# 2021-22

##### Organisers

Leonardo T. Rolla and Vedran Sohinger in collaboration with Sasha Sodin (Queen MaryLink opens in a new window), Jess Jay and Benjamin Lees (BristolLink opens in a new window).

##### Seminars in Term 1

**Oct 13** - Alexander Povolotsky, Joint Institute for Nuclear Research, Dubna, Russia.

* Title: *Generalized TASEP between KPZ and jamming regimes

** Abstract:** Totally Asymmetric Simple Exclusion Process (TASEP) with generalized update is an integrable stochastic model of interacting particles, which differs from the standard TASEP by the presence of an additional interaction controlling the degree of particle clustering. As the strength of the interaction varies, the system suffers the transition from the regime in which the fluctuations of the particle flow are described by standard random processes associated with the Kardar-Parisi-Zhang (KPZ) universality class, to the jamming regime, in which all particles stick to one cluster and move synchronously as the simple random walk. I will focus on the limiting laws of fluctuations of distances travelled by tagged particles both in the KPZ and in the transitional regime for two types of initial conditions. In particular new transitional processes interpolating between the known limiting cases will be discussed.

**Oct 20** - Eveliina Peltola, University of Bonn & Aalto University

* Title:* Large deviations of SLEs, real rational functions, and zeta-regularized determinants of Laplacians

* Abstract: *When studying large deviations (LDP) of Schramm-Loewner evolution (SLE) curves, we recently introduced a ''Loewner potential'' that describes the rate function for the LDP. This object turned out to have several intrinsic, and perhaps surprising, connections to various fields. For instance, it has a simple expression in terms of zeta regularized determinants of Laplace-Beltrami operators. On the other hand, minima of the Loewner potential solve a nonlinear first order PDE that arises in a semiclassical limit of certain correlation functions in conformal field theory,

arguably also related to isomonodromic systems. Finally, and perhaps most interestingly, the Loewner potential minimizers classify rational functions with real critical points, thereby providing a novel proof for

a version of the now well-known Shapiro-Shapiro conjecture in real enumerative geometry. This talk is based on joint work with Yilin Wang (MIT).

**Oct 27** - Matan Harel, Northeastern University

** Title:** Quantitative estimates on the effect of disorder on low-dimensional lattice models

** Abstract: **In their seminal work, Imry and Ma predicted that the addition of an arbitrarily small random external field to a low-dimensional statistical physics model causes the usual first-order phase transition to be `rounded-off.' This phenomenon was proven rigorously by Aizenman and Wehr in 1989 for a vastly general class of spin systems and random perturbations. Recently, the effect was quantified for the random-field Ising model, proving that it exhibits exponential decay of correlations at all temperatures. Unfortunately, the analysis relies on the monotonicity (FKG) properties which are not present in many other classical models of interest. This talk will present quantitative versions of the Aizenman-Wehr theorems for general spin systems with random disorder, including Potts, spin O(n), spin glasses. This is joint work with Paul Dario and Ron Peled.

**Nov 03** - Ariel Yadin, Ben-Gurion University

** Title:** Realizations of random walk entropy

** Abstract:** Random walk entropy is a numerical measure of the behaviour of the random walk at infinity. Out of all random walks on groups generated by d elements, the free group has the maximal entropy. One may ask naturally which intermediate values between 0 and the full entropy of the free group can be realized as entropies of random walks on groups. This question is still open. We analyze a related question, which is a "stochastic" version of the above open question. Here we are able to provide a full answer for quotients of the free group, and even a bit further than that. Generalizing results of Bowen (Inventiones 2014), we show that all possible "IRS entropy" values can be realized on the free group. These notions will be precisely explained during the talk. Based on joint works with Yair Hartman and Liran Ron-George.

**Nov 10** - Slim Kammoun, Université de Toulouse III (France).

** Title:** Longest Common Subsequence of Random Permutations

** Abstract: **Bukh and Zhou conjectured that the expectation of the length of the longest common subsequence (LCS) of two i.i.d random permutations of size $n$ is greater than $\sqrt{n}$.

This problem is related to the Ulam-Hammersley problem; Ulam conjectured that the expectation of the length of the longest increasing subsection (LIS) for a uniform permutation behaves like $c\sqrt{n}$. The conjecture was solved in 1977, but few results are known for non-uniform permutations. The LIS and LCS are closely related, and solving the conjecture of Bukh and Zhou is equivalent to minimize the expected value of LIS for random permutations that can be written as $\rho_n\circ\sigma_n^{-1}$ where $\sigma_n$ and $\rho_n$ are i.i.d. random permutations.

We recall the classical results for the uniform case as well as partial answers for the conjugation invariant case.

**Nov 17** - Fabio Toninelli, Technical University of Vienna.

** Title:** Diffusion in the curl of the 2-dimensional Gaussian Free Field

** Abstract:** I will discuss the large time behaviour of a Brownian diffusion in two dimensions, whose drift is divergence−free, ergodic and given by the curl of the 2−dimensional Gaussian Free Field. Together with G. Cannizzaro and L. Haundschmid, we prove the conjecture by B. Toth and B. Valko that the mean square displacement is of order t√log t. The same type of superdiffusive behaviour has been predicted to occur for a wide variety of (self)−interacting diffusions in dimension d = 2: the diffusion of a tracer particle in a fluid, self−repelling polymers and random walks, Brownian particles in divergence−free random environments, and, more recently, the 2−dimensional critical Anisotropic KPZ equation. To the best of our authors’ knowledge, ours is the first instance in which √log t superdiffusion is rigorously established in this universality class.

**Nov 24** - Sarah Penington, University of Bath.

** Title:** Genealogy of the N-particle branching random walk with polynomial tails

** Abstract:** The N-particle branching random walk is a discrete time branching particle system with selection consisting of N particles located on the real line. At every time step, each particle is replaced by two offspring, and each offspring particle makes a jump from its parent's location, independently from the other jumps, according to a given jump distribution. Then only the N rightmost particles survive; the other particles are removed from the system to keep the population size constant. I will discuss recent results and open conjectures about the long-term behaviour of this particle system when N, the number of particles, is large. In the case where the jump distribution has regularly varying tails, building on earlier work of J. Bérard and P. Maillard, we prove that at a typical large time the genealogy is given by a star-shaped coalescent, and that almost the whole population is near the leftmost particle on the relevant space scale. Based on joint work with Matt Roberts and Zsófia Talyigás.

**Dec 01** - Ofer Zeitouni, Weizmann Institute of Science & NYU

** Title:** High moments of partition function for 2D polymers in the weak disorder regime.

** Abstract:** Consider

Caravena, Sun and Zygouras showed that for all

Lygkonis and Zygouras have recently shown that for

*There exists *

With the same method, we also prove that the estimate [above] holds for all

**Dec 08** - Cyril Labbé, Paris Dauphine University.

** Title:** Mixing points on an interval

** Abstract: **Consider N points on the unit interval. Resample each point at rateone uniformly in between its two nearest neighbours. The question is: how much time is needed to reach equilibrium (the uniform measureonthe simplex) starting from the worst initial condition ? I will present results in collaboration with Pietro Caputoand Hubert Lacoin where we identified the asymptotic of themixingtimesand showed a cutoff phenomenon for the distance to equilibrium.

##### Seminars in Term 2

**Jan 12** - Zied Ammari, Université Rennes 1.

** Title: **On well-posedness for the Gross-Pitaevskii and Hartree Hierarchy Equations

** Abstract: **Gross-Pitaevskii and Hartree hierarchies are infinite systems of coupled PDEs related to mean field theory of Bose gases. Due to their physical and mathematical relevance, the issues of well-posedness and uniqueness for these hierarchies have recently been studied thoroughly using specific nonlinear and combinatorial techniques. In this talk I will introduce a new approach based on a duality between hierarchies and Liouville equations. Several new results are obtained as an outcome of this approach.

**Jan 19** - Shahar Mendelson, University of Warwick.

** Title:** Approximating Lₚ balls via sampling

** Abstract: **Let X be a centred random vector in Rⁿ. The Lₚ norms that X endows on Rⁿ are defined by ‖v‖_L__ₚ= (E|<X,v>|ᵖ)¹/ᵖ. The goal is to approximate those Lₚ norms, and the given data consists of N independent sample points X₁,...,X_N distributed as X. More accurately, one would like to construct data−dependent functionals ϕₚ,ε which satisfy with (very) high probability, that for every v in Rⁿ, (1−ε) ϕₚ,ε ≤ E|<X,v>|ᵖ ≤ (1+ε) ϕₚ,ε. I will show that the functionals \frac{1}{N}∑ⱼ∈J |<Xⱼ,v>|ᵖ are a good choice, where the set of indices J is obtained from {1,...,N} by removing the cε²N largest values of |<Xⱼ,v>|. Under mild assumptions on X, only N=(cᵖ)ε⁻²n measurements are required, and the probability that the functional performs well is at least 1−2\exp(−cε² N).

**Jan 26** - Benjamin Lees, University of Bristol

** Title: **The random path representation of classical spin systems

** Abstract: **Many models in statistical mechanics can be written in terms of a collection of (random) geometric objects. This may consist of a "soup" of random walks/loops, random currents, or random transpositions. I will introduce a model of random paths that includes several interesting models when its parameters are chosen appropriately. One such example is the spin O(N) model. Unlike similar representations of this model, the random path model is defined in terms of a product of local terms which allows many nice techniques, such as reflection positivity, to be used. I will show how to use this model to prove exponential decay of correlations for spin O(N) with an external field. The idea of the proof is very simple and applies much more generally than the alternative Lee-Yang method.

**Feb 2 **- Vedran Sohinger, University of Warwick

** Title:** Interacting loop ensembles and Bose gases

** Abstract: **We study interacting quantum Bose gases in thermal equilibrium on a lattice. In this framework, we establish convergence of the grand-canonical Gibbs states to their mean-field (classical field) and large-mass (classical particle) limit. Our analysis is based on representations in terms of ensembles of interacting random loops, namely the Ginibre loop ensemble for quantum Bose gases and the Symanzik loop ensemble for classical scalar field theories. For small enough interactions, we obtain corresponding results in the infinite volume limit by means of cluster expansions. This is joint work with Juerg Froehlich, Antti Knowles, and Benjamin Schlein.

**Feb 9 **- Oriane Blondel, CNRS-Lyon

** Title: **Kinetically constrained models out of equilibrium

*Kinetically constrained models are interacting particle systems on Z^d, in which particles can appear/disappear only if a given local constraint is satisfied. This condition complexifies significantly the dynamics. In particular, it deprives the system of monotonicity properties, which leaves us with few tools to study the dynamics when it is initially not at equilibrium. I will review the results and techniques we have in this direction.*

**Abstract:****Feb 16** - Pietro Caputo, Roma Tre University

*Rapid mixing of Gibbs samplers: Coupling, Spectral Independence, and Entropy factorizations.*

**Title:** ** Abstract: **We discuss some recent developments in the analysis of convergence to stationarity for the Gibbs sampler of general spin systems on arbitrary graphs. These are based on two recently introduced concepts: Spectral Independence and Block Factorization of Entropy. We show that if a system is spectrally independent then its entropy functional satisfies a general block factorization, which in turn implies a modified log-Sobolev inequality and a tight control of the mixing time for the Glauber dynamics as well as for any other heat bath block dynamics. Moreover, we show that the existence of a contractive coupling for a local Markov chain implies that the system is spectrally independent. As a corollary, we obtain new optimal bounds on the mixing time of a large class of sampling algorithms for the ferromagnetic Ising/Potts models in the so-called tree-uniqueness regime, including non-local Markov chains such as the Swendsen-Wang dynamics. The methods also apply to spin systems with hard constraints such as q-colorings of a graph and the hard-core gas. Based on some recent joint works with Antonio Blanca, Zongchen Chen, Daniel Parisi, Alistair Sinclair, Daniel Stefankovic, and Eric Vigoda.

**Feb 23** - Peter Nandori, Yeshiva University (New York)

** Title: **Flexibility of the central limit theorem in smooth dynamical systems

** Abstract: **We say that a diffeomorphism that preserves a smooth probability

**measure satisfies the CLT if the ergodic sums of all sufficiently smooth functions converge to a Gaussian law. Many diffeomorphisms are known to satisfy the CLT under the usual scaling sqrt(n). Most of these examples also share many other chaotic properties, such as ergodicity, mixing, positive entropy, K property, Bernoulli property. In this talk, I will present new examples of diffeomorphisms that satisfy the CLT but in a more exotic way. For example, the scaling may be regularly varying with index 1, or several ergodic properties from the above list may fail. The main idea of the construction goes back to random walks in random sceneries. This is a joint work with D. Dolgopyat, C. Dong and A. Kanigowski.**

**Mar 2** - Frank den Hollander, Leiden University

** Title: **Spatial populations with seed-bank

** Abstract: **In this lecture we consider a spatial population with seed-bank. Individuals carry one of two types, live in colonies labelled by a countable Abelian group playing the role of geographic space, and are subject to resampling and mi- gration as long as they are active. Each colony has a seed-bank into which individuals can retreat to become dormant, suspending their resampling and migration until they become active again. Our goal is to understand in what way the seed-bank enhances genetic diversity.

When individuals become dormant they adopt a random colour that deter- mines their wake-up time. The system of continuum stochastic differential equations describing the population in the large-colony-size limit has a unique strong solution that converges to an equilibrium parametrised by the initial type densities. This equilibrium exhibits a dichotomy between two phases: coexistence (= locally multi-type equilibrium) versus clustering (= locally mono-type equilibrium). We identify the parameter regimes for which these two phases occur. We also establish the finite-systems scheme, i.e., identify how a finite truncation of the system (both in the geographic space and in the seed-bank) behaves as both the time and the truncation level tend to infinity, properly tuned together.

This is joint work with Andreas Greven (Erlangen) and Margriet Oomen (Leiden).

**Mar 9** - Marta Sanz-Solé, University of Barcelona

** Title: **Global solutions to stochastic wave equations with superlinear coefficients.

** Abstract: **We consider the nonlinear stochastic wave equation on ℝᵈ, d=1,2,3,

∂_t² u(t,x) − Δₓ u(t,x) = b(u(t,x)) + σ(u(t,x)) ∂_tW(t,x), (t,x) ∈ (0,T] × ℝᵈ,

with given initial conditions. The process ∂_tW is a space−time white noise if d=1, while for d=2,3 it is white in time and coloured in space. For coefficients b and σ satisfying |b(z)| ≤ b₁ + b₂ |z| (ln |z|)ᵟ¹, |σ(z)| ≤ σ₁ + σ₂ |z| (ln |z|)ᵟ², when |z|→∞, we find conditions on the superlinear exponents and also on the noise when d=2,3, ensuring global existence (and uniqueness) of a random field solution. Examples of relevant noises are also provided. Recent results by M. Foondun and E. Nualart (2021) provide some partial information on the critical values of the growth exponents. The proof relies on sharp moment estimates of the solution, and of increments in time and space of them, of a sequence of stochastic wave equations closely related to the above equation, with globally Lipschitz continuous coefficients. The research is joint work with A. Millet. We were motivated by a paper from R. Dalang, D. Khoshnevisan and T. Zhang (2019) where a similar question is addressed for a nonlinear stochastic heat equation on [0,1].

**Mar 16** - Sabine Jansen, Ludwig Maximilian University Munich

** Title: **Cluster expansions: necessary and sufficient convergence conditions

** Abstract: **Correlation functions of Gibbs measures in general cannot be computed explicitly, but at low density / for weak interactions they are amenable to a power series expansion around the ideal gas (Poisson point process). The convergence of these series has been studied at least since the 60s, but sufficient convergence conditions keep being developed, raising the question if there is a theoretical limit to the improvements - an if and only if condition that cannot be surpassed.

The talk presents such an if and only if condition for repulsive interactions. It is based on Kirkwood-Salsburg equations and alternating sign properties and it generalizes a result by Bissacot, Fernández and Procacci. For possibly attractive potentials, the criterion yields new sufficient convergence conditions. Based on joint work with Leonid Kolesnikov (arXiv:2112.13134 [math.ph]).

##### Seminars in Term 3

**May 04 - **Daniel Valesin, University of Warwick

* Title:* The contact process on random d-regular graphs, static and dynamic

* Abstract:* We consider the contact process on random d−regular graphs, briefly presenting earlier work on the case where the graph is static, and focusing on more recent work where the graph evolves simultaneously with (and independently of) the contact process. In both cases, the analysis involves fixing the infection rate of the contact process and the graph parameters, taking the number of vertices N to infinity and studying the asymptotic behaviour of τ_N, the time it takes for the infection to disappear. Concerning the static graph, we prove that this behaviour undergoes a phase transition: there is a value λ_c such that τ_N is of order log(N) if λ < λ_c, whereas τ_N grows exponentially with N if λ > λ_c. The latter situation is called the metastable regime. Turning to the dynamic graph setting, our choice of graph evolution is a Markovian edge−switching mechanism, whose rate is chosen so that the evolving local landscape seen by a fixed vertex approaches a limiting dynamic graph process. We again show the existence of a metastable regime for the contact process on these graphs, and notably, we show that this regime occurs for values of λ that would be subcritical in the static graph. Joint work with Jean−Christophe Mourrat (static graphs) and with Gabriel Baptista da Silva and Roberto Imbuzeiro Oliveira (dynamic graphs).

**May 11 -** Kevin Yang, Stanford University

* Title:* KPZ and Boltzmann-Gibbs Principle

* Abstract:* Fluctuation theory for hydrodynamic limits of interacting particle systems in (1+1)-dimensions is conjectured to generally be governed by linear and (KPZ) quadratic corrections. Boltzmann-Gibbs principles are supposed to provide a general method of rigorously establishing something of this type. Much is known for stationary systems with sufficiently explicit invariant measures, while little is known for non-stationary systems outside a class of “partially integrable” models such as ASEP. In this talk, we broadly discuss KPZ equation fluctuations for general non-stationary systems. We then specialize to a perturbation of ASEP, where the main problem is deriving the aforementioned linear-type corrections with a method that should extend quite generally.

**May 18 -** Franco Severo, ETH Zürich

* Title:* On the off-critical level sets of smooth Gaussian fields

* Abstract:* We consider the level sets of smooth Gaussian fields on $\mathbb{R}^d$ below a parameter $\ell\in \mathbb{R}$. As $\ell$ varies this defines a percolation model, whose critical point is denoted by $\ell_c$. In this talk we will discuss the behaviour of these level sets on the off-critical regime, i.e. for $\ell \neq \ell_c$. Our main result states that, for fields with positive and sufficiently fast decaying correlations, the connection probabilities decay exponentially for $\ell<\ell_c$ and percolation occurs in sufficiently thick 2D slabs for $\ell>\ell_c$. This result, often referred to as (subcritical and supercritical, respectively) sharpness of phase transition, is typically the starting point for the study of finer properties of the off-critical phases. The result follows from a global comparison with a truncated and discretised version of the model, which may be of independent interest. The proof of this comparison relies on an interpolation scheme that integrates out the long-range and infinitesimal correlations of the model while compensating them with a small change in the parameter $\ell$.

**May 25 -** Will FitzGerald, University of Manchester

* Title:* Fredholm Pfaffians in random matrices and interacting particle systems

* Abstract:* I will describe how probabilistic methods can be used to find asymptotics for Fredholm determinants and Fredholm Pfaffians. This has applications to the tail behaviour of the largest real eigenvalue for non-Hermitian random matrices, persistence probabilities for coalescing and annihilating Brownian motions and the probability that random polynomials have no real roots.

**June 01 -** Alexander Marynych, Taras Shevchenko National University of Kyiv

* Title:* Lah distribution: properties and applications

* Abstract:* Motivated by a problem arising in the analysis of convex hulls of high-dimensional random walks, we introduce a new class of discrete probability distributions, which we call Lah distributions, and which involve Stirling numbers of both kinds. We provide a combinatorial interpretation of the Lah distributions in terms of random compositions and records and prove various limit theorems for them. This talk is based on a recent joint work with Zakhar Kabluchko (Münster, Germany).

**June 08 -** Pierre Tarrès, NYU-Shanghai

* Title:* The *-Edge Reinforced random walk, bayesian statistics and statistical physics

* Abstract: *We will introduce recent non-reversible generalizations of the Edge-Reinforced Random Walk and its motivation in Bayesian statistics for variable order Markov Chains. The process is again partially exchangeable in the sense of Diaconis and Freedman (1982), and its mixing measure can be explicitly computed. It can also be associated to a continuous process called the *-Vertex Reinforced Random Walk, which itself is in general not exchangeable. We will also discuss some properties of that process.Based on joint work with S. Bacallado and C. Sabot.

**June 15 -** Makiko Sasada, University of Tokyo

**Title:**** **KdV- and Toda-type discrete locally-defined dynamics and generalized Pitman’s transform

* Abstract:* The Korteweg-de Vries equation (KdV equation) and the Toda lattice are typical and well-known classical integrable systems. For the KdV equation, the (almost-sure) well-posedness of a solution starting from a general ergodic random field on the line is still an open problem, though the invariance, as well as the well-posedness of a solution, of the white noise was proved recently by Killip, Murphy and Visan recently. In this talk, I will consider discretized versions of KdV equation and Toda lattice on the infinite one-dimensional lattice. These systems are understood as "deterministic vertex model”, which are discretely indexed in space and time, and their deterministic dynamics is defined locally via lattice equations. They have another formulation via the generalized Pitman’s transform, which is a new and crucial observation for our result. We show that there exists a unique solution to the initial value problem when the given data lies within a certain class, which includes the support of many shift ergodic measures. Also, a detailed balance criterion is presented that, amongst the measures that describe spatially independent and identically/alternately distributed configurations, characterizes those that are temporally invariant in distribution. This talk is based on a joint work with David Croydon and Satoshi Tsujimoto.

**June 22 -** Alice Guionnet, ENS-Lyon

* Title:* Large deviations for the largest eigenvalue of random matrices

* Abstract:* The understanding of the typical behaviour as well as the fluctuations of the spectrum has made tremendous progresses during the last thirty years. Yet, the study of large deviations has been so far restricted to integrable cases where the joint law of the eigenvalues is explicit or to ''heavy tails'' cases where large deviations are created by large entries. In this talk I will discuss the large deviations of the largest eigenvalue in the case of sub-Gaussian entries. This talk is based on joint works with F. Augeri, N. Cook, R. Ducatez and J. Husson..

**June 29 -** Anna Ben-Hamou, Sorbonne University Pierre and Marie Curie

* Title:* Cutoff for permuted Markov chains

* Abstract:* For a given finite Markov chain with uniform stationary distribution, and a given permutation on the state-space, we consider the Markov chain which alternates between random jumps according to the initial chain, and deterministic jumps according to the permutation. In this framework, Chatterjee and Diaconis (2020) showed that when the permutation satisfies some expansion condition with respect to the chain, then the mixing time is logarithmic, and that this expansion condition is satisfied by almost all permutations. We will see that the mixing time can even be characterized much more precisely: for almost all permutations, the permuted chain has cutoff, at a time which only depends on the entropic rate of the initial chain.