Abstracts: Mini Courses
Michael Damron: Geodesics in first-passage percolation
Geodesics in first-passage percolation. First-passage percolation (FPP) is a model of a random metric on the vertices of a graph, typically Z^d. Assigning i.i.d., nonnegative random variables to the edges of Z^d, we consider the weighted graph metric. The main questions in FPP concern comparing large-scale properties of this metric space to those of, say, R^d, or Z^d with the l^1 metric.
This course will focus on directional properties of infinite geodesics, which are infinite paths all of whose segments are length-minimizing, and their relation to Busemann functions. The four lectures will roughly cover (A) Hoffman's '05 proof of existence of disjoint infinite geodesics (B) Newman's work and conjectures under curvature assumptions (C) the use of Busemann functions in studying directional properties of geodesics and (D) construction of Busemann gradient fields and verification of some directional properties of geodesics from the work of Damron-Hanson '13.
Ron Peled: The spin and loop O(n) models
The (classical) spin O(n) model is a model on a d-dimensional lattice in which a vector on the (n-1)-dimensional sphere is assigned to every lattice site and the vectors at adjacent sites interact ferromagnetically via their inner product. Special cases include the Ising model (n=1), the XY model (n=2) and the Heisenberg model (n=3). We will discuss questions of long-range order (spontaneous magnetization) and decay of correlations in the spin O(n) model for different combinations of the lattice dimension d and the spin dimension n. Among the topics presented are the Mermin-Wagner theorem, the Kosterlitz-Thouless transition, the infra-red bound and Polyakov's conjecture on the two-dimensional Heisenberg model.
The loop O(n) model is a model for a random configuration of disjoint loops on the hexagonal lattice. The model is parameterized by two real numbers n, x>=0 and assigns probability proportional to x^{number of edges} n^{number of loops} to each configuration. Special cases include self-avoiding walk (n=0), the Ising model (n=1), critical percolation (n=x=1), dimer model (n=1, x=infinity), integer-valued (n=2) and tree-valued (integer n>=3) Lipschitz functions and the hard hexagon model (n=infinity). The object of study in the model is the typical structure of loops. We will review the connection of the model with the spin O(n) model and discuss its conjectured phase diagram, emphasizing the many open problems remaining. We then elaborate on recent results for the self-avoiding walk case and for large values of n.
Invited Talks
Krzysztof Burdzy: Twin peaks
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 estimate the distance between two peaks on a large discrete torus, assuming that the random labeling is conditioned on having exactly two peaks.
Joint work with Sara Billey, Soumik Pal, Lerna Pehlivan and Bruce Sagan.
Manuel Cabezas: Scaling limit for the ant in a high-dimensional labyrinth
It is believed that in high dimensions, a large critical percolation cluster should scale to the so-called integrated super Brownian excursion (ISBE). Moreover, it is also believed that a simple random walk in the critical percolation cluster should scale to the Brownian motion on the ISBE. In this talk I will present a result that gives conditions for a general sequence of random subgraphs of Z^d under which the random walk on these graphs scales to the Brownian motion on the ISBE. We will show how to apply this general theorem in the case where the graphs are obtained as the trace of critical branching random walks in Z^d, d>12. Joint work with Gerard Ben Arous and Alexander Fribergh
Elisabetta Candellero: Percolation and isoperimetric inequalities
In this talk we will discuss some relations between percolation on a given graph G and its geometry. There are several interesting questions relating various properties of G such as growth (or dimension) and the process of percolation on it. In particular we will look for conditions under which its critical percolation threshold is non-trivial, that is: p_c(G) is strictly between zero and one. In a very influential paper on this subject, Benjamini and Schramm asked whether it was true that for every graph satisfying dim(G) > 1, one has p_c(G) < 1. We will explain this question in detail and present some recent results that have been obtained in this direction. This talk is based on a joint work with Augusto Teixeira (IMPA, Rio de Janeiro, Brazil).
Pietro Caputo: Random walk on sparse random directed graphs
A finite Markov chain exhibits cutoff if its distance from stationarity remains close to the initial value for a certain number of iterations and then abruptly drops to near zero on a much shorter time scale. Originally discovered in the context of card shuffling by Aldous and Diaconis in the 80's, this remarkable phenomenon is now rigorously established for many reversible chains. Here we consider random walks on random directed graphs with given degree sequences. In this nonreversible setting, even understanding the stationary distribution represents a challenge. In the sparse regime, we establish the cutoff phenomenon, determine its time window and prove that the cutoff profile approaches a universal gaussian shape. Moreover, we determine an explicit recursive equation characterizing the stationary distribution. This is joint work with Charles Bordenave and Justin Salez.
Jack Hanson: Bigeodesics in first-passage percolation
In two-dimensional first-passage percolation, a number of longstanding conjectures exist regarding the behavior of infinite geodesics (infinite paths whose finite subsegments are point-to-point geodesics). Notable among them is the claim that, under mild assumptions on the distribution of the edge weights, there should a.s. be no doubly infinite geodesic ("bigeodesic"). In the '90s, Licea and Newman showed that, under an unproven curvature assumption on the model's "limiting shape" (which describes the shape of large balls in the random metric), every infinite geodesic a.s. has asymptotic direction, and there is a full-measure set D of [0, 2p) such that for any \theta_1, \theta_2 in D, there is no bigeodesic with ends directed in directions \theta_1 and \theta_2. We will discuss new results on the bigeodesic conjecture showing, under the assumption that the limiting shape's boundary is differentiable, there is a.s. no bigeodesic with one end directed in any deterministic direction. This rules out existence of ground state pairs whose interface has a deterministic direction in the related disordered ferromagnet model.
Dima Ioffe: Low temperature interfaces, ordered random walks and Ferrari-Spohn diffusions
I shall discuss scaling limits for a class of ordered random walks subject to positivity constraint and self-potentials, which look like generalized area tilts. Such polymers arise, for instance, as effective models for:
(a) Phase segregation lines in 2D Ising model with negative b.c. and positive magnetic fields. (b) Level lines of 2+1 discrete SOS models coupled with Bernoulli bulk fields.
The limiting objects happen to be ergodic Ferrari-Spohn diffusions, conditioned on non-intersection. Invariant measures for n such diffusions are given in terms of Slater determinants constructed from n first eigenfunctions of appropriate Sturm-Liouville operators.
Based on joint works with S. Shlosman, Y.Velenik and V.Wachtel
Elena Kosygina: A zero-one law for recurrence and transience of frog processes
We provide sufficient conditions for the validity of a dichotomy, i.e. zero-one law, between recurrence and transience of frog models on a large class of non-random and on some random graphs. In particular, the results cover frog models with i.i.d. numbers of frogs per site where the frog dynamics are given by quasi-transitive Markov chains or by random walks in a common random environment including super-critical percolation clusters on Zd. We also give a sufficient and almost sharp condition for recurrence of uniformly elliptic frog processes on Zd. Its proof uses the general zero-one law. This is a joint work with Martin Zerner (Universitaet Tuebingen, Germany).
Fabio Martinelli: Bootstrap percolation and kinetically constrained spin models: critical lengths and mixing time scales
In recent years, a great deal of progress has been made in understanding the behaviour of a particular class of monotone cellular automata, commonly known as bootstrap percolation. In particular, if one considers only two-dimensional automata, then we now have a fairly precise understanding of the typical evolution of these processes, starting from p-random initial conditions of infected sites. Given a bootstrap model, one can consider the associated kinetically constrained spin model in which the state (infected or healthy) of a vertex is resampled (independently) at rate 1 by tossing a p-coin if it could be infected in the next step by the bootstrap process, and remains in its current state otherwise. Here p is the probability of infection. The main interest in KCM's stems from the fact that, as p \to 0, they mimick some of the most stricking features of the glass transition, a major and still largely open problem in condensed matter physics.
In this talk, motivated by recent universality results for bootstrap percolation, I'll discuss some "universality conjectures" concerning the growth of the (random) infection time of the origin in a KCM as p \to 0.
Joint project with R. Morris (IMPA) and C. Toninelli (Paris VII).
Yinon Spinka: Long-range order in random 3-colorings of Z^d
Consider a random coloring of a bounded domain in Zd with the probability of each coloring F proportional to exp(-\beta*N(F)), where \beta>0 is a parameter (representing the inverse temperature) and N(F) is the number of nearest neighboring pairs colored by the same color. This is the anti-ferromagnetic 3-state Potts model of statistical physics, used to describe magnetic interactions in a spin system. The Kotecky conjecture is that in such a model, for d=3 and high enough \beta, a sampled coloring will typically exhibit long-range order, placing the same color at most of either the even or odd vertices of the domain. We give the first rigorous proof of this fact for large d. This extends previous works of Peled and of Galvin, Kahn, Randall and Sorkin, who treated the case \beta=infinity.
Joint work with Ohad Feldheim.
Alain-Sol Sznitman: Level-set percolation of the Gaussian free field on transient trees
In this talk we will discuss some comparisons between level-set percolation of the Gaussian free field and percolation of the vacant set of random interlacements on transient weighted graphs, with a special interest for the case of transient trees. Some of the results presented in this talk come from a recent work in collaboration with Angelo Abächerli.
Vincent Tassion: Critical oriented percolation in the plane
We consider oriented Bernoulli percolation on the square lattice. This process exhibits a well-known phase transition at a critical edge density p_c. In this talk, we will discuss the geometric properties of the process at criticality. Based on a Russo-Seymour-Welsh type result and an adapted version of the Box-Crossing property, we prove new results concerning the critical behavior. First, we establish that the probability that the origin is connected to distance n decays polynomially fast in n. This contrasts with the sub-critical regime, where the decay is exponential. We also prove that the critical cluster of 0 conditioned to survive to distance n has a typical width w(n) which satisfies w(n)>n^{2/5} and w(n)=o(n). These sub-linear polynomial fluctuations contrast with the super-critical regime where w(n) is known to behave linearly in n. All our results extend to the graphical representation of the one-dimensional contact process. This talk is based on a joint work with Hugo Duminil-Copin and Augusto Teixeira.
Daniel Ueltschi: First-passage percolation and the decay of transverse correlations in quantum spin systems
The random loop representations of quantum Heisenberg models were introduced twenty years ago by Tóth and Aizenman-Nachtergaele. I will describe an extension of these representations, and invoke results from first-passage percolation to get a proof of exponential decay of transverse quantum correlations. (Joint work with J. Björnberg)
Contributed Talks
Linxiao Chen: Interface structure in the half planar Ising-triangulations
We present a construction of the half-planar limit of random planar triangulations of a polygon, coupled to an Ising model on its faces. Our approach is based on the resolution of the loop equation associated to the model, and the use of peeling processes. Our method also yields naturally several results on the interface structure of the limit.
Rene Conijn: Largest clusters in percolation
Consider an n x n-box in the triangular lattice. The asymptotic behaviour, as n tends to infinity, of the largest percolation clusters in this box was well studied by Borgs, Chayes, Kesten and Spencer in (1999 and 2001). However some questions remained open. If we restrict ourself to critical percolation the size of the largest cluster is of the order n^(91/48). The first natural question is: does there exist a limiting distribution for the size of the largest cluster scaled by its order? Furthermore if it exists: what is its support and does it have atoms? In this talk we show that the limiting distribution exists and state some properties. Additionally we comment on related results for the Ising model. Based on joint work with Rob van den Berg, Federico Camia and Demeter Kiss.
Dirk Erhard: On a scaling limit of the stochastic heat equation with exclusion interaction
This talk is about the equation \partial u(x,t)/\partial t = \Delta u(x,t) + [\xi(x,t)-\rho]u(x,t), x\in \Z^d, t\geq 0. Here, \Delta is the discrete Laplacian and the \xi-field is a stationary and ergodic dynamic random environment with mean $\rho$ that drives the equation. I will focus on the case where \xi is given in terms of a simple symmetric exclusion process, i.e., $\xi can be described by a field of simple random walks that move independently from each other subject to the rule that no two random walks are allowed to occupy the same site at the same time. I will discuss the behaviour of the equation when time and space are suitably scaled by some parameter N that tends to infinity. It turns out that in dimension two and three a renormalisation has to be carried out in order to see a non-trivial limit. This is joint work in progress with Martin Hairer.
Clément Erignoux: Hydrodynamic limit of a non-gradient model for collective dynamics
Extensive work has been put in the modelling of animal collective dynamics in the last decades, building on the work of Viscek&Al (1995). These empirical approaches have unveiled several interesting phenomenon regarding phase transitions and separations. However, most of the theoretical background in collective dynamics modelling relies on mean-field approximations. I will present a lattice model where interactions between partices happen on a purely microscopic level, and describe some of the challenges in the proof of its hydrodynamic limit.
Maxime Gagnebin: Delocalization of General Height Functions
We consider random functions of a $n\times n$ torus with one point fixed at zero. The functions $f$ are weighted according to a Boltzmann weight $U(f)$, which gives rise to the usual Boltzman measure on functions. Computing the variance of the function at a certain point has only been done for few potential. In this work, we show that the variance grows at least like $\sqrt{\log n}$ under very weak assumption on the potential. Joint work with R.Peled and P.Milos
Carl-Erik Gauthier: Self attracting diffusions on a sphere and application to a periodic case
In this talk, I will prove the almost-sure convergence for the self-attracting diffusion on the unit sphere $$dX_t=\sigma dW_{t}(X_t)-a\int_{0}^{t}\nabla_{\mathbb{S}^n}V_{X_s}(X_t) dsdt,\qquad X_0=x\in\mathbb{S}^n $$ where $\sigma >0$, $a 0$.
Matthias Gorny: A Curie-Weiss model of Self-Organized Criticality
In their famous 1987 article, Per Bak, Chao Tang and Kurt Wiesenfeld showed that certain complex systems, composed of a large number of dynamically interacting elements, are naturally attracted by critical points, without any external intervention. This phenomenon, called self-organized criticality (SOC), can be observed empirically or simulated on a computer in various models. However the mathematical analysis of these models turns out to be extremely difficult. Even models whose definition seems simple, such as the models describing the dynamics of a sandpile, are not well understood mathematically. In my presentation, I will introduce a model of SOC which is built by modifying the generalized Ising Curie-Weiss model. I will present a fluctuation theorem which proves that this model indeed exhibits SOC: the sum $S_{n}$ of the random variables behaves as in the typical critical generalized Ising Curie-Weiss model, i.e., the fluctuations are of order $n^{3/4}$ and the limiting law is $C \exp(-\lambda x^{4})\,dx$ where $C$ and $\lambda$ are suitable positive constants. Finally I will introduce associated dynamic models of SOC.
Watthanan Jatuviriyapornchai: Coarsening dynamics in condensing zero-range processes and size-biased birth death chains
Zero-range processes with decreasing jump rates are well known to exhibit a condensation transition under certain conditions on the jump rates, and the dynamics of this transition continues to be a subject of current research interest. Starting from homogeneous initial conditions, the time evolution of the condensed phase exhibits an interesting coarsening phenomenon of mass transport between cluster sites characterized by a power law. We revisit the approach in [C. Godreche, J. Phys. A: Math. Gen., 36(23) 6313 (2003)] to derive effective single site dynamics which form a non-linear birth death chain describing the coarsening behaviour. We extend these results to a larger class of parameter values, and introduce a size-biased version of the single site process, which provides an effective tool to analyze the dynamics of the condensed phase without finite size effects and is the main novelty of this paper. Our results are based on a few heuristic assumptions and exact computations, and are corroborated by detailed simulation data.
Matthew Junge: Frogs, bullets, and recursion
We present a novel technique for proving a random variable is infinite. First, cook up a recursive distributional equation, then transform via the probability generating function. New theorems for two notoriously difficult processes---the frog model and the bullet problem---showcase the power of this approach.
Pierre Monmarché: Functional inequalities approach for chains of interacting particle
Consider a chain (or more generally a finite graph) of N particles that interact with their neighbours in a deterministic way, and suppose only a few of them (for instance only the edges) are subject to a Brownian excitation. Then the invariant measure of the process is generally not explicit, and it was an open question whether it satisfied a functional inequality such as the Poincare or log-Sobolev one, and whether this could imply an exponential convergence to equilibrium. We solve the problem in the case of zero mass particles.
Chikara Nakamura: Laws of the Iterated Logarithm for Random Walk on the Random Conductance Model
In this talk, we will discuss laws of the iterated logarithm (LIL) for random walks on random conductance models (RCM). In the recent progress of random walk on random conductance models, long time Gaussian estimates of heat kernels have been obtained for various random walks on RCMs such as the super critical percolation cluster, i.i.d. random conductance models, vacant sets of random interlacements and level sets of Gaussian Free Field. We use the heat kernel estimates, and derive LIL and so-called another LIL which describes liminf behavior of random walks under the assumption that random walk on RCM enjoys the long time Gaussian heat kernel estimates. This talk is based on a joint work with T. Kumagai.
Izumi Okada: Geometric structures of favorite points and late points of simple random walks and high points of Gaussian free fields in two dimensions
We are working on the relation between the local time of random walks and Gaussian free fields, especially asymptotic properties of points where the local time are large or small namely, so-called favorite points or late points and where the value of the Gaussian free field are large or small namely, so-called high points in two dimensions. In this talk, I will introduce the results about the geometric structures of three points.
Thomas Rafferty: A dynamic transition in the relaxation time of a condensing zero-range process
The relaxation time of a Markov process controls the asymptotic rate of convergence towards its stationary distribution, which is also a key estimate in calculating hydrodynamic limits of such processes. We consider zero-range processes that exhibit condensation due to site defects (slow exit rates), where above a critical density a non-zero fraction of the particles accumulate on the defect sites. We compute the relaxation time for the dynamics on the complete graph, and show that there is a dynamical transition in the system size as the density crosses the critical density. A key step in this calculation is to compare the zero-range process with a birth-death chain generated via a decomposition of the state space. A method previously used to calculate relaxation times for non-condensing zero-range processes and the Ising model with Kawasaki dynamics. This is joint work with Stefan Grosskinsky and Paul Chleboun.
Helge Schäfer: On the number of cycles in random permutations without long cycles
We consider random permutations without long cycles. It is shown that the number of cycles converges to a Gaussian limit as in the classical case. We provide asymptotic expansions for expected value and variance. This is joint work with V. Betz.
Kevin Tanguy: Some superconcentration inequalities for extrema of stationary Gaussian processes
We present some concentration inequalities for extrema of stationary Gaussian processes. It provides non-asymptotic tail inequalities which fully reflect the fluctuation rate, and as such improve upon standard Gaussian concentration. The arguments rely on the hypercontractive approach developed by Chatterjee for superconcentration variance bounds.
Debleena Thacker: A New Approach to Classical and Modern Urn models
In this work the authors use the method of embedding into random recursive trees to study classical and generalized balanced urn models with non-negative replacement matrices, for both finite and infinitely many colors. In this paper, we show that for a balanced urn model, the law induced by randomly chosen color is same as the law induced by a branching Markov chain on a random recursive tree. We use this embedding to calculate the co-variance between the proportions of any two colors when the replacement matrix is irreducible, aperiodic, positive recurrent and geometrically ergodic. This proves the strong law of large numbers for the color count statistics. Several other results are obtained for the marginal distribution of the randomly chosen color.
Florian Völlering: Entrance Laws of Annihilating Brownian motions
We consider independent Brownian motions which annihilate instantly when they meet. This is straight forward as long as we start with a finite number of annihilating Brownian motions(aBMs). However, if we let the set of starting points become increasingly dense it is not obvious whether the resulting systems of aBMs converge and what the possible limit points are. The set of all possible limit points are the entrance laws, and the aBMs come down from infinity instantly in the sense that for all positive times there are locally only finitely many aBMs. I will provide a full classification of the entrance laws and illustrate with examples how these can be obtained via finite approximations.
Wei Wu: Space-time first passage percolation and wandering exponent
I will describe a new space-time first passage percolation model on $\mathbb{R}^2$, for which we prove the wandering exponent is bounded below by 3/5. Joint work with Yuri Bakhtin.