### Session S04 - Random Walks and Related Topics

## Talks

Thursday, July 15, 11:00 ~ 11:30 UTC-3

## Genealogy and spatial distribution of the $N$-particle branching random walk with polynomial tails

### Sarah Penington

#### University of Bath, United Kingdom - This email address is being protected from spambots. You need JavaScript enabled to view it.

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 about the long-term behaviour of this particle system in the case where the jump distribution has regularly varying tails and the number of particles is large, building on earlier work of J. Bérard and P. Maillard. We prove that at a typical large time the genealogy of the population is given by a star-shaped coalescent, and that almost the whole population is near the leftmost particle on the relevant space scale.

Joint work with Matt Roberts (University of Bath) and Zsófia Talyigás (University of Bath).

Thursday, July 15, 11:40 ~ 12:10 UTC-3

## Branching Brownian motion with self-repulsion

### Anton Bovier

#### Universität Bonn, Germany - This email address is being protected from spambots. You need JavaScript enabled to view it.

We consider a model of branching Brownian motion with self repulsion. Self-repulsion is introduced via change of measure that penalises particles spending time in an $\e$-neighbourhood of each other. We derive a simplified version of the model where only branching events are penalised. This model is almost exactly solvable and we derive a precise description of the particle numbers and branching times. In the limit of weak penalty, an interesting universal time-inhomogeneous branching process emerges. The position of the maximum is governed by a F-KPP type reaction-diffusion equation with a time dependent reaction term.

Joint work with Lisa Hartung (Gutenberg-University Mainz, Germany).

Thursday, July 15, 12:20 ~ 12:50 UTC-3

## REFINED LARGE DEVIATION PRINCIPLE FOR BRANCHING BROWNIAN MOTION CONDITIONED TO HAVE A LOW MAXIMUM

### Lisa Hartung

#### JGU Mainz, Germany - This email address is being protected from spambots. You need JavaScript enabled to view it.

Conditioning a branching Brownian motion to have an atypically low maximum leads to a suppression of the branching mechanism. We consider a branching Brownian motion conditioned to have a maximum below $\sqrt{2}\alpha t$ with $\alpha<1$. We study the precise effects of an early/late first branching time and a low/high first branching location under this condition. We do so by imposing additional constraints on the first branching time and location. We obtain large deviation estimates, as well as the optimal first branching time and location given the additional constraints.

Joint work with Yanjia Bai (Bonn University, Germany).

Thursday, July 15, 13:00 UTC-3

## A limit law for the most favorite point of a simple random walk on a regular tree

### Oren Louidor

#### Technion - Israel's Institute of Technology, Israel - This email address is being protected from spambots. You need JavaScript enabled to view it.

We consider a continuous-time random walk on a regular tree of depth $n$ and study its most favorite point among the leaf vertices. We prove that, for the walk started from a leaf vertex and stopped upon hitting the root, under suitable scaling and centering, the maximal time spent at any leaf converges, as $n$ tends to infinity, to a randomly-shifted Gumbel law. The random shift is characterized using a derivative-martingale like object associated with the square-root local-time process on the tree.

Joint work with Marek Biskup (UCLA, United States).

Thursday, July 15, 13:40 UTC-3

## Exceptional points of random walks in planar domains

### Marek Biskup

#### UCLA, United States - This email address is being protected from spambots. You need JavaScript enabled to view it.

I will consider exceptional sets of associated with the local time of the simple random walk in finite subsets of the square lattice. These sets approximate a nice bounded continuum planar domain in the scaling limit; the walk moves as the ordinary simple random walk inside the domain and, whenever it exits, it returns via a uniformly-chosen boundary edge in the next step. For the walk run up to a positive multiple of the cover time, I will show that the sets of suitably defined thick and thin points as well as the set of avoided (a.k.a. late) points are asymptotically distributed according to versions of the Liouville Quantum Gravity in the underlying continuum domain. The conclusions are cleanest when the walk is parametrized by the local time spent at the “boundary vertex” with non-trivial corrections to the limit law arising in the conversion to the actual time.

Joint work with Yoshihiro Abe (Chiba University, Japan) and Sangchul Lee (UCLA, United States).

Thursday, July 15, 14:20 ~ 14:50 UTC-3

## Extremal distance and conformal radius of a $CLE(4)$ loop

### Avelio Sepúlveda

#### Universidad de Chile, Chile - This email address is being protected from spambots. You need JavaScript enabled to view it.

In this talk, we will discuss the geometry of the loop of a $CLE(4)$ surrounding the origin. In particular, we show how to compute the joint law between the conformal radius seen from the origin of the domain surrounded by the loop together with the extremal distance from this loop to the boundary. This joint law is related to certain random times of the Brownian motion, more precisely the first exit time of the set $[-2\pi, 2\pi]$ together with the last hit of $0$. We will explain where this relationship comes from together with a new characterization of this coupling.

Joint work with Juhan Aru (École polytechnique fédérale de Lausanne, Switzerland) and Titus Lupu (CNRS, France).

Thursday, July 15, 15:00 ~ 15:30 UTC-3

## Non-intersecting Brownian motions and random matrices

### Daniel Remenik

#### Universidad de Chile, Chile - This email address is being protected from spambots. You need JavaScript enabled to view it.

A well known result shows that the rescaled maximal height of a system of N non-intersecting Brownian bridges starting and ending at the origin converges, as N goes to infinity, to the Tracy-Widom GOE random variable from random matrix theory. In this talk I will discuss some recent extensions of this result, involving the distribution of the height of a fixed number of paths and of the maximal height on an interval, as well as the asymptotic distribution in the case when a few paths at the top start and end at arbitrary locations. I will describe results which connect these distributions to other random matrix ensembles as well as to KPZ fluctuations and certain PDEs.

Friday, July 16, 11:00 UTC-3

## Finding geodesics on graphs using reinforcement learning

### Daniel Kious

#### University of Bath, United Kingdom - This email address is being protected from spambots. You need JavaScript enabled to view it.

The premise of our talk will be the fact that ants are believed to be able to find shortest paths between their nest and the sources of food by successive random explorations, without any mean of communication other than the pheromones they leave behind them.

We will discuss a work in collaboration with Bruno Schapira and C\'ecile Mailler in which we introduce a general probabilistic model for this phenomenon, based on reinforcement-learning. We will present various variants of the model, with slightly different reinforcement mechanisms, and show that these small differences in the rules yield significantly different large-time behaviors. In the version called the loop-erased ant process, we are able to prove that the ants manage to find the shortest paths on all series-parallel graphs.

Joint work with Cecile Mailler (Bath, UK), Bruno Schapira (Marseille, France).

Friday, July 16, 11:40 ~ 12:10 UTC-3

## Critical exponents for a percolation model on transient graphs induced by random walks

### Alexander Drewitz

#### Universität zu Köln, Germany - This email address is being protected from spambots. You need JavaScript enabled to view it.

We consider the bond percolation problem on a transient weighted graph induced by the excursion sets of the Gaussian free field on the corresponding cable system. Owing to the continuity of this setup and the strong Markov property of the field on the one hand, and the links with potential theory for the associated diffusion on the other, we rigorously determine the behavior of various key quantities related to the (near-)critical regime for this model. In particular, our results apply in case the base graph is the three-dimensional cubic lattice. They unveil the associated critical exponents, which are explicit but not mean-field and consistent with predictions from scaling theory below the upper-critical dimension.

Joint work with Alexis Prévost (U Cambridge) and Pierre-François Rodriguez (Imperial College).

Friday, July 16, 12:20 ~ 12:50 UTC-3

## Random walks in dynamic random environments with non-uniform mixing

### Marcelo Hilario

#### Universidade Federal de Minas Gerais, Brazil - This email address is being protected from spambots. You need JavaScript enabled to view it.

We will discuss some recent progress on the limiting behavior of random walks in dynamic random environments. These are random walks whose transition kernel depends on a random environment that also evolves stochastically in time. We will mainly consider stationary one-dimensional random environments that exhibit bad mixing conditions, meaning that the mixing rates are either slow non-uniform over the initial configuration. We will show how renormalization techniques may be used to deal with this class of models in order to obtain results such as annealed law of large numbers and large deviation estimates. Under certain conditions, with the aid of a regeneration structure, we are also able to deduce annealed central limit theorems. Our techniques apply to environments given by interacting particle systems such as the exclusion process.

Joint work with Oriane Blondel,, Frank den Hollander (University of Leiden, The Netherlands), Daniel Kious (University of Bath, UK), Renato dos Santos (Federal University of Minas Gerais, Brazil), Vladas Sidoravicius and Augusto Teixeira (IMPA, Brazil).

Friday, July 16, 13:00 ~ 13:30 UTC-3

## Balanced excited random walk in dimension $d=2$

### Alejandro Ramírez

#### Pontificia Universidad Católica, Chile - This email address is being protected from spambots. You need JavaScript enabled to view it.

We obtain bounds on the growth as a function of time of the range of the balanced excited random walk in dimension $d=2$ introduced by Benjamini, Kozma and Shapira in 2010.

Joint work with Mark Holmes (University of Melbourne, Australia) and Omer Angel (University of British Columbia, Canada).

Friday, July 16, 13:40 ~ 14:10 UTC-3

## On renewal contact processes: some new results

### Maria Eulália Vares

#### Instituto de Matemática / Universidade Federal do Rio de Janeiro, Brasil - This email address is being protected from spambots. You need JavaScript enabled to view it.

This talk will be mostly based on [1]. We refine previous results concerning the renewal contact processes and significantly widen the family of interarrival times for which the critical value can be shown to be strictly positive. The result now holds for any spatial dimension $d\ge 1$ and the decreasing failure rate assumption present in [2] is removed, among other improvements. For heavy tailed interarrival times we provide some further description of the processes, including a complete convergence theorem and an examination of how, conditioned on survival, the process can be asymptotically predicted knowing the renewal processes.

[1] L.R. Fontes, T.S. Mountford, D. Ungaretti, M.E. Vares. Renewal contact processes: phase transition and survival (arXiv:2101.06207 [math.PR])

[2] L.R. Fontes, T.S. Mountford, M.E. Vares. Contact process under renewals II. Stoch. Proc. Appl., v. 130, p. 1103-1118, 2020.

Joint work with Luiz Renato Fontes (USP, Brazil), Thomas S. Mountford (EPFL, Switzerland), Daniel Ungaretti (USP, Brazil).

Friday, July 16, 14:20 ~ 14:50 UTC-3

## The parabolic Anderson model on a Galton-Watson tree

### Renato Soares dos Santos

#### UFMG, Brazil - This email address is being protected from spambots. You need JavaScript enabled to view it.

We consider the non-negative solution to the parabolic Anderson model (i.e., the heat equation with random multiplicative potential) on a supercritical Galton-Watson tree with bounded degrees. The potential is taken i.i.d. with double-exponential upper tails. We obtain second-order asymptotics of the rescaled total mass of the solution in terms of a variational formula. Analysis of the variational formula under further assumptions suggests mass concentration in subtrees with minimal degrees.

Joint work with Frank den Hollander (Leiden University) and Wolfgang König (TU Berlin, WIAS Berlin).

Friday, July 16, 15:00 ~ 15:30 UTC-3

## The Stochastic Heat Equation with Lévy white noise and its continuum polymer counterpart

### Hubert Lacoin

#### IMPA, Brazil - This email address is being protected from spambots. You need JavaScript enabled to view it.

The object of the talk is the following stochastic PDE \[\partial_t u = \Delta u + \xi \cdot u \] in $[0,\infty)×\mathbb R^d$. $\Delta$ denotes the usual Laplacian in $\mathbb R^d$ and $\xi$ is a space-time white noise. The equation above has been extensively studied in the case where $\xi$ is a Gaussian White noise. In that case, the equation is well defined only when $d=1$. In that case, a probability measure on space time trajectory associated to the solutions of the equation has been introduced as the continuum directed polymer model.

In our talk, we consider the case where $\xi$ is a Lévy white noise with no diffusive part. We identify necessary and sufficient conditions on $\xi$ for the existence of solution to the equation, and discuss its intermittency properties, which are intimately related to the localization properties of the associated continuum polymer.

Joint work with Quentin Berger (Université de Paris) and Carsten Chong (Columbia University).

Monday, July 19, 16:00 UTC-3

## Geometry of Gaussian multiplicative chaos in the Wiener space

### Chiranjib Mukherjee

#### Universität Münster, Germany - This email address is being protected from spambots. You need JavaScript enabled to view it.

We report on a recent joint work with Y. Bröker (Münster) where we develop an approach for investigating geometric properties of Gaussian multiplicative chaos (GMC) in an infinite dimensional set up. The base space is chosen to be the space of continuous functions endowed with Wiener measure, and the Gaussian field is a space-time Gaussian noise integrated against Brownian paths. In this set up, we show that in any dimension $d\geq 1$ and for any inverse temperature $\gamma>0$, the volume of a GMC ball, uniformly around all paths, decays exponentially with an explicit decay rate. For $d\geq 3$ and high temperature, the decay rate is just the principal eigenvalue of the Dirichlet Laplacian of the ball, which reflects a similar behavior of the free Brownian path. Incidentally, this is also the regime when our GMC attains very high values on all paths, making these points thick under the GMC measure. For any $d\in \mathbb N$, and small temperatures, the rate is given by an additional energy functional minimized over (probability measures on) a translation- invariant compactification constructed together with Varadhan. Quantifying exponential decay rates of GMC balls are natural infinite dimensional extensions of similar behavior captured by thescaling exponents of $2d$ Liouville quantum gravity, which has been studied intensively over the last few years.

Joint work with Yannic Bröker (University of Münster, Germany).

Monday, July 19, 16:40 ~ 17:10 UTC-3

## The shattering phase and metastability for spin glasses

### Gerard BEN AROUS

#### Courant Institute of Mathematical Sciences, New York University, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.

I will present a recent joint work with Aukosh Jagannath, where we explore the relationship between replica symmetry breaking, shattering, and metastability.

To this end, we study the static and dynamic behaviour of spherical pure p-spin glasses *above* the replica symmetry breaking temperature. In this regime, we find that there are at least two distinct temperatures related to non-trivial behaviour.

First we prove that there is a regime of temperatures in which the spherical p-spin model exhibits a shattering phase. We then find that metastable states exist up to an even higher temperature, as predicted by Barrat–Burioni–Mezard, which is expected to be higher than the phase boundary for the shattering phase.

We first develop a Thouless–Anderson–Palmer decomposition which builds on the recent work of Subag. We then present a series of questions and conjectures regarding the sharp phase boundaries for shattering and slow mixing.

Joint work with Aukosh Jagannath (University of Waterloo).

Monday, July 19, 17:20 ~ 17:50 UTC-3

## The cutoff phenomenon in total variation for nonlinear Langevin systems with stable type noise

### Juan Carlos Pardo

#### Centro de Investigación en Matemáticas, Mexico - This email address is being protected from spambots. You need JavaScript enabled to view it.

This paper provides an extended case study of the cutoff phenomenon for a prototypical class of nonlinear Langevin systems with a single stable state perturbed by an additive pure jump L\'evy noise of small amplitude, where the driving noise process is of layered stable type.

Under a drift coercivity condition the associated family of solutions turns out to be exponentially ergodic with equilibrium distribution in total variation distance which extends a result from Peng and Zhang (2018) to arbitrary moments. The main results establish the cutoff phenomenon with respect to the total variation, under a sufficient smoothing condition of Blumenthal-Getoor index larger than 3/2. That is to say, in this setting we identify a deterministic time scale which tends to infinity in the limit of small noise and a respective time window during which the total variation distance between the current state and its equilibrium essentially collapses as the noise amplitude tends to zero. In addition, we extend the dynamical characterization under which the latter phenomenon can be described by the convergence of such distance to a unique profile function first established in Barrera and Jara (2020) to the L\'evy case for nonlinear drift. This leads to sufficient conditions, which can be verified in examples, such as gradient systems subject to small symmetric alpha-stable noise for index larger than 3/2.

Joint work with Gerardo Barrera (University of Helsinki, Finland) and Michael Hoegele (Universidad de los Andes. Bogot\'a, Colombia).

Monday, July 19, 18:00 ~ 18:30 UTC-3

## From First Passage Percolation to Topology Learning: Theory and Methods.

### Pablo Groisman

#### Universidad de Buenos Aires, Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.

First Passage Percolation (FPP) has been studied for more than fifty years since the seminal work of Hammersley and Welsh in which they introduce it as a model of fluid flow through a porous (random) medium. Typically the model is known in advance and one wants to understand its behavior. Here we reverse the roles and use FPP to understand data. We know the behavior and want to understand the model (data). We will show that we can use FPP models to learn geometry, topology and chaotic dynamical systems.

More precisely, for N iid points in a manifold we introduce the Fermat distance which is nothing but the one introduced by Howard and Newman to consider Euclidean models of FPP and we study the set of points equipped with this distance as a metric space. We prove the convergence in the Gromov-Hausdorff sense to a metric space determined by the manifold and the density that produced the points. Then we use this result to show that we can use the Fermat distance to learn the topology of the manifold through persistent homology and to validate chaotic dynamical systems models .

Joint work with Facundo Sapienza (Berkley, USA), Matthieu Jonckheere (Universidad de Buenos Aires, Argentina), Ximena Fernández (Swansea University, UK), Eugenio Borghini (Universidad de Buenos Aires, Argentina) and Gabriel Mindlin (Universidad de Buenos Aires, Argentina).

Monday, July 19, 18:40 ~ 19:10 UTC-3

## Interacting particles diffusing in spatially heterogeneous system

### Lea Popovic

#### Concordia University, Montreal, Canada - This email address is being protected from spambots. You need JavaScript enabled to view it.

We propose a novel measure valued process which models the behaviour of interacting particles in terms of chemical reaction networks in spatially heterogeneous systems. It models reaction dynamics between different molecular species and continuous movement of molecules in space. Reactions rates at a spatial location are proportional to the mass of different species present locally and to a location specific chemical rate, which may be a function of the local or global species mass as well. We obtain asymptotic limits for the process, with appropriate rescaling depending on the abundance of different molecular types. When the mass of all species scales the same way we get a deterministic limit, whose long-term behaviour depends on the mobility of types and localization of reactions. When the mass of some species in the scaling limit is discrete while the mass of the others is continuous, we obtain a new type of spatial random evolution process. This process can be shown, in some situations, to correspond to a measure-valued piecewise deterministic Markov process in which the discrete mass of the process evolves stochastically, and the continuous mass evolves in a deterministic way between consecutive jump times of the discrete part.

Joint work with Amandine Véber (École Polytechnique/CNRS Paris).

Monday, July 19, 19:20 UTC-3

## Phase transition for percolation on randomly stretched lattices

### Augusto Teixeira

#### IMPA - Rio de Janeiro, Brazil - This email address is being protected from spambots. You need JavaScript enabled to view it.

In this talk we study the existence/absence of phase transitions for Bernoulli percolation on a class of random planar graphs. More precisely, the graphs we consider have vertex sets given by $\mathbb{Z}^2$ and we start by adding all horizontal edges connecting nearest neighbor vertices. This gives us a disconnected graph, composed of infinitely many copies of $\mathbb{Z}$, with the trivial behavior $p_c(\mathbb{Z}) = 1$. We now add to $G$ vertical lines of edges at $\{X_i\} \times \mathbb{Z}$, where the points $X_i$ are given by an i.i.d. integer-valued renewal process with inter arrivals distributed as $T$. This graph $G$ now looks like a randomly stretched version of the nearest neighbor graph on $\mathbb{Z}^2$. In this talk we show an interesting phenomenon relating the existence of phase transition for percolation on $G$ with the moments of the variable $T$. Namely, if $E(T^{1+\epsilon})$ is finite, then $G$ almost surely features a non-trivial phase transition. While if $E(T^{1-\epsilon})$ is infinite, then $p_c(G) = 1$.

Joint work with Marcelo Hilário (Universidade Federal de Minas Gerais, Brazil), Remy Sanchis (Universidade Federal de Minas Gerais, Brazil) and Marcos Sá (IMPA, Brazil).

## Posters

## Quantifying Central Limit Theorems for Heavy-Tailed Random Walks

### Medeiros Chiarini

#### IMPA, Brazil - This email address is being protected from spambots. You need JavaScript enabled to view it.

The central limit theorem (CLT) is one of the most important results in probability theory. Therefore, it is not surprising that its quantitative estimates are of interest. In this talk we discuss how to obtain such estimates when the underlying random variables do not fall in the classical CLT, but instead in the stable setting. Moreover, we will discuss how to obtain its respective potential kernel estimates. The discussion will be carried in dimension one and follows from an expansion of the law of the random walk in terms of a collection of stable processes.

Joint work with Milton Jara (IMPA, Brazil) and Wioletta Ruszel (Utrecht University, Netherlands).

## The Flip-Murder process exhibits a kind of first order phase transition

### Alex Ramos

#### Federal University of Pernambuco, Brazil - This email address is being protected from spambots. You need JavaScript enabled to view it.

On the theory of interacting particle systems, the supposition that the set of particles does not change in the process of interaction is lore crystallized. This supposition, which we call constant-length, is not the only possible one. A. Toom and V. Malyshev[1,2] have considered another approximation, which we call variable-length. Here, we shall examine a random process motivated by this new paradigm. The process we have studied here and called Flip-Murder, it has discrete-time and its states are bi-infinite sequences, whose particles take only two values, denoted here as minus and plus. Our operator is a composition of the following two operators. The first operator, called flip, turns every minus into plus with probability $\beta$ independently from what happens at other places. The second operator, called murder, acts in the following way: whenever a plus is a left neighbor of a minus, this plus disappears with probability $\alpha$ independently from what happens at other places. We prove that our process exhibits regions of ergodicity and non-ergodicity on the parameter space; a kind of first order phase transition and an invariant measure whose density of plus is less than one. Moreover, we have performed some numerical investigations.

[1] MALYSHEV, V. A. Quantum grammars. Journal of Mathematical Physics, v.41, n.7, p. 4508-4520, 2000.

[2] TOOM, A. Non-ergodicity in a 1-D particle process with variable length. Journal of Statistical Physics, v. 115, n. 3-4, p. 895-924, 2004.

Joint work with L. T. Costa(Rural Federal University of Pernambuco).