continuum objects that appear as scaling limits of random maps are very irregular spaces, by no means close to being smooth Riemannian manifolds. This makes their study even more interinsting, since it is necessary to find the geometric notions that still make sense in this context, like geodesic paths.

By contrast with these "continuous" notions, we will see that the study of scaling limits of random maps relies strongly on tools of a purely combinatorial nature. We will also discuss conjectures which, quite surprisingly, connect these scaling limits with conformally invariant random fields in the plane.
RGM 5th February 2015
16:00 to 17:00
J-C Mourrat The dynamic phi^4 model in the plane
The dynamic phi^4 model is a non-linear stochastic PDE which involves a cubic power of the solution. In dimensions 3 and less, solutions are expected to have the same local regularity as the linearised equation, for which the law of the Gaussian free field is invariant. Hence, in dimensions 2 and 3, some renormalisation needs to be performed in order to define the cubic power of the solution. In the (full) plane, I will explain how to do this and show that the stochastic PDE has a well-defined solution for all times. If time permits, I will also sketch a proof that the model is the scaling limit of a near-critical Ising model with long-range interactions. Joint work with Hendrik Weber.
RGM 11th March 2015
12:30 to 13:30
On random Hamilton-Jacobi equation and "other" KPZ
RGMW03 16th March 2015
10:00 to 11:00
Compensated Fragmentations
A new class of fragmentation type processes is introduced, in which the accumulation of small dislocations which would instantaneously shatter the entire mass into dust, is compensated by an adequate dilation of fragments. A main feature of these processes is that their evolution is govern by a dislocation measure which can be much more general than usual.
RGMW03 16th March 2015
11:30 to 12:30
Random trees constructed by aggregation
Co-author: Nicolas Curien (Université Paris-Sud)

Starting from a sequence of positive numbers (a_n), we build an increasing sequence of random trees (T_n) by deciding that T_1 is a segment of length a_1, and then, recursively, attaching at step n a segment of length a_n on a uniform point of the tree T_{n-1}. We will see how the sequence (a_n) influences the geometric properties of the limiting tree: compactness, Hausdorff dimension, self-similarity.

RGMW03 16th March 2015
14:00 to 15:00
W Kendall Google maps and improper Poisson line processes
I will review recent work on the construction of random metric spaces using a certain kind of improper Poisson line process.
RGMW03 16th March 2015
15:30 to 16:30
A line-breaking construction of the stable trees
Co-author: Benedicte Haas (Universite Paris-Dauphine)

Consider a critical Galton-Watson tree whose offspring distribution lies in the domain of attraction of a stable law of parameter \alpha \in (1,2], conditioned to have total progeny n. The stable tree with parameter \alpha \in (1,2] is the scaling limit of such a tree, where the \alpha=2 case is Aldous' Brownian continuum random tree. In this talk, I will discuss a new, simple construction of the \alpha-stable tree for \alpha \in (1,2]. We obtain it as the completion of an increasing sequence of \mathbb{R}-trees built by gluing together line-segments one by one. The lengths of these line-segments are related to the increments of an increasing \mathbb{R}_+-valued Markov chain. For \alpha = 2, we recover Aldous' line-breaking construction of the Brownian continuum random tree based on an inhomogeneous Poisson process.

RGMW03 17th March 2015
10:00 to 11:00
The Compulsive Gambler process
Co-authors: Dan Lanoue (U.C. Berkeley), Justin Salez (Paris 7)

In the Compulsive Gambler process there are $n$ agents who meet pairwise at random times ($i$ and $j$ meet at times of a rate-$\nu_{ij}$ Poisson process) and, upon meeting, play an instantaneous fair game in which one wins the other's money. The process seems pedagogically interesting as being intermediate between coalescent-tree models and interacting particle models, and because of the variety of techniques available for its study. Some techniques are rather obvious (martingale structure; comparison with Kingman coalescent) while others are more subtle (an ``exchangeable over the money elements" property, and a ``token process" construction reminiscent of the Donnelly-Kurtz look-down construction). One can study both kinds of $n \to \infty$ limit. The process can be defined under weak assumptions on a countable discrete space (nearest-neighbor interaction on trees, or long-range interaction on the $d$-dimensional lattice) and there is also a continuous-space extension called the Metric Coalescent.

RGMW03 17th March 2015
11:30 to 12:30
Delocalization of two-dimensional random surfaces with hard-core constraints
Co-author: Piotr Milos (University of Warsaw)

We study the fluctuations of random surfaces on a two-dimensional discrete torus. The random surfaces we consider are defined via a nearest-neighbor pair potential which we require to be twice continuously differentiable on a (possibly infinite) interval and infinity outside of this interval. This includes the case of the so-called hammock potential, when the random surface is uniformly chosen from the set of all surfaces satisfying a Lipschitz constraint. Our main result is that these surfaces delocalize, having fluctuations whose variance is at least of order log n, where n is the side length of the torus. The main tool in our analysis is an adaptation to the lattice setting of an algorithm of Richthammer, who developed a variant of a Mermin-Wagner-type argument applicable to hard-core constraints. We rely also on the reflection positivity of the random surface model. The result answers a question mentioned by Brascamp, Lieb and Lebowitz on the hammock potential and a quest ion of Velenik. All terms will be explained in the talk. Joint work with Piotr Milos.

RGMW03 17th March 2015
14:00 to 15:00
Small-particle limits in a regularized Laplacian random growth model
Co-authors: Fredrik Johansson Viklund (Uppsala University), Alan Sola (University of Cambridge)

In 1998 Hastings and Levitov proposed a one-parameter family of models for planar random growth in which clusters are represented as compositions of conformal mappings. This family includes physically occurring processes such as diffusion-limited aggregation (DLA), dielectric breakdown and the Eden model for biological cell growth. In the simplest case of the model (corresponding to the parameter alpha=0), James Norris and I showed how the Brownian web arises in the limit resulting from small particle size and rapid aggregation. In particular this implies that beyond a certain time, all newly aggregating particles share a single common ancestor. I shall show how small changes in alpha result in the emergence of branching structures within the model so that, beyond a certain time, the number of common ancestors is a random number whose distribution can be obtained. This is based on joint work with Fredrik Johansson Viklund (Uppsala) and Alan Sola (Cambridge).

RGMW03 17th March 2015
15:30 to 16:30
Local graph coloring
Co-authors: Oded Schramm (), David B Wilson ()

How can we color the vertices of a graph by a local rule based on i.i.d. vertex labels? More precisely, suppose that the color of vertex v is determined by examining the labels within a finite (but perhaps random and unbounded) distance R of v, with the same rule applied at each vertex. (The coloring is then said to be a finitary factor of the i.i.d. labels). Focusing on Z^d, we investigate what can be said about the random variable R if the coloring is required to be proper, i.e. if adjacent vertices must have different colors. Depending on the dimension and the number of colors, the optimal tail decay is either a power law, or a tower of exponentials. I will briefly discuss generalizations to shifts of finite type and finitely dependent processes.

RGMW03 18th March 2015
10:00 to 11:00
Branching Brownian motion, the Brownian net and selection in spatially structured populations
Co-authors: Nic Freeman (University of Bristol), Daniel Straulino (University of Oxford)

Our motivation in this work is to understand the influence of the spatial structure of a population on the efficacy of natural selection acting upon it. From a biological perspective, what is interesting is that whereas when population density is high, the probability that a selectively favoured genetic type takes over a population is independent of spatial dimension, when population density is low, this is no longer the case and spatial dimension plays an important role. The proofs are of independent mathematical interest: for example, in one dimension we find a new route to the Brownian net, from a continuum model of branching and coalescing lineages. In the biologically most interesting setting of two spatial dimensions, as we rescale our continuum model there is a finely balanced tradeoff between branching and coalescing lineages, eventually resulting in a branching Brownian motion.

RGMW03 18th March 2015
11:30 to 12:30
Rigorous results for a population model with selection
We consider a model of a population of fixed size $N$ in which each individual acquires beneficial mutations at rate $\mu$. Each individual dies at rate one, and when a death occurs, an individual is chosen with probability proportional to the individual's fitness to give birth. We obtain rigorous results for the rate at which mutations accumulate in the population, the distribution of the fitnesses of individuals in the population at a given time, and the genealogy of the population. Our results confirm predictions of Desai and Fisher (2007), Desai, Walczak, and Fisher (2013), and Neher and Hallatschek (2013).
RGMW03 18th March 2015
14:00 to 15:00
A Veber Genealogies with recombination in spatial population genetics
Co-author: Alison Etheridge (University of Oxford)

Discrete or continuous, the spatial structure of a population has an effect on the evolution of its genetic diversity. In recent studies, the random process of recombination (by which certain portions of a chromosome of interest are inherited from one's father and the complement from one's mother) has been used to reconstruct the recent past of a population. This reconstruction is based on the properties of the genealogical trees corresponding to such populations. We shall consider two examples (in continuous space) in which it is possible to use the information left by recombination to infer quantities such that the dispersal rate of a gene, or to test the presence of rare but recurrent catastrophes. (Partially supported by the European project INTEGER).

RGMW03 18th March 2015
15:30 to 16:30
Branching processes with competition by pruning of Levy trees
Co-authors: Joaquin Fontbona (U. Chile), Maria Clara Fittipaldi (U. Chile), L. Doering (U. Zurich), L. Mytnik (Technion), L. Zambotti (UPMC)

There are several ways to describe the evolution of a population with no interactions between individuals. One approach is to use the local time process of a forrest of Lévy trees, or, following the work of Dawson and Li, one can construct the whole population flow as the solution to a certain system of Lévy driven stochastic differential equation. The equivalence between these two constructions is a generalization of the well-known Ray-Knight Theorem.

When one wants to introduce a form of competition in the population, the situation becomes more involved. The stochastic differential approach still works (with an added negative drift term) and the purpose of this talk is to present a novel construction based on the interactive pruning of the Lévy forrest.

The case of a positive drift, which corresponds to an interactive immigration, is also of interest as it is related to the question of existence of exceptional times for Generalized Fleming-Viot processes with mutations at which the number of genetic types in the population is finite.

Based on joint works with : a) L. Doering, L. Mytnik and L. Zambotti and b) J. Fontbona and M.C. Fittipaldi

RGMW03 19th March 2015
10:00 to 11:00
A multi-scale refinement of the second moment method Co-authors: L.P
Arguin, D. Belius, A. Bovier and M. Schmidt I will present a version of the second moment method which is particularly efficient to analyze the extremes of random fields where multiple scales can be identified. The method emerged from work on the extremes of branching Brownian motion, joint with Louis-Pierre Arguin (CUNY) and Anton Bovier (Bonn), and from work with David Belius (NYU) on the cover time by planar Brownian motion. I will also discuss a model which interpolates between Derrida's REM and branching random walks, thereby neatly showing how an increasing number of scales affects the extremes of random fields (joint with Marius Schmidt, Frankfurt). Time permitting, I will conclude with some pointers on a procedure of local projections which allows, in a number of models, to generate scales from first principles.
RGMW03 19th March 2015
11:30 to 12:30
Log-correlated Gaussian fields: study of the Gibbs measure
Co-author: Louis-Pierre ARUIN (CUNY)

Gaussian fields with logarithmically decaying correlations, such as branching Brownian motion and the two-dimensional Gaussian free field, are conjectured to form universality class of extreme value statistics (notably in the work of Carpentier & Le Doussal and Fyodorov & Bouchaud). This class is the borderline case between the class of IID random variables, and models where correlations start to affect the statistics. In this talk, I will describe a general approach based on rigorous works in spin glass theory to describe features of the Gibbs measure of these Gaussian fields. I will focus on the two-dimensional discrete Gaussian free field. At low temperature, we show that the normalized covariance of two points sampled from the Gibbs measure is either 0 or 1. This is used to prove that the joint distribution of the Gibbs weights converges in a suitable sense to that of a Poisson-Dirichlet variable.

RGMW03 19th March 2015
14:00 to 15:00
On the complex cascade and the complex branching Brownian motion
Co-authors: Rhodes (Université Paris Est), Vargas (ENS Paris)

In 2013, Lacoin, Rhodes and Vargas introduced the complex Gaussian multiplicative chaos and exhibited for this model a phase transition diagram. This diagram contains three phases and three frontiers. In this talk we will study the frontier I/II in the context of the complex dyadic Gaussian branching random walk (complex cascade) and the phase II in the context of the complex branching Brownian motion.

RGMW03 19th March 2015
15:30 to 16:30
L-P Arguin Maxima of log-correlated Gaussian fields and of the Riemann Zeta function on the critical line
Co-authors: David Belius (NYU), Adam Harper (Cambridge)

A recent conjecture of Fyodorov, Hiary & Keating states that the maxima of the Riemann Zeta function on a bounded interval of the critical line behave similarly to the maxima of a specific class of Gaussian fields, the so-called log-correlated Gaussian fields. These include important examples such as branching Brownian motion and the 2D Gaussian free field. In this talk, we will highlight the connections between the number theory problem and the probabilistic models. We will outline the proof of the conjecture in the case of a randomized model of the Zeta function. We will discuss possible approaches to the problem for the function itself. This is joint work with D. Belius (NYU) and A. Harper (Cambridge).

RGMW03 20th March 2015
10:00 to 11:00
Scale-free percolation
Co-authors: Mia Deijfen (Stockholm University), Gerard Hooghiemstra (Delft University of Technology)

We propose and study a random graph model on the hypercubic lattice that interpolates between models of scale-free random graphs and long-range percolation. In our model, each vertex $x$ has a weight $W_x$, where the weights of different vertices are i.i.d.\ random variables. Given the weights, the edge between $x$ and $y$ is, independently of all other edges, occupied with probability $1-{\mathrm{e}}^{-\lambda W_xW_y/|x-y|^{\alpha}}$, where (a) $\lambda$ is the percolation parameter, (b) $|x-y|$ is the Euclidean distance between $x$ and $y$, and (c) $\alpha$ is a long-range parameter. The most interesting behavior can be observed when the random weights have a power-law distribution, i.e., when $\mathbb{P}(W_x>w)$ is regularly varying with exponent $1-\tau$ for some $\tau>1$. In this case, we see that the degrees are infinite a.s.\ when $\gamma =\alpha(\tau-1)/d \leq 1$ or $\alpha\leq d$, while the degrees have a power-law distribution with exponent $\gamma$ when $\gamma>1$. Our main results describe phase transitions in the positivity of the percolation critical value and in the graph distances in the percolation cluster as $\gamma$ varies. Our results interpolate between those proved in inhomogeneous random graphs, where a wealth of further results is known, and those in long-range percolation. We also discuss many open problems, inspired both by recent work on long-range percolation (i.e., $W_x=1$ for every $x$), and on inhomogeneous random graphs (i.e., the model on the complete graph of size $n$ and where $|x-y|=n$ for every $x\neq y$).

RGMW03 20th March 2015
11:30 to 12:30
Rate of convergence of the mean of sub-additive ergodic processes
Co-authors: Michael Damron (Indiana University), Jack T. Hanson (Indiana University)

For a subadditive ergodic sequence ${X_{m,n}}$, Kingman's theorem gives convergence for the terms $X_{0,n}/n$ to some non-random number $g$. In this talk, I will discuss the convergence rate of the mean $\mathbb EX_{0,n}/n$ to $g$. This rate turns out to be related to the size of the random fluctuations of $X_{0,n}$; that is, the variance of $X_{0,n}$, and the main theorems I will present give a lower bound on the convergence rate in terms of a variance exponent. The main assumptions are that the sequence is not diffusive (the variance does not grow linearly) and that it has a weak dependence structure. Various examples, including first and last passage percolation, bin packing, and longest common subsequence fall into this class. This is joint work with Michael Damron and Jack Hanson.

RGMW03 20th March 2015
14:00 to 15:00
Uniformity of the late points of random walk on $\mathbb{Z}_d^n$ for $d \geq 3$
Co-author: Jason Miller (MIT)

Let $X$ be a simple random walk in $\mathbb{Z}_n^d$ and let $t_{\rm{cov}}$ be the expected amount of time it takes for $X$ to visit all of the vertices of $\mathbb{Z}_n^d$. For $\alpha\in (0,1)$, the set $\mathcal{L}_\alpha$ of $\alpha$-late points consists of those $x\in \mathbb{Z}_n^d$ which are visited for the first time by $X$ after time $\alpha t_{\rm{cov}}$. Oliveira and Prata (2011) showed that the distribution of $\mathcal{L}_1$ is close in total variation to a uniformly random set. The value $\alpha=1$ is special, because $|\mathcal{L}_1|$ is of order 1 uniformly in $n$, while for $\alpha<1$ the size of $\mathcal{L}_\alpha$ is of order $n^{d-\alpha d}$. In joint work with Jason Miller we study the structure of $\mathcal{L}_\alpha$ for values of $\alpha<1$. In particular we show that there exist $\alpha_0<\alpha_1 \in(0,1)$ such that for all $\alpha>\alpha_1$ the set $\mathcal{L}_\alpha$ looks uniformly random, while for $\alpha<\alpha_0$ it does not (in the total variation sense). In this talk I will try to explain the main ideas of our proof and what are the next steps in this direction.

RGMW03 20th March 2015
15:30 to 16:30
Maxima of logarithmically correlated fields
Co-authors: Jian Ding (University of Chicago), Rishideep Roy (University of Chicago)

I will describe sufficient conditions that ensure the convergence in distribution of the centered maximum for logarithmically correlated Gaussian fields. This class includes the GFF, MBRW, and the circular logarithmic REM. No assumption is made on a Markov property of the fields.

RGMW04 20th April 2015
10:00 to 11:00
Compact Brownian surfaces
Co-author: Jérémie Bettinelli (CNRS and IECN (Nancy))

We will show how to prove that rescaled uniform quandrangulations on oriented compact surface with a (possibly non-simple) boundary converge to random metric spaces with the same topology. This is done by performing suitable surgery operations on conditioned versions of the Brownian map.

RGMW04 20th April 2015
11:30 to 12:30
Blossoming trees and the scaling limit of maps
Co-authors: Louigi Addario-Berry (Mc Gill University), Olivier Bernardi (Brandeis University), Gwendal Collet (TU Wien), Éric Fusy (CNRS), Dominique Poulalhon (Université Paris 7)

In the last years, numerous families of planar maps have been shown to converge to the Brownian map introduced by Miermont and Le Gall. Most of these results rely on some bijections with labeled trees mobiles due to Schaeffer and Bouttier, di Francesco and Guitter. In this talk, I'll present another class of bijections between so-called blossoming trees and maps. These bijections have been established 15 years ago but it is not since only recently that we managed to use them to track down the distances in the maps as a function of the trees. This link relies on some canonical « leftmost paths », which behave well both in the map and in the tree. As an example of the possible outcomes of these bijections, I'll prove that the scaling limit of simple maps (that is maps without loops nor multiple edges) is also the Brownian map. I'll emphasize the combinatorial construction which lies at the heart of this proof.

RGMW04 20th April 2015
14:00 to 15:00
T Budd Scaling constants and the lazy peeling of infinite Boltzmann planar maps
Recently Curien and Le Gall derived precise scaling limits of the volume and perimeter of the explored region during a ("simple") peeling process of uniform infinite planar triangulations (UIPT) and quadrangulations (UIPQ). We show that the same limits may be obtained for a slightly modified "lazy" peeling process in the more general setting of infinite Boltzmann planar maps (IBPM) with arbitrary (regular critical) weight sequences. Combining the scaling constants involved with previous results by Miermont on graph distances in Boltzmann planar maps, we show how one may obtain (at least at a heuristic level) simple expressions for all constants appearing in the relative scaling of the following quantities associated to an IBPM: volume, perimeter, graph distance, dual graph distance, first-passage time, and hop count. Finally we will comment on how one may recover the simple peeling process from the lazy one.
RGMW04 20th April 2015
15:30 to 16:30
Critical exponents in FK-weighted planar maps
Co-authors: Nathanael Berestycki (University of Cambridge), Benoit Laslier (University of Cambridge)

In this paper we consider random planar maps weighted by the self-dual Fortuin--Kastelyn model with parameter q in (0,4). Using a bijection due to Sheffield and a connection to planar Brownian motion in a cone we obtain rigorously the value of critical exponents associated with the length of cluster interfaces, which is shown to be $$\frac{4}{\pi} arccos\left(\frac{\sqrt{2-\sqrt{q}}}{2}\right).$$ Similar results are obtained for the area. Applying the KPZ formula we find that this value is consistent with the dimension of SLE curves and SLE duality. Various isoperimetric relationships of independent interest are also derived.

RGMW04 21st April 2015
10:00 to 11:00
Scaling limits of the uniform spanning tree
Co-authors: David Croydon (University of Warwick), Takashi Kumagai (RIMS, Kyoto)

The uniform spanning tree (UST) has played a major role in recent developments in probability. In particular the study of its scaling limit led to the discovery of SLE by Oded Schramm.

In this talk I will discuss the geometry of the UST in 2 dimensions, and what we can say about its scaling limit.

RGMW04 21st April 2015
11:30 to 12:30
A hidden quantum group for pure partition functions of multiple SLEs
Co-author: Eveliina Peltola (University of Helsinki)

A classification result of Schramm identifies the candidates for scaling limit random curves in critical planar models by their conformal invariance and domain Markov property: in simply connected domains with curves connecting two boundary points the curves are chordal SLEs. The classification of corresponding multiple curves is more involved, due to the presence of nontrivial conformal moduli: instead of a unique law of a curve, there is a finite dimensional convex set of laws consistent with the requirements. The growth process construction of multiple SLE curves relies on partition functions, which must solve a system of partial differential equations. We present a method based on the representation theory of a quantum group, with help of which we explicitly construct a basis of solutions to the partial differential equations corresponding to the extremal points of the convex set.

RGMW04 21st April 2015
14:00 to 15:00
Self-avoiding Walk and Connective Constant
Co-author: Geoffrey Grimmett (University of Cambridge)

A self-avoiding walk (SAW) is a path on a graph that revisits no vertex. The connective constant of a graph is defined to be the exponential growth rate of the number of n-step SAWs with respect to n. We prove that sqrt{d-1} is a universal lower bound for connective constants of any infinite, connected, transitive, simple, d-regular graph. We also prove that the connective constant of a Cayley graph decreases strictly when a new relator is added to the group and increases strictly when a non-trivial word is declared to be a generator. I will also present a locality result regarding to the connective constants proved by defining a linearly increasing harmonic function on Cayley graphs. In particular, the connective constant is local for all solvable groups. Joint work with Geoffrey Grimmett.

RGMW04 21st April 2015
15:30 to 16:30
I Kortchemski Looptrees
A looptree of a plane tree is the graph obtained by replacing each vertex of the tree by a discrete loop of length equal to its degree, and by gluing these loops according to the tree structure. We will be interested in the scaling limits, for the Gromov-Hausdorff topology, of looptrees associated with different classes of random trees: - random trees built by preferential attachment: in this case, the scaling limit is the Brownian rabbit (joint work with N. Curien, T. Duquesne and I. Manolescu); - critical Galton-Watson trees with finite variance: in this case the scaling limit is a multiple of the CRT (joint work with N. Curien and B. Haas) - critical Galton-Watson trees with infinite variance and heavy tail offspring distribution: in this case, the scaling limits are the so-called stable looptrees, which are informally the dual graphs of stable Lévy trees. We will see that the scaling limit of the boundary of large critical site percolation clusters on the UIPT is the random stable looptree of index 3/2 (joint works with N. Curien).
RGMW04 22nd April 2015
09:00 to 10:00
On the geometry of discrete and continuous random planar maps
Co-author: Curien, Nicolas (Université Paris-Sud)

We discuss some recent results concerning the geometry of discrete and continuous random planar maps. In the continuous setting, we consider the so-called Brownian plane, which is an infinite-volume version of the Brownian map and is conjectured to be the universal scaling limit of many discrete random lattices such as the UIPT (uniform infinite planar triangulation) or the UIPQ (uniform infinite planar quadrangulation). The hull of radius r in the Brownian plane is obtained by filling in the holes in the ball of radius r centered at the distinguished point. We obtain a complete description of the process of hull volumes, as well as several explicit formulas for related distributions. In the discrete setting of the UIPT or the UIPQ, we derive similar results via a detailed study of the peeling process already inverstigated by Angel. We also apply our results to first-passage percolation on these infinite random lattices. This is a joint work with Nicolas Curien.

RGMW04 22nd April 2015
10:00 to 11:00
G Schaeffer On classes of planar maps with $\alpha$-orientations having geometric interpretations
Fixed out-degree orientations or $\alpha$-orientations play a central rôle in the so-called master bijections relating various families of planar maps to simple varieties of trees. I will discuss several known cases in which these orientations also provide purely combinatorial ways to compute geometric representations.
RGMW04 22nd April 2015
11:30 to 12:30
G Borot Nesting statistics in the O(n) loop model on random lattices
Co-author: Jeremie Bouttier (CEA Saclay and ENS Paris)

We investigate how deeply nested are the loops in the O(n) model on random maps. In particular, we find that the number P of loops separating two points in a planar map in the dense phase with V >> 1 vertices is typically of order c(n) \ln V for a universal constant c(n), and we compute the large deviations of P. The formula we obtain shows similarity to the CLE_{\kappa} nesting properties for n = 2\cos\pi(1 - 4/\kappa). The results can be extended to all topologies using the methods of topological recursion.

RGMW04 22nd April 2015
14:00 to 15:00
O Bernardi Differential equations for colored maps
Coauthor: Mireille Bousquet-Melou (CNRS)

We study the Potts model on planar maps. The partition function of this model is the generating function of colored maps counted according to the number of monochromatic edges and dichromatic edges. We characterize this partition function by a simple system of differential equations. Some special cases, such as properly 4-colored maps, have particularly simple equations waiting for a more direct combinatorial explanation.

RGMW04 22nd April 2015
15:30 to 16:30
O Gurel Gurevich Recurrence of planar graph limits
Co-author: Asaf Nacmias (Tel Aviv University)

What does a random planar triangulation on n vertices looks like? More precisely, what does the local neighbourhood of a fixed vertex in such a triangulation looks like? When n goes to infinity, the resulting object is a random rooted graph called the Uniform Infinite Planar Triangulation (UIPT). Angel, Benjamini and Schramm conjectured that the UIPT and similar objects are recurrent, that is, a simple random walk on the UIPT returns to its starting vertex almost surely. In a joint work with Asaf Nachmias, we prove this conjecture. The proof uses the electrical network theory of random walks and the celebrated Koebe-Andreev-Thurston circle packing theorem. We will give an outline of the proof and explain the connection between the circle packing of a graph and the behaviour of a random walk on that graph.

RGMW04 23rd April 2015
09:00 to 10:00
The uniform spanning forest of planar graphs
The free uniform spanning forest (FUSF) of an infinite graph G is obtained as the weak limit of the law of a uniform spanning tree on G_n, where G_n is a finite exhaustion of G. It is easy to see that the FUSF is supported on spanning graphs of G with no cycles, but it need not be connected. Indeed, a classical result of Pemantle ('91) asserts that when G=Z^d, the FUSF is almost surely a connected tree if and only if d=1,2,3,4.

In this talk we will show that if G is a plane graph with bounded degrees, then the FUSF is almost surely connected, answering a question of Benjamini, Lyons, Peres and Schramm ('01). An essential part of the proof is the circle packing theorem.

Joint work with Tom Hutchcroft.

RGMW04 23rd April 2015
10:00 to 11:00
Characterstic polynomials of random matrices and logarithmically correlated processes
I will discuss relations between logarithmically-correlated Gaussian processes and the characteristic polynomials of large random $N \times N$ matrices, either from the Circular Unitary (CUE) or from the Gaussian Unitary (GUE) ensembles. Such relations help to address the problem of characterising the distribution of the global maximum of the modulus of such polynomials, and of the Riemann $\zeta\left(\frac{1}{2}+it\right)$ over some intervals of $t$ containing of the order of $\log{t}$ zeroes. I will show how to arrive to an explicit expression for the asymptotic probability density of the maximum by combining the rigorous Fisher-Hartwig asymptotics with the heuristic {\it freezing transition} scenario for logarithmically correlated processes. Although the general idea behind the method is the same for both CUE and GUE, the latter case is much more technically challenging. In particular I will show how the conjectured {\it self-duality} in the freezing transition scenario plays the crucial role in selecting the form of the maximum distribution for GUE case. The found probability densities will be compared to the results of direct numerical simulations of the maxima. The presentation is mainly based on joint works with Ghaith Hiary, Jon Keating, Boris Khoruzhenko, and Nick Simm.
RGMW04 23rd April 2015
11:30 to 12:30
Pinning and disorder relevance for the lattice Gaussian free field
Co-author: Giambattista Giacomin (Université Paris Diderot)

We present a rigorous study of the localization transition for a Gaussian free field on Zd interacting with a quenched disordered substrate that acts on the interface when the interface height is close to zero. The substrate has the tendency to localize or repel the interface at different sites and one can show that a localization-delocalization transition takes place when varying the average pinning potential h: the free energy density is zero in the delocalized regime, that is for h smaller than a threshold hc, and it is positive for h>hc. For d=3 we compute hc and we show that the transition happens at the same value as for the annealed model. However we can show that the critical behavior of the quenched model differs from the one of the annealed one. While the phase transition of the annealed model is of first order, we show that the quenched free energy is bounded above by (h-hc)2+ times a positive constant and that, for Gaussian disorder, the quadrat ic behavior is sharp. Therefore this provides an example in which a {\sl relevant disorder critical exponent} can be made explicit: in theoretical physics disorder is said to be {\sl relevant} when the disorder changes the critical behavior of a system and, while there are cases in which it is known that disorder is relevant, the exact critical behavior is typically unknown. For d=2 we are not able to decide whether the quenched and annealed critical points coincide, but we provide an upper bound for the difference between them.

RGMW04 23rd April 2015
14:00 to 15:00
Planar lattices do not recover from forest fires
Co-authors: Demeter Kiss (University of Cambridge) and Vladas Sidoravicius (IMPA)

Self-destructive percolation with parameters p, delta is obtained by taking a site percolation configuration with parameter p, closing all sites belonging to the infinite cluster, then opening every site with probability delta, independently of the rest. Call theta(p,delta) the probability that the origin is in an infinite cluster in the configuration thus obtained. For two dimensional lattices, we show the existence of delta > 0 such that, for any p > p_c , theta(p,delta) = 0. This proves a conjecture of van den Berg and Brouwer, who introduced the model. Our results also imply the non-existence of the infinite parameter forest-fire model on planar lattices.

RGMW04 23rd April 2015
15:30 to 16:30
The exact $k$-SAT threshold for large $k$
Co-authors: Jian Ding (University of Chicago), Allan Sly (University of California--Berkeley)

We establish the random $k$-SAT threshold conjecture for all $k$ exceeding an absolute constant $k_0$. That is, there is a single critical value $\alpha_*(k)$ such that a random $k$-SAT formula at clause-to-variable ratio $\alpha$ is with high probability satisfiable for $\alpha$ less than $\alpha_*(k)$, and unsatisfiable for $\alpha$ greater than $\alpha_*(k)$. The threshold $\alpha_*(k)$ matches the explicit prediction derived by statistical physicists on the basis of the one-step replica symmetry breaking (1RSB) heuristic. In the talk I will describe the main obstacles in computing the threshold, and explain how they are overcome in our proof. Joint work with Jian Ding and Allan Sly.

RGMW04 24th April 2015
10:00 to 11:00
Squarings of rectangles
Co-author: Nicholas Leavitt (McGill University)

Growing random trees, maps, and squarings. We use a growth procedure for binary trees due to Luczak and Winkler, a bijection between binary trees and irreducible quadrangulations of the hexagon due to Fusy, Poulalhon and Schaeffer, and the classical angular mapping between quadrangulations and maps, to define a growth procedure for maps. The growth procedure is local, in that every map is obtained from its predecessor by an operation that only modifies vertices lying on a common face with some fixed vertex. The sequence of maps has an almost sure limit G; we show that G is the distributional local limit of large, uniformly random 3-connected graphs.

A classical result of Brooks, Smith, Stone and Tutte associates squarings of rectangles to edge-rooted planar graphs. Our map growth procedure induces a growing sequence of squarings, which we show has an almost sure limit: an infinite squaring of a finite rectangle, which almost surely has a unique point of accumulation. We know almost nothing about the limit, but it should be in some way related to Liouville quantum gravity.

RGMW04 24th April 2015
11:30 to 12:30
Parabolic and Hyperbolic Unimodular maps
Co-authors: Tom Hutchcraft (UBC), Asaf Nachmias (Tel Aviv), Gourab Ray (Cambridge)

We show that for a unimodular random planar map, many geometric and probabilistic properties are equivalent. These include local and global properties: Negative mean curvature, invariant non-amenability, gap between the critical and uniqueness parameters for percolation, distinction between free and wired uniform spanning forests, and more.

RGMW04 24th April 2015
14:00 to 15:00
Liouville Quantum gravity on Riemann surfaces
I will present the rigorous construction of Liouville quantum field theory on different Riemann surfaces (sphere, disk, torus, etc...). These constructions, which are based on Polyakov's path integral, yield non trivial conformal field theories. As an output of the construction, we introduce the so-called Liouville measures which we conjecture to be the scaling limit of the volume form of finite maps conformally embedded in the sphere. If time permits, I will present the semi-classical approximation of these measures. Based on joint works with F. David, Y. Huang, H. Lacoin, A. Kupiainen, R. Rhodes.
RGMW04 24th April 2015
15:30 to 16:30
Scaling limits of random planar maps and growth-fragmentations
Co-authors: Jean Bertoin (University Zürich), Igor Kortchemski (CNRS and École Polytechnique)

We prove a scaling limit result for the structure of cycles at heights in random Boltzmann triangulations with a boundary. The limit process is described as a compensated fragmentation process of index $-1/2$ with explicit parameters. The proof is based on the analysis of the peeling by layers algorithm in random triangulations. However, contrary to previous works on the subject we let the exploration branch and explore different components. The analysis heavily relies on a martingale structure inside random planar triangulations and a recent scaling limits result for discrete time Markov chains. One motivation is to give a new construction of the Brownian map from a compensated growth-fragmentation process.

RGMW05 15th June 2015
10:00 to 11:00
W Werner Some news from the loop-soup front
Co-author: Wei Qian (ETH Zürich)

We will present some new features of loop-soups.

RGMW05 15th June 2015
11:30 to 12:30
Conformal representations of Random Maps and Surfaces
Motivated by the quest to find a somewhat conformal map from the Riemann sphere to a Liouville quantum-gravity sphere, I will talk about uniformization of discrete maps (based on joint work with Don Marshall), and discuss an analog of Mario Bonk's carpet uniformization in the setting of the Conformal Loop Ensemble CLE (based on joint work with Brent Werness).
RGMW05 15th June 2015
14:00 to 15:00
J Miller Liouville quantum gravity and the Brownian map
Co-author: Scott Sheffield (MIT)

Over the past few decades, two natural random surface models have emerged within physics and mathematics. The first is Liouville quantum gravity, which has roots in string theory and conformal field theory. The second is the Brownian map, which has roots in planar map combinatorics. We show that the Brownian map is equivalent to Liouville quantum gravity with parameter $\gamma = \sqrt{8/3}$.

RGMW05 15th June 2015
15:30 to 16:30
Essential spanning forests on periodic planar graphs
The laplacian on a periodic planar graph has a rich algebraic and integrable structure, which we usually don't see when we do standard potential theory. We discuss the combinatorial, algebraic and integrable features of the laplacian, and in particular interpret combinatorially the points of the "spectral curve" of the laplacian in terms of probability measures on spanning trees and forests.
RGMW05 16th June 2015
09:00 to 10:00
Competitive erosion is conformally invariant
Co-author: Shirshendu Ganguly (University of Washington)

We study a graph-theoretic model of interface dynamics called {\bf competitive erosion}. Each vertex of the graph is occupied by a particle, which can be either red or blue. New red and blue particles are emitted alternately from their respective sources and perform random walk. On encountering a particle of the opposite color they remove it and occupy its position. This is a finite competitive version of the celebrated Internal DLA growth model first analyzed by Lawler, Bramson and Griffeath in 1992. We establish conformal invariance of competitive erosion on discretizations of smooth, simply connected planar domains. This is done by showing that at stationarity, with high probability the blue and the red regions are separated by an orthogonal circular arc on the disc and more generally by a hyperbolic geodesic. (Joint work with Shirshendu Ganguly, available at http://arxiv.org/abs/1503.06989 ).

RGMW05 16th June 2015
10:10 to 11:10
A-S Sznitman On Disconnection, random walks, random interlacements, and the Gaussian free field
In this talk we will discuss some large deviations estimates related to the question of understanding how a simple random walk in dimension 3 and above can insulate a macroscopic body. Some of these results have been obtained in collaboration with Xinyi Li.
RGMW05 16th June 2015
11:30 to 12:30
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RGMW05 16th June 2015
14:00 to 15:00
B Werness Convergence of discrete holomorphic functions on non-uniform lattices
The theory of discrete holomorphic functions has been studied by researchers from a diverse set of fields from classical complex analysts to applied computer scientists. In the field of conformally invariant random processes, discrete analyticity has found a particularly central role as the convergence of discrete analytic functions to their continuum counterparts is the key step in the showing convergence of discrete random processes to Schramm--Loewner Evolutions.

In this talk, we will discuss recent work that proves that discrete analytic functions converge to their continuum counterparts on lattices with only local control on the geometry. We will then discuss potential applications of this result to the study conformally invariant random processes on random surface models.

RGMW05 16th June 2015
15:30 to 16:30
Conformal restriction: the chordal and the radial
Co-authors: Greg Lawler (Math. Department of Chicago University), Oded Schramm (Microsoft Research), Wendelin Werner (Math. Department of ETH)

When people tried to understand two-dimensional statistical physics models, it is realized that any conformally invariant process satisfying a certain restriction property has corssing or intersection exponents. Conformal field theory has been extremely successful in predicting the exact values of critical exponents describing the bahvoir of two-dimensional systems from statistical physics. The main goal of this talk is to review the restriction property and related critical exponents. First, we will introduce Brownian intersection exponents. Second, we discuss Conformal Restriction---the chordal case and the radial case. Third, we explain the idea of the proofs. Finally, we give some relation between conformal restriction sets and intersection exponents.

RGMW05 17th June 2015
09:00 to 10:00
Scaling window of Bernoulli percolation on Z^d
We will discuss the notion of scaling window for Bernoulli percolation on Z^d. While the notion is classical in two dimensions, the systematic study of the size of the window in higher dimension has not been done by now. We will present a few progress in this direction.
RGMW05 17th June 2015
10:10 to 11:10
A (slightly) new look at the backbone
Co-author: Jean-Christophe MOURRAT (ENS Lyon)

In this talk, I will start by discussing one of Greg's many influential papers: "One-arm exponent for critical 2D percolation" by Lawler, Schramm and Werner. I will then focus on the particular case of the backbone exponent with some new insights on this topic. Joint work with Jean-Christophe Mourrat.

RGMW05 17th June 2015
11:30 to 12:30
A random walk proof of Kirchhoff's matrix tree theorem
Kirchhoff's matrix tree theorem relates the number of spanning trees in a graph to the determinant of a matrix derived from the graph. There are a number of proofs of Kirchhoff's theorem known, most of which are combinatorial in nature. In this talk we will present a relatively elementary random walk-based proof of Kirchhoff's theorem due to Greg Lawler which follows from his proof of Wilson's algorithm. Moreover, these same ideas can be applied to other computations related to general Markov chains and processes on a finite state space. Based in part on joint work with Larissa Richards (Toronto) and Dan Stroock (MIT).
RGMW05 17th June 2015
14:00 to 15:00
Return of the Multiplicative Coalescent
Co-authors: David Aldous (UC Berkeley), Mathieu Merle (Université Paris 7), Justin Salez (Université Paris 7)

No one steps in the same river twice. Sometimes life makes a (almost) second chance possible. I will talk about my thesis work. The choice of the topic will be explained at the talk.

RGMW05 17th June 2015
15:30 to 16:30
Where Planar Simple Random Walk Loses its Rotational Symmetry
Local central limit theorems for random walks come in several shapes and forms. Lawler's books contain a number of versions which are very precise at and somewhat beyond the typical range for random walk. In this talk, I will give a detailed description of what happens for a large range of atypical points for simple random walk and will show where and how the walk loses its approximate rotational symmetry.
RGMW05 18th June 2015
10:00 to 11:00
SLE Quantum Multifractality
RGMW05 18th June 2015
11:30 to 12:30
Radial SLE martingale-observables
Co-author: Nikolai Makarov (Caltech)

After implementing a version of radial conformal field theory in the OPE family of statistical fields generated by background charge modification of the Gaussian free field, I present an analytical and probabilistic proof of a well-known statement in physics that the correlation functions of such fields under the insertion of one-leg operator form a collection of radial SLE martingale-observables. In the construction of one-leg operator, the so-called neutrality conditions on the charges play an important role. I explain two neutrality conditions: first, the linear combination of bosonic fields is to be a well-defined Fock space field; second, the Coulomb gas correlation function is to be conformally invariant. To reconcile these two neutrality conditions, one needs to place the background charge at the marked interior point, the target point of SLE. This is a joint work with Nikolai Makarov.

RGMW05 18th June 2015
14:00 to 15:00
Scaling limit of the probability that loop-erased random walk uses a given edge
Co-authors: Christian Benes (CUNY), Greg Lawler (University of Chicago)

I will discuss a proof of the following result: The probability that a loop-erased random walk (LERW) uses a given edge in the interior of a lattice approximation of a simply connected domain converges in the scaling limit to a constant times the SLE(2) Green's function, an explicit conformally covariant quantity. I will also indicate how this result is related to convergence of LERW to SLE(2) the natural parameterization.

This is based on joint work with Christian Benes and Greg Lawler and work in progress with Greg Lawler.

RGMW05 18th June 2015
15:20 to 16:20
Welding of the Backward SLE and Tip of the Forward SLE
Co-author: Steffen Rohde (University of Washington)

Let $\kappa\in(0,4]$. A backward chordal SLE$_\kappa$ process generates a conformal welding $\phi$, which is a random auto-homeomorphism of $\mathbb R$ that satisfies $\phi^{-1}=\phi$ and has a single fixed point: $0$. Using a stochastic coupling technique, we proved that the welding $\phi$ satisfies the following symmetry: Let $h(z)=-1/z$. Then $h\circ \phi\circ h$ has the same law as $\phi$. Combining this symmetry result with the forward/backward SLE symmetry and the conformal removability of forward SLE curve, we then derived some ergodic property of the tip of a forward SLE$_\kappa$ curve for $\kappa\in(0,4)$.

RGMW05 18th June 2015
16:30 to 17:30
Boundary Measures and Natural Time Parameterization for SLE
Among Greg's many important contributions to the Schramm-Loewner evolution, he and his co-authors have in recent years been largely responsible for the construction and analysis of the natural time parameterization. The term ``natural'' refers to the parameterization favored by probabilists in the discrete curve setting, one unit of time per lattice site, and although it has not yet been shown that this discrete time parameterization converges it is still possible to define and study what should be the corresponding time parameterization for the continuum SLE curve. I will review many of the results in this direction that are due to Greg et al, along with some related results of my own in the boundary case, and then discuss some work in progress for extending the results on the boundary.
RGMW05 19th June 2015
10:00 to 11:00
C Burdzy Twin peaks
Co-authors: Sara Billey (University of Washington), Soumik Pal (University of Washington), Lerna Pehlivan (University of Washington), Bruce Sagan (Michigan State University)

I will discuss some questions and results on random labelings of graphs conditioned on having a small number of peaks (local maxima). The main open question is to determine the distance between two peaks on a large discrete torus, assuming that the random labeling is conditioned on having exactly two peaks.

RGMW05 19th June 2015
11:30 to 12:30
Loewner curvature
Co-author: Steffen Rohde (University of Washington)

Inspired by the geometric understanding of the SLE trace, there has been interest in studying how the deterministic Loewner equation encodes geometric properties of 2-dim sets into the 1-dim data of the driving function. Working in this vein, we define a new notion of curvature, called Loewner curvature, so-named because it captures key behavior of the trace curve of the Loewner equation. The Loewner curvature is defined for (nice enough) curves that begin at a marked boundary point of a Jordan domain and grow towards a second marked boundary point. We show that if this curvature is small, then the curve must remain a simple curve.

RGMW05 19th June 2015
14:00 to 15:00
From Internal DLA to self-interacting walks
I will present a few self-interacting random-walk-type models that somehow sit in between internal diffusion-limited aggregation and reinforced walks, associated to a strategy for proving a shape theorem for once-reinforced RW; and I will show some results in that direction.
RGMW06 9th July 2018
10:00 to 11:00
Sourav Chatterjee An introduction to gauge theories for probabilists: Part I
In this lecture series, I will introduce the basic framework of quantum Yang-Mills theories and lattice gauge theories from the probabilist’s point of view. A summary of available rigorous results and open problems will be given. This will be followed by a discussion of gauge-string duality, large N lattice gauge theories, and some recent results. In the first lecture, I will introduce the physical description of quantum Yang-Mills theories. I will also briefly discuss Wilson loops, quark confinement, and perturbative expansions.
RGMW06 9th July 2018
11:15 to 12:15
Jason Miller Random walk on random planar maps I
We will describe some recent developments on the study of random walks on random planar maps. We first review the continuum constructions from Liouville quantum gravity as a mating of trees. We will then explain how one can analyze the behavior of random walk on the mated-CRT map, a random planar map model defined out of the continuum tree-mating constructions. Finally, we will explain how these results can be transferred to a wide variety of discrete random planar map models. This is based on joint works with Bertrand Duplantier, Ewain Gwynne, and Scott Sheffield.
RGMW06 9th July 2018
13:45 to 14:30
Jian Ding Percolation for level-sets of Gaussian free fields on metric graphs
In this talk, I will present a new result on the chemical distance for percolation of level-sets for Gaussian free fields on metric graphs in two dimensions. I will try to present a sketch of our proof, which is based on analyzing a certain exploration martingale. Time permitting, I will also discuss some other potential applications as well as related open problems. Based on joint work with Mateo Wirth.
RGMW06 9th July 2018
14:35 to 15:20
Vincent Tassion The phase transition for Boolean percolation
We consider Boolean percolation in dimension d. Around every point of a Poisson point process of intensity lambda, draw a ball of random radius, independently for different points. We investigate the connection probabilities in the subcritical regime and use the randomized algorithm method to prove that the phase transition in lambda is sharp. Interestingly, for this process, sharpness of the phase transition does not imply exponential decay of connection probabilities in the subcritical regime, and its meaning depends on the  law of the radii. In this talk, we will focus on this specific feature of Boolean percolation. This talk is based on a joint work with H. Duminil-Copin and A. Raoufi.
RGMW06 10th July 2018
09:10 to 09:55
Nina Holden Cardy embedding of uniform triangulations
The Cardy embedding is a discrete conformal embedding for random planar maps which is based on percolation observables. We present a series of works in progress with the goal of showing convergence of uniform triangulations to $\sqrt{8/3}$-LQG under this embedding. The project is a collaboration with Xin Sun, and also based on our joint works with Bernardi, Garban, Gwynne, Lawler, Li, and Sepulveda.
RGMW06 10th July 2018
10:00 to 11:00
Gregory Miermont Exploring random maps: slicing, peeling and layering - 1
The combinatorial theory of maps, or graphs on surfaces, is rich of many different approaches (recursive decompositions, algebraic approches, matrix integrals, bijective approaches) which often have probabilistic counterparts that are of interest when one wants to study geometric aspects of random maps. In these lectures, I will review parts of this theory by focusing on three different decompositions of maps, namely, the slice decomposition, the peeling process, and the decomposition in layers, and by showing how these decompositions can be used to give access to quite different geometric properties of random maps.




RGMW06 10th July 2018
11:15 to 12:15
Sourav Chatterjee An introduction to gauge theories for probabilists: Part II
In the second lecture of the series, I will introduce lattice gauge theories. A brief review of available rigorous results and open problems will be given.
RGMW06 10th July 2018
13:45 to 14:30
Ofer Zeitouni On the Liouville heat kernel and Liouville graph distance (joint with Ding and Zhang)
RGMW06 10th July 2018
14:35 to 15:20
Wei Qian Uniqueness of the welding problem for SLE and LQG
Fix $\kappa \in (0,8)$ and suppose that $\eta$ is an SLE$_\kappa$ curve in $\mathbb{H}$ from $0$ to $\infty$. We show that if $\varphi \colon \mathbb{H} \to \mathbb{H}$ is a homeomorphism which is conformal on $\mathbb{H} \setminus \eta$ and $\varphi(\eta)$, $\eta$ are equal in distribution then $\varphi$ is a conformal automorphism of $\mathbb{H}$. Applying this result for $\kappa=4$ establishes that the welding operation for critical ($\gamma=2$) Liouville quantum gravity (LQG) is well-defined. Applying it for $\kappa \in (4,8)$ gives a new proof that the welding of two looptrees of quantum disks to produce an SLE$_\kappa$ on top of an independent $4/\sqrt{\kappa}$-LQG surface is well-defined. These results are special cases of a more general uniqueness result which applies to any non-space-filling SLE-type curve (e.g., the exotic SLE$_\kappa^\beta(\rho)$ processes). This is a joint work with Oliver McEnteggart and Jason Miller.
RGMW06 11th July 2018
09:10 to 09:55
Gilles Schaeffer The combinatorics of Hurwitz numbers and increasing quadrangulations
RGMW06 11th July 2018
10:00 to 11:00
Jason Miller Random walk on random planar maps II
We will describe some recent developments on the study of random walks on random planar maps. We first review the continuum constructions from Liouville quantum gravity as a mating of trees. We will then explain how one can analyze the behavior of random walk on the mated-CRT map, a random planar map model defined out of the continuum tree-mating constructions. Finally, we will explain how these results can be transferred to a wide variety of discrete random planar map models. This is based on joint works with Bertrand Duplantier, Ewain Gwynne, and Scott Sheffield.
RGMW06 11th July 2018
11:15 to 12:15
Antti Kupiainen Quantum Liouville Theory
I will discuss prospects for deriving quantum theory (Virasoro representations, spectrum and bootstrap) from the probabilistic formulation of Liouville conformal field theory.




RGMW06 12th July 2018
09:10 to 09:55
Tom Hutchcroft An operator-theoretic approach to nonamenable percolation
I will discuss progress on and implications of the following new conjecture: For any nonamenable transitive graph G=(V,E), the matrix of connection probabilities in critical Bernoulli percolation is bounded as an operator from L^2(V) to L^2(V).
RGMW06 12th July 2018
10:00 to 11:00
Sourav Chatterjee An introduction to gauge theories for probabilists: Part III
In the third lecture of the series, I will talk about the master loop equation and gauge-string duality. Recent rigorous results will be discussed and proof sketches will be given. As in the previous lectures, many open problems will be stated.
RGMW06 12th July 2018
11:15 to 12:15
Gregory Miermont Exploring random maps: slicing, peeling and layering - 2
The combinatorial theory of maps, or graphs on surfaces, is rich of many different approaches (recursive decompositions, algebraic approches, matrix integrals, bijective approaches) which often have probabilistic counterparts that are of interest when one wants to study geometric aspects of random maps. In these lectures, I will review parts of this theory by focusing on three different decompositions of maps, namely, the slice decomposition, the peeling process, and the decomposition in layers, and by showing how these decompositions can be used to give access to quite different geometric properties of random maps.




RGMW06 12th July 2018
13:45 to 14:30
Julien Dubedat Stochastic Ricci Flow
The Ricci Flow on surfaces evolves the Riemannian metric to the constant curvature metric in the same conformal class. It has a natural interpretation as an infinite dimensional gradient flow. We describe the corresponding Langevin dynamics and its relation to Liouville theory. Joint work with Hao Shen.
RGMW06 12th July 2018
14:35 to 15:20
Igor Kortchemski Condensation in critical Cauchy Bienaymé-Galton-Watson trees
We will be interested in the structure of large Bienaymé-Galton-Watson random trees whose offspring distribution is critical and falls within the domain of attraction of a stable law of index α=1. In stark contrast to the case α∈(1,2], we show that a condensation phenomenon occurs: in such trees, one vertex with macroscopic degree emerges. One of the main tools is a limit theorem for centered downwards skip-free random walks whose steps are in the domain of attraction of a Cauchy distribution, when conditioned on a late entrance in the negative real line. As an application, we study the geometry of the boundary of random planar maps in a specific regime (called non-generic of parameter 3/2) and support the conjecture that faces in Le Gall & Miermont's 3/2-stable maps are self-avoiding. This is joint work with Loïc Richier.
RGMW06 13th July 2018
09:10 to 09:30
Cyril Marzouk Geometry of large random planar maps with a prescribed degree sequence
I will discuss some recent progress and still ongoing work about the scaling limit of the following configuration-like model on random planar maps: for every integer n, we are given n deterministic (even) integers and we sample a planar map uniformly at random amongst those maps with n faces and these prescribed degrees. Under a `no macroscopic face' assumption, these maps converge in distribution after suitable scaling towards the celebrated Brownian map, in the Gromov-Hausdorff-Prokhorov sense. This model covers that of p-angulations when all the integers are equal to some p, which we can allow to vary with n, without constraint; it also applies to so-called Boltzmann random maps and yields a CLT for planar maps.




RGMW06 13th July 2018
09:35 to 09:55
Joonas Turunen Critical Ising model on random triangulations of the disk: enumeration and limits
In this talk, I consider Boltzmann random triangulations coupled to the Ising model on their faces, under Dobrushin boundary conditions and at the critical point. First, the partition function is computed and the perimeter exponent shown to be 7/3 instead of the exponent 5/2 for uniform triangulations. Then, I sketch the  construction of the local limit in distribution when the two components of the Dobrushin boundary tend to infinity one after the other, using the peeling process along an Ising interface. In particular, the main interface in the local limit touches the (infinite) boundary almost surely only finitely many times, a behavior opposite to that of the Bernoulli percolation on uniform maps. Some scaling limits closely related to the perimeters of clusters are also discussed. This is based on a joint work with Linxiao Chen.




RGMW06 13th July 2018
10:00 to 11:00
Gregory Miermont Exploring random maps: slicing, peeling and layering - 3
The combinatorial theory of maps, or graphs on surfaces, is rich of many different approaches (recursive decompositions, algebraic approches, matrix integrals, bijective approaches) which often have probabilistic counterparts that are of interest when one wants to study geometric aspects of random maps. In these lectures, I will review parts of this theory by focusing on three different decompositions of maps, namely, the slice decomposition, the peeling process, and the decomposition in layers, and by showing how these decompositions can be used to give access to quite different geometric properties of random maps.




RGMW06 13th July 2018
11:15 to 12:15
Jason Miller Random walk on random planar maps III
We will describe some recent developments on the study of random walks on random planar maps. We first review the continuum constructions from Liouville quantum gravity as a mating of trees. We will then explain how one can analyze the behavior of random walk on the mated-CRT map, a random planar map model defined out of the continuum tree-mating constructions. Finally, we will explain how these results can be transferred to a wide variety of discrete random planar map models. This is based on joint works with Bertrand Duplantier, Ewain Gwynne, and Scott Sheffield.
RGMW06 13th July 2018
13:45 to 14:30
Eveliina Peltola Multiple SLEs, discrete interfaces, and crossing probabilities
Multiple SLEs are conformally invariant measures on families of curves, that naturally correspond to scaling limits of interfaces in critical planar lattice models with alternating (”generalized Dobrushin”) boundary conditions. I discuss classification of these measures and how the convergence for discrete interfaces in many models is obtained as a consequence. When viewed as measures with total mass, the multiple SLEs can also be related to probabilities of crossing events in lattice models. The talk is based on joint works with Hao Wu (Yau Mathematical Sciences Center, Tsinghua University) and Vincent Beffara (Université Grenoble Alpes, Institut Fourier).
RGMW06 13th July 2018
14:35 to 15:20
Jason Schweinsberg Yaglom-type limit theorems for branching Brownian motion with absorption
We consider one-dimensional branching Brownian motion in which particles are absorbed at the origin. We assume that when a particle branches, the offspring distribution is supercritical, but the particles are given a critical drift towards the origin so that the process eventually goes extinct with probability one. We establish precise asymptotics for the probability that the process survives for a large time t, improving upon a result of Kesten (1978) and Berestycki, Berestycki, and Schweinsberg (2014). We also prove a Yaglom-type limit theorem for the behavior of the process conditioned to survive for an unusually long time, which also improves upon results of Kesten (1978). An important tool in the proofs of these results is the convergence of branching Brownian motion with absorption to a continuous state branching process.




RGMW06 16th July 2018
10:00 to 11:00
Wendelin Werner Conformal loop ensembles on Liouville quantum gravity 1
In these three lectures, I plan to describe our ongoing research program with Jason Miller and Scott Sheffield that deals with the natural interplay between Conformal Loop Ensembles and the Liouville Quantum gravity area measures. I will explain some features in the continuous world of CLE and LQG/GFF that do hopefully enlighten some of the basic features of the Quantum gravity approach to critical systems, and give a roadmap to understand the scaling limit of some decorated discrete planar maps models.
RGMW06 16th July 2018
11:15 to 12:15
Allan Sly Phase transitions of Random Constraint Satisfaction Problems - 1
Random constraint satisfaction problems encode many interesting questions in the study of random graphs such as the chromatic and independence numbers. Ideas from statistical physics provide a detailed description of phase transitions and properties of these models. We will discuss the one step replica symmetry breaking transition that many such models undergo and the Satisfiability Threshold for the random K-SAT model.




RGMW06 16th July 2018
13:45 to 14:30
Ellen Powell A characterisation of the Gaussian free field
I will present a joint work with Nathanaël Berestycki and Goirab Ray in which we prove that a random distribution in two dimensions which is conformally invariant and satisfies a natural domain Markov property is a multiple of the Gaussian free field. This result holds subject only to a fourth moment assumption.
RGMW06 16th July 2018
14:35 to 14:55
Thomas Budzinski Simple random walk on supercritical causal maps
We study the simple random walk on supercritical causal maps, obtained from a supercritical Galton-Watson tree by adding at each level a cycle joining neighbour vertices. In particular, we will prove that, if
the underlying tree has no leaf, the simple random walk has almost sure positive speed.





RGMW06 16th July 2018
15:00 to 15:20
Lukas Schoug A multifractal SLE_kappa(rho) boundary spectrum
In a recent work, we have derived an almost sure multifractal boundary spectrum for SLE_kappa(rho)-processes. Finding the upper bound on the dimension is fairly standard and is done with the help of Girsanov's theorem. The lower bound on the dimension, however, is a bit more cumbersome. To find this, we use the imaginary geometry coupling of SLE and GFF to find a sufficient two-point estimate. In this talk, we will mainly focus on construction of the perfect points and the method of finding the two-point estimate.
RGMW06 17th July 2018
09:10 to 09:55
Tim Budd Lattice walks & peeling of planar maps
RGMW06 17th July 2018
10:00 to 11:00
Allan Sly Phase transitions of Random Constraint Satisfaction Problems - 2
Random constraint satisfaction problems encode many interesting questions in the study of random graphs such as the chromatic and independence numbers. Ideas from statistical physics provide a detailed description of phase transitions and properties of these models. We will discuss the one step replica symmetry breaking transition that many such models undergo and the Satisfiability Threshold for the random K-SAT model.




RGMW06 17th July 2018
11:15 to 12:15
Wendelin Werner Conformal loop ensembles on Liouville quantum gravity 2
In these three lectures, I plan to describe our ongoing research program with Jason Miller and Scott Sheffield that deals with the natural interplay between Conformal Loop Ensembles and the Liouville Quantum gravity area measures. I will explain some features in the continuous world of CLE and LQG/GFF that do hopefully enlighten some of the basic features of the Quantum gravity approach to critical systems, and give a roadmap to understand the scaling limit of some decorated discrete planar maps models.
RGMW06 17th July 2018
13:45 to 14:30
Yilin Wang Geometric descriptions of the Loewner energy
The Loewner energy of a simple loop on the Riemann sphere is defined to be the Dirichlet energy of its driving function which is reminiscent in the SLE theory. It was shown in a joint work with Steffen Rohde that the definition is independent of the parametrization of the loop, therefore provides a Moebius invariant quantity on free loops which vanishes only on the circles. In this talk, I will present intrinsic interpretations of the Loewner energy (without involving the iteration of conformal distortions given by the Loewner flow), using the zeta-regularizations of determinants of Laplacians and show that the class of finite energy loops coincides with the Weil-Petersson class of the universal Teichmueller space.
RGMW06 17th July 2018
14:35 to 15:20
Marcin Lis Circle patterns and critical Ising models
A circle pattern is an embedding of a planar graph in which each face is inscribed in a circle. I will define and prove magnetic criticality of a large family of Ising models on planar graphs whose dual is a circle pattern. The new construction includes as a special case the critical isoradial Ising models of Baxter.
RGMW06 18th July 2018
09:10 to 09:30
Danny Nam Cutoff for the Swendsen-Wang dynamics
The Swendsen-Wang dynamics is an MCMC sampler of the Ising/Potts model, which recolors many vertices at once based on the random-cluster representation of the model. Although widely used in practice due to efficiency, the mixing time of the Swendsen-Wang dynamics is far from being well-understood, mainly because of its non-local behavior. In this talk, we prove cutoff phenomenon for the Swendsen-Wang dynamics on the lattice at high enough temperatures, meaning that the Markov chain exhibits a sharp transition from mixed€ to well-mixed.

Joint work with Allan Sly.




RGMW06 18th July 2018
09:35 to 09:55
Antoine Jego Thick Points of Random Walk and the Gaussian Free Field
We consider the thick points of random walk, i.e. points where the local time is a fraction of the maximum. In two dimensions, we answer a question of Dembo, Peres, Rosen and Zeitouni and compute the number of thick points of planar random walk, assuming that the increments are symmetric and have a finite moment of order two. The proof provides a streamlined argument based on the connection to the Gaussian free field and works in a very general setting including isoradial graphs. In higher dimensions, we show that the number of thick points converges to a nondegenerate random variable and that the maximum of the local times converges to a randomly shifted Gumbel distribution.




RGMW06 18th July 2018
10:00 to 11:00
Wendelin Werner Conformal loop ensembles on Liouville quantum gravity 3
In these three lectures, I plan to describe our ongoing research program with Jason Miller and Scott Sheffield that deals with the natural interplay between Conformal Loop Ensembles and the Liouville Quantum gravity area measures. I will explain some features in the continuous world of CLE and LQG/GFF that do hopefully enlighten some of the basic features of the Quantum gravity approach to critical systems, and give a roadmap to understand the scaling limit of some decorated discrete planar maps models.
RGMW06 18th July 2018
11:15 to 12:15
Allan Sly Phase transitions of Random Constraint Satisfaction Problems - 3
Random constraint satisfaction problems encode many interesting questions in the study of random graphs such as the chromatic and independence numbers. Ideas from statistical physics provide a detailed description of phase transitions and properties of these models. We will discuss the one step replica symmetry breaking transition that many such models undergo and the Satisfiability Threshold for the random K-SAT model.




RGMW06 19th July 2018
09:10 to 09:55
Perla Sousi Capacity of random walk and Wiener sausage in 4 dimensions
In four dimensions we prove a non-conventional CLT for the capacity of the range of simple random walk and a strong law of large numbers for the capacity of the Wiener sausage. This is joint work with Amine Asselah and Bruno Schapira.
RGMW06 19th July 2018
10:00 to 11:00
Vincent Vargas The semiclassical limit of Liouville conformal field theory
RGMW06 19th July 2018
11:15 to 12:15
Beatrice de Tiliere The Z-Dirac and massive Laplacian operators in the Z-invariant Ising model
RGMW06 19th July 2018
13:45 to 14:30
Ewain Gwynne The fractal dimension of Liouville quantum gravity: monotonicity, universality, and bounds
We show that for each $\gamma \in (0,2)$, there is an exponent $d_\gamma > 2$, the ``fractal dimension of $\gamma$-Liouville quantum gravity (LQG)", which describes the ball volume growth exponent for certain random planar maps in the $\gamma$-LQG universality class, the graph-distance displacement exponent for random walk on these random planar maps, the exponent for the Liouville heat kernel, and exponents for various continuum approximations of $\gamma$-LQG distances such as Liouville graph distance and Liouville first passage percolation. This builds on work of Ding-Zeitouni-Zhang (2018). We also show that $d_\gamma$ is a continuous, strictly increasing function of $\gamma$ and prove upper and lower bounds for $d_\gamma$ which in some cases greatly improve on previously known bounds for the aforementioned exponents. For example, for $\gamma=\sqrt 2$ (which corresponds to spanning-tree weighted planar maps) our bounds give $3.4641 \leq d_{\sqrt 2} \leq 3.63299$ and in the limiting case we get $4.77485 \leq \lim_{\gamma\rightarrow 2^-} d_\gamma \leq 4.89898$. Based on joint works with Jian Ding, Nina Holden, Tom Hutchcroft, Jason Miller, and Xin Sun.
RGMW06 19th July 2018
14:35 to 15:20
Adrien Kassel Quantum spanning forests
I will introduce a new integrable statistical physics model, which we call a quantum spanning forest (QSF) and which provides a probabilistic framework for studying spanning-tree like structures coupled to a connection (with holonomies taking values in a unitary group of arbritrary rank), both on finite and infinite graphs. I will explain that QSFs form a special case of a new general family of probability measures which we call determinantal subspace processes (DSP), and for which we develop an independent theory. Finally, I will describe some relationships between QSFs and holonomies of random walk and the covariant Gaussian free field via the study of electrical networks with holonomy. This is joint work with Thierry Lévy.
RGMW06 20th July 2018
09:10 to 09:55
Guillaume Remy Exact formulas on Gaussian multiplicative chaos and Liouville theory
We will present recent progress that has been made to prove exact formulas on the Gaussian multiplicative chaos (GMC) measures. We will give the law of the total mass of the GMC measure on the unit circle (the Fyodorov-Bouchaud formula) and on the unit interval (in collaboration with T. Zhu). The techniques of proof come from the link between GMC and Liouville conformal field theory studied by David-Kupiainen-Rhodes-Vargas. If time permits we will also discuss the connections with the quantum sphere and the quantum disk of the Duplantier-Miller-Sheffield approach to Liouville quantum gravity.
RGMW06 20th July 2018
10:00 to 11:00
Nicolas Curien Random stable maps : geometry and percolation
Random stable maps are discrete random Boltzmann maps with large faces that are conjecturally linked to the CLE. We review some recent results on the geometry of such graphs and their duals, and on the behavior of Bernoulli percolations on these objects. The phenomenons that appear are the analogs of those we encoutered (or conjectured) for the Euclidean CLE. In particular, the critical bond percolation process creates a duality between the dense and dilute phase of random stable maps. The talk is based on joint works with Timothy Budd, Cyril Marzouk and Loïc Richier.
RGMW06 20th July 2018
11:15 to 12:15
Remi Rhodes Towards quantum Kähler geometry
We propose a natural framework for probabilistic Kähler geometry on a one-dimensional complex manifold based on a path integral involving the Liouville action and the Mabuchi K-energy. Both functionals play an important role respectively in Riemannian geometry (in the case of surfaces) and Kähler geometry. The Weyl anomaly of this path integral, which encodes the way it reacts to changes of background geometry, displays the standard Liouville anomaly plus an additional K-energy term. Motivations come from theoretical physics where these type of path integrals arise as a model for fluctuating metrics on surfaces when coupling (small) massive perturbations of conformal field theories to quantum gravity as advocated by A. Bilal, F. Ferrari, S. Klevtsov and S. Zelditch in a series of physics papers. Interestingly, our computations show that quantum corrections perturb the classical Mabuchi K-energy and produce a quantum Mabuchi K-energy: this type of correction is reminiscent of the quantum Liouville theory. Our construction is probabilistic and relies on a variant of Gaussian multiplicative chaos (GMC), the Derivative GMC (DGMC for short). The technical backbone of our construction consists in two estimates on (derivative and standard) GMC which are of independent interest in probability theory. Firstly, we show that these DGMC random variables possess negative exponential moments and secondly we derive optimal small deviations estimates for the GMC associated with a recentered Gaussian Free Field.
RGMW06 20th July 2018
13:45 to 14:05
Joshua Pfeffer External DLA on a spanning-tree-weighted random planar map
External diffusion limited aggregation (DLA) is a widely studied subject in the physics literature, with many manifestations in nature; but it is not well-understood mathematically in any environment. We consider external DLA on an infinite spanning-tree-weighted random planar map. We prove that the growth exponent for the external diameter of the DLA cluster exists and is equal to $2/d _{\sqrt{2}}$, where $d_{\sqrt{2}}$ denotes the ``fractal dimension of $\sqrt{2}$-Liouville quantum gravity (LQG)''---or, equivalently, the ball volume growth exponent for the spanning-tree weighted map. Our proof is based on the fact that the complement of an external DLA cluster on a spanning-tree weighted map is a spanning-tree weighted map with boundary, which allows us to reduce our problem to proving certain estimates for distances in random planar maps with boundary. This is joint work with Ewain Gwynne.




RGMW06 20th July 2018
14:10 to 14:30
Tunan Zhu Distribution of gaussian multiplicative chaos on the unit interval
Starting from a log-correlated field one can define by a standard regularization technique the associated Gaussian multiplicative chaos (GMC) measure with density formally given by the exponential of the log-correlated field. Very recently exact formulas have been obtained for specific GMC measures. On the Riemann sphere a proof of the celebrated DOZZ formula has been given by Kupiainen-Rhodes-Vargas and for the GMC on the unit circle the Fyodorov-Bouchaud formula has been recently proven by Remy. In this talk we will present additional results on GMC measures associated to a log-correlated field on the unit interval [0,1]. We will present a very general formula for the real moments of the total mass of GMC with log-singularities in 0 and 1. This proves a set of conjectures given by Fyodorov, Le Doussal, Rosso and Ostrovsky. As a corollary, this gives the distribution of the total mass.




University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons