Justin Salez

Recent progress on the cutoff phenomenon

The cutoff phenomenon is an abrupt transition from out of equilibrium to equilibrium undergone by certain Markov processes in the limit where the size of the state space tends to infinity: instead of decaying gradually over time, their distance to equilibrium remains close to its maximal value for a while and suddenly drops to zero as the time parameter reaches a critical threshold. Discovered four decades ago in the context of card shuffling, this surprising phenomenon has since then been observed in a variety of models, from random walks on groups or complex networks to interacting particle systems. It is now believed to be universal among fast-mixing high-dimensional processes. Yet, current proofs are heavily model-dependent, and identifying the general conditions that trigger a cutoff remains one of the biggest challenges in the quantitative analysis of finite Markov chains. The purpose of this mini-course is to provide a self-contained introduction to this fascinating question, and to describe its recently uncovered relations with curvature and concentration.

Perla Sousi

Capacity for branching random walks and percolation

The capacity of a set is a classical notion in potential theory and it is a measure of the size of a set as seen by a random walk or Brownian motion. Recently Zhu defined the notion of branching capacity as the analogue of capacity in the context of a branching random walk. In this talk I will describe joint work with Amine Asselah and Bruno Schapira where we introduce a notion of capacity of a set for critical bond percolation and I will explain how it shares similar properties as in the case of branching random walks.

Guillaume Conchon-Kerjan

The random walk on the exclusion process in dimensions ≥ 5

We study a random walk on the d-dimensional lattice (d ≥ 5) whose movements are randomly driven by an underlying symmetric Simple Exclusion Process (SEP). In this set-up, standard techniques from static environments no longer apply. Moreover, the SEP is conservative and induces long-range space-time correlations. With the loss of monotonicity in dimensions above one, only perturbative results have been derived thus far.

We prove a law of large numbers in a wide regime of parameters. To do so, we track the measure of the environment conditioned on the past trajectory of the walker. This measure can be bounded in some strong sense between two inhomogeneous products of Bernoulli measures, that differ from the homogeneous one in an 'summable' way if the dimension is large enough. Joint work (in progress) with Daniel Kious and Rémy Poudevigne.

Christina Goldschmidt

The stable trees revisited

Consider the family tree of a branching process with offspring distribution (p_k, k ≥ 0) of mean 1 and with a heavy tail such that p_k ~ c k^-α-1 as k → ∞, for some constant c > 0 and α ϵ (1,2). (This implies that the offspring distribution is in the domain of attraction of an α-stable distribution.) Now condition the tree to have exactly n vertices. It is a well-known theorem (originally due to Duquesne) that distances in the tree vary as n^1/α and, on rescaling them by this factor, we obtain a limit in distribution as n → ∞ called the stable tree. In this talk, I’ll discuss a new (simple) construction of the stable trees, and indicate how to give a proof of the scaling limit theorem using it. This is joint work with Liam Hill.

Júlia Komjáthy

How communities help epidemics survive

Real-world contact networks are rarely just homogeneous collections of individuals: they contain households, workplaces, classrooms, social groups, and many other overlapping communities. In this talk, I will discuss a toy model for understanding how such local community structure changes the spread of an epidemic.

The network model will be a random intersection graph, where individuals belong to one or more microscopic-to-mesoscopic communities, and two individuals are connected if they share a community. On this graph we run a Markovian Susceptible-Infected-Susceptible epidemic, or equivalently the contact process: infected vertices recover at rate 1, while healthy vertices are infected by each infected neighbour at rate lambda.

The main question is how the presence of communities changes the epidemic threshold. More precisely, we compare the contact process on a random intersection graph with the corresponding "community-free" random graph having the same/similar degree distribution. How does community structure affect the critical infection rate for long survival? And when the infection does survive for a long time, how can we describe the metastable density of infected individuals in terms of the community-size and membership distributions? In this model, these questions can be answered quite explicitly, giving a mathematically clean picture of how small local clusters can have a macroscopic effect on epidemic persistence.

Pierre-Francois Rodriguez

Switching and the one-arm exponent for the GFF in three dimensions

For the metric graph Gaussian free field in three dimensions, the one-arm probability admits sharp two-sided estimates: the lower bound is due to Ding–Wirth, and matching upper bounds were recently obtained by Drewitz–Prévost–Rodriguez and separately by Cai–Ding. In this talk I will present a new, shorter proof of the more difficult upper bound, using ideas from these works and - crucially - Werner's switching identity for loop soups, which I will also explain.

Tyler Helmuth

Off-giant criticality in the mean-field arboreal gas

The arboreal gas is Bernoulli percolation conditioned on the event that each connected component is a tree. On the complete graph K_N, there is a phase transition: a subcritical phase in which all components (trees) are small, and a supercritical phase in which a giant tree exists. Unlike percolation, the supercritical phase has many mesoscopic trees of size N^2/3 in addition to the unique giant. I will discuss a new proof of this result by comparatively soft means, as well as some conjectures that this proof suggests.

Hugo Vanneuville

The Ising model, noise sensitivity, and considerations regarding curvature

Consider a spin model, for instance the Ising model, and let it evolve in time. Let t > 0. Which of these two alterations of the system has the strongest effect, compared to just letting the system evolve for a time t?

1) Flipping a spin at time 0 and letting the system evolve for a time t.

2) Letting the system evolve for a time t and then flipping a spin.

Answering such a question amounts to studying "curvature properties" of the dynamics. We will discuss this question and explain why it is useful in order to study noise sensitivity-type questions. Noise sensitivity, as introduced by Benjamini, Kalai and Schramm in 1999 in the context of Bernoulli percolation, refers to the fact that some events mix extremely quickly: if the law of the system at time 0 is the equilibrium measure conditioned on such an event, the conditioning is forgotten in an infinitesimal amount of time. Percolation events are examples of such events. Joint work with Vincent Tassion.

Paul Dario

Delocalisation for the low temperature long-range Gaussian chain

In this talk, we will discuss the discrete long-range Gaussian chain with 1/r^alpha interactions. I will introduce the model, its history and phase diagram. In this direction, a first notable result is the existence of a roughening phase transition for α = 2 established by Kjaer-Hilhorst and Fröhlich-Zegarlinski. For α > 2, the model is not expected to undergo a phase transition and a few important results have been recently obtained: Garban characterized the fluctuations of the chain at high temperature (and in fact fully identified its scaling limit) and Coquille-van Enter-Le Ny-Ruszel showed the (qualitative) delocalisation of the chain at every inverse temperature. After discussing these results in more details, I will present some quantitative estimates in the low temperature regime with range exponent α > 2 obtained in an joint work with L. Coquille and A. le Ny.

Ahmed Bou-Rabee

Sharpness and critical scaling of parking

Place a car or a parking spot at each site of the lattice, independently at random. Every car then moves by simple random walk until it reaches an empty spot, where it parks. When spots outnumber cars, I will show that the time a car spends driving has a stretched exponential tail with exponent d/(d+2), and that its expectation diverges at an explicit rate as the density of cars increases to the critical value. At the critical density, every car still parks and every spot is still filled, and yet every site is crossed infinitely often: the expected number of crossings of a site by time n grows like n^(4-d/4) in dimensions at most three, and like log n in higher dimensions. This confirms the conjectured n^1/2 growth on the square lattice and answers several further questions of Damron, Gravner, Junge, Lyu, and Sivakoff (2019). The main tool is a comparison with the divisible sandpile, whose odometer solves an optimal stopping problem for random walk in random scenery. This is joint work with Antal Járai and Christoforos Panagiotis.