Marie Albenque
Simple maps (also) converge towards the Brownian map
In the last years, numerous families of planar maps (embeddings of planar graphs in the sphere) have been shown to converge to the Brownian map introduced by Miermont and Le Gall. Here we prove that simple maps (that is maps without loops nor multiple edges) also converge to the Brownian map when their number of edges goes to infinity.
This work relies first on a bijection between simple maps with a triangular outer faces and oriented binary trees. Then the distance in the maps can be studied with the help of some canonical “rightmost paths”, which behave well both in the map and in the tree. I’ll emphasize the combinatorial constructions that play a major role in this work.
This is joint work with Olivier Bernardi, Gwendal Collet and Eric Fusy.
Omer Angel
Random walks on transient planar graphs
I will describe results on random walks on transient planar graphs. In the general case, I will describe a connection between the Poisson and Martin boundaries and the topological boundary obtained by circle packing the graph in the unit disk. (Joint with Martin Barlow, Ori Gurel-Gurevich and Asaf Nachmias).
Anne-Laure Basdevant
The shape of large balls in highly supercritical percolation
In this talk, I will describe a connection between the distances in the infinite percolation cluster on the plane and the discrete-time TASEP on . This will show that, when the parameter of the percolation model goes to one, large balls in the cluster are asymptotically shaped near the axes like arcs of parabola.
Dima Chelkak
Kac-Ward formula, Q-determinants and XOR-Ising loops
In this talk we discuss a yet another proof (probably, simplest known) of the Kac-Ward formula for the Ising model partition function and its version involving Q-determinants. Being treated in full generality, this generalization would allow one to use the `` connections formalism’’ for the double Ising model and to prove Wilson’s conjecture (convergence of XOR-Ising loops to appropriate level lines of the GFF) in the same spirit as it was done by Kenyon and Dubedat for double-dimers. Another application is a new approach to (not necessarily critical) spin correlations in the single Ising model. Based on a joint work in progress with David Cimasoni (Université de Genève) and Adrien Kassel (ETH Zürich).
Loren Coquille
On the Gibbs states of the non-critical Potts model on
All Gibbs states of the supercritical q-state Potts model on are convex combinations of the q pure phases; in particular, they are all translation invariant. We recently proved this theorem with Hugo Duminil-Copin (Geneva), Dima Ioffe (Haifa) and Yvan Velenik (Geneva).
I will explain the basic concepts underlying this result and present the heuristics of the proof, which consists of considering the model in large finite boxes with arbitrary boundary condition, and proving that the center of the box lies deeply inside a pure phase with high probability. Our estimate of the finite-volume error term is of essentially optimal order, which stems from the Brownian scaling of fluctuating interfaces. The results hold at any supercritical value of the inverse temperature .
Sylvie Corteel
Dimers on Rail Yard Graphs
We introduce a general model of dimer coverings of certain plane bipartite graphs, which we call Rail Yard Graphs (RYG). Using a transfer matrix approach and the celebrated boson-fermion correspondence, the model can be reformulated as a Schur process (i.e. a random sequence of integer partitions). We obtain explicit expressions for the partition function and for the inverse Kasteleyn matrix, which yields all dimer correlation functions. Plane partitions, domino tilings of the Aztec diamond and pyramid partitions arise as particular cases of our model. This is joint work with Cédric Boutillier (Paris 6), Jérémie Bouttier (CEA), Guillaume Chapuy (CNRS) and Sanjay Ramassamy (Brown U.)
Hugo Duminil-Copin
Continuity of the phase transition for Random-Cluster models with
(joint work with V. Sidoravicius and V. Tassion)
We show that the phase transition of the nearest-neighbor ferromagnetic $q$-state Potts model on is continuous for , in the sense that there exists a unique Gibbs state, or equivalently that there is no ordering for the critical Gibbs states with monochromatic boundary conditions.
The proof uses the random-cluster representation and is based on two ingredients:
- Studying parafermionic observables on a discrete Riemann surface, it is shown that the two-point function for the free state decays sub-exponentially fast for cluster-weights .
- A new result proving the equivalence of several properties of
critical random-cluster models:
- the absence of infinite-cluster for wired boundary conditions,
- the uniqueness of infinite-volume measures,
- the sub-exponential decay of the two-point function for free boundary conditions,
- a Russo-Seymour-Welsh type result on crossing probabilities in rectangles with arbitrary boundary conditions.
The result leads to a number of consequences concerning the scaling limit of the model. It shows that the family of interfaces (for instance for Dobrushin boundary conditions) are tight when taking the scaling limit and that any sub-sequential limit can be parametrized by a Loewner chain. We also study the effect of boundary conditions on these sub-sequential limits. We will also mention applications for arm-events.
Alan Hammond
Exceptional times in dynamical percolation, and the Incipient Infinite Cluster
Critical percolation on planar lattices such as the hexagonal lattice is known to have no infinite open clusters. The term “incipient infinite cluster” was introduced by physicists to describe the large open clusters that are present at the critical value. Kesten gave the term a precise mathematical meaning, as the weak limit of conditionings of critical percolation on larger and larger open clusters containing the origin.
Antal Jarai has shown that several natural means of conditioning into existence an infinite open cluster containing the origin in critical percolation all give the incipient infinite cluster, lending the concept a sense of being natural. One possible route leading to the incipient is via dynamical percolation, in which the status of each hexagon in percolation is independently updated at exponential rate one to become open or closed with probability one-half. In this way, dynamical percolation has critical percolation has its invariant measure. There are exceptional times at which the open cluster of the origin in dynamical percolation has an infinite cluster, and this set of times has been proved by Garban, Pete and Schramm to have Hausdorff dimension . In this talk, I will discuss joint work with Gabor Pete and Oded Schramm in which the question of how the incipient infinite cluster arises in dynamical percolation is examined. While the configuration at the first positive exceptional time does not have the law of the incipient, a certain selection procedure which in essence picks a uniform exceptional time does locate a random configuration having this law.
Clément Hongler
Ising Model — making probabilistic sense of Conformal Field Theory ideas
We will consider the 2D Ising model at or near the critical point. The scaling limit of this model is conjecturally described by a minimal model of Conformal Field Theory. We will discuss various aspects of this description, and focus on making rigorous, probabilistic sense of these ideas.
Based on joint works with S. Benoist, D. Chelkak, H. Duminil-Copin, K. Izyurov, F. Johansson Viklund, A. Kemppainen, K. Kytölä and S.Smirnov.
Adrien Kassel
Natural loop models on (graphs on) surfaces
We describe some natural discrete loop models constructed from loop-erased random walk, spanning trees, or abelian sandpiles and explicit relations between them. We further describe the structure and geometry of these processes and discuss their model-dependence (in particular lattice dependence) versus universal features (existence of a continuous scaling limit in the universality class).
Richard Kenyon
Dimers, graphs and integrability.
We discuss the integrable structure underlying the planar dimer model, and, time permitting, part of the analogous structure for the planar graph Laplacian.
Zhongyang Li
1-2 model, Path Probability and Phase Transition
A 1-2 model is a probability measure on edge subsets of a hexagonal lattice, satisfying the condition that each vertex is incident to 1 or 2 present edges.
We construct a measure-preserving correspondence between 1-2 model configurations on the hexagonal lattice and dimer configurations on a decorated graph, and prove an closed form to compute the probability that a path appears in a 1-2 model configuration. With the help of the mass transport principle, we prove that almost surely there is no infinite paths in a 1-2 model configuration for any translation-invariant Gibbs measure. We also prove that almost surely there is at most 1 infinite homogeneous cluster in a 1-2 model configuration under any translation-invaraint Gibbs measure.
By investigating the behaviour of edge-edge correlations with changing parameters, we prove that there is a sharp phase transition in the 1-2 model, with critical parameter given by the condition that the spectral curve intersect the unit torus at a unique real point.
Jason Miller and Scott Sheffield
Quantum Loewner Evolution
We will describe a new universal family of growth processes called “Quantum Loewner Evolution” (QLE) and explain how QLE can be used to relate -Liouville quantum gravity with the Brownian map. We will also explain how QLE is related to diffusion limited aggregation, first passage percolation, and the dielectric breakdown model.
Gabor Pete
The scaling limit of the Minimal Spanning Tree in the plane
For independent percolation on planar lattices, Kesten’s scaling relation (1987) gives the near-critical window and the off-critical exponent in terms of critical exponents. Moreover, in a joint work started long ago with Christophe Garban and Oded Schramm, completed only recently (arXiv:1305.5526 and arXiv:1309.0269), we managed to build the scaling limit of the near-critical percolation ensemble and related models from the critical scaling limit. In this talk, I will describe some of the ideas in the construction of the continuum Minimal Spanning Tree. This is an interesting new object that is invariant under rotations, scalings, and translations, but probably not under general conformal maps.
Vincent Tassion
A universal behavior for planar Divide and Color percolation
We study the Divide and Color model on a planar lattice , defined as follows. First, sample a Bernoulli bond percolation, at a subcritical parameter . This yields a random partition of into finite clusters, and we say that two clusters at distance 1 from each other are adjacent. In a second step, assign one color to each cluster independently of the others. The color is chosen to be black with probability and white with probability . For fixed , we observe an infinite path of adjacent black clusters, as soon as the parameter exceeds a critical density . We focus on the behavior of when tends to . Considering that the critical behavior of bond percolation should be universal, Beffara and Camia conjectured that should converge , for any choice of . We prove it for the square lattice.
Wu Hao
Intersections of SLE paths
SLE curves are introduced by Oded Schramm as the candidate of the scaling limit of discrete models. In this talk, we first describe basic properties of SLE curves and their relation with discrete models. Then we summarize the Hausdorff dimension results related to SLE curves, in particular the new results about the dimension of cut points and double points. Third we introduce Imaginary Geometry, and from there give the idea of the proof of the dimension results.
