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Timetable (RGMW04)

Random Planar Structures and Statistical Mechanics

Monday 20th April 2015 to Friday 24th April 2015

Monday 20th April 2015
09:00 to 09:50 Registration
09:50 to 10:00 Welcome from Christie Marr (INI Deputy Director) INI 1
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.

INI 1
11:00 to 11:30 Morning Coffee
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.

INI 1
12:30 to 13:30 Lunch at Wolfson Court
14:00 to 15:00 T Budd (University of Copenhagen)
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.
INI 1
15:00 to 15:30 Afternoon Tea
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.

INI 1
16:30 to 17:30 Welcome Wine Reception
Tuesday 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.

INI 1
11:00 to 11:30 Morning Coffee
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.

INI 1
12:30 to 13:30 Lunch at Wolfson Court
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.

INI 1
15:00 to 15:30 Afternoon Tea
15:30 to 16:30 I Kortchemski (CNRS (Centre national de la recherche scientifique))
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).
INI 1
Wednesday 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.

INI 1
10:00 to 11:00 G Schaeffer (École Polytechnique)
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.
INI 1
11:00 to 11:30 Morning Coffee
11:30 to 12:30 G Borot (Max-Planck-Institut fur Mathematik, Bonn)
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.

INI 1
12:30 to 13:30 Lunch at Wolfson Court
14:00 to 15:00 O Bernardi (Brandeis University)
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.

INI 1
15:00 to 15:30 Afternoon Tea
15:30 to 16:30 O Gurel Gurevich (Hebrew University of Jerusalem)
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.

INI 1
19:30 to 22:00 Conference Dinner at Trinity College
Thursday 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.

INI 1
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.
INI 1
11:00 to 11:30 Morning Coffee
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.

INI 1
12:30 to 13:30 Lunch at Wolfson Court
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.

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

INI 1
Friday 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.

INI 1
11:00 to 11:30 Morning Coffee
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.

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

INI 1
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons