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Timetable (RGMW02)

Conformally Invariant Scaling Limits

Monday 26th January 2015 to Friday 30th January 2015

Monday 26th January 2015
09:00 to 09:50 Registration
09:50 to 10:00 Welcome from John Toland (INI Director)
10:00 to 11:00 W Werner (ETH Zürich)
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).

INI 1
11:00 to 11:30 Morning Coffee
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.
INI 1
12:30 to 13:30 Lunch at Wolfson Court
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.

INI 1
14:30 to 15:00 Afternoon Tea
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.

INI 1
17:00 to 18:00 Welcome Wine Reception
Tuesday 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.

INI 1
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.

INI 1
11:00 to 11:30 Morning Coffee
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.
INI 1
12:30 to 13:30 Lunch at Wolfson Court
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.

INI 1
14:30 to 15:00 Afternoon Tea
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)

INI 1
Wednesday 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.
INI 1
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.

INI 1
11:00 to 11:30 Morning Coffee
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.
INI 1
12:30 to 13:30 Lunch at Wolfson Court
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.
INI 1
14:30 to 15:00 Afternoon Tea
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.

INI 1
19:30 to 22:00 Conference Dinner at Cambridge Union Society hosted by Cambridge Dining Co.
Thursday 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.

INI 1
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.

INI 1
11:00 to 11:30 Morning Coffee
11:30 to 12:30 V Beffara ([ENS - Lyon, Universität Bonn])
Drawing maps
I will discuss various natural ways of drawing maps, and some of the mathematics behind a few nice pictures.
INI 1
12:30 to 13:30 Lunch at Wolfson Court
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).

INI 1
Friday 30th January 2015
09:00 to 10:00 D Chelkak ([ETH Zürich, Russian Academy of Sciences])
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).
INI 1
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.
INI 1
11:00 to 11:30 Morning Coffee
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.

INI 1
12:30 to 13:30 Lunch at Wolfson Court
13:30 to 14:30 G Pete ([Hungarian Academy of Sciences & Technical University of Budapest])
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}).
INI 1
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons