### Session S06 - Interacting Stochastic Systems

## Talks

Tuesday, July 13, 12:00 ~ 12:35 UTC-3

## The contact process on random hyperbolic graphs

### Daniel Valesin

#### University of Groningen, The Netherlands - This email address is being protected from spambots. You need JavaScript enabled to view it.

We consider the contact process on a random graph embedded in hyperbolic space (introduced by Krioukov, Papadopoulos, Kitsak, Vahdat and Boguñá in 2010), in the regime where the degree distribution obeys a power law with finite mean and infinite second moment. We show that the process exhibits metastable behavior (prolonged persistence of the infection) regardless of the value of the infection rate $\lambda$. We also find that the exponent of the metastable infection density when $\lambda$ is close to zero. This coincides with the the corresponding quantity for the contact process on another random graph (namely, the configuration model with power law degree distribution), suggesting universality of this exponent.

Joint work with Amitai Linker (University of Cologne, Germany), Dieter Mitsche (University of Lyon, France) and Bruno Schapira (Aix-Marseille Université, France).

Tuesday, July 13, 12:35 ~ 13:10 UTC-3

## The time constant of finitary random interlacements

### Sarai Hernandez-Torres

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

The finitary random interlacement $\text{FRI}(u, T)$ is a Poisson point process of geometrically killed random walks on $\mathbb{Z}^d$, with $d \geq 3$. The parameter $u$ modulates the intensity of the point process, while $T$ is the expected path length. Although the process lacks global monotonicity on $T$, $\text{FRI}(u, T)$ exhibits a phase transition. For $T > T^{*}(u)$, $\text{FRI}(u, T)$ defines a unique infinite connected subgraph of $\mathbb{Z}^d$ with a chemical distance. We focus on the asymptotic behavior of this chemical distance and—in particular—the time constant function. This function is a normalized limit of the chemical distance between the origin and a sequence of vertices growing in a fixed direction. In this sense, the time constant function defines an asymptotic norm. Our main result is on its continuity (as a function of the parameters of $\text{FRI}$).

Joint work with Eviatar B. Procaccia (Technion, Israel) and Ron Rosenthal (Technion, Israel).

Tuesday, July 13, 13:25 ~ 14:00 UTC-3

## From generalized Ray-Knight theorems to functional limit theorems for some models of self-interacting random walks on $\mathbb{Z}$

### Elena Kosygina

#### Baruch College and the CUNY Graduate Center, United States - This email address is being protected from spambots. You need JavaScript enabled to view it.

Generalized Ray-Knight theorems for local times proved to be a very useful tool for studying the long time behavior of several models of self-interacting random walks on $\mathbb{Z}$. Examples include some classes of reinforced random walks, excited random walks, rotor walks with defects. An excited random walk on $\mathbb{Z}$ is a nearest-neighbor random walk whose probability $\omega_x(i)$ to jump to the right from site $x$ at time $n$ depends on $x$ and on the number of visits, $i$, to $x$ up to time $n$. The collection $\omega_x(i)$, where $x$ is an integer and $i$ is a positive integer, is sometimes called the ``cookie environment'' due to the following informal interpretation. On each site of $\mathbb{Z}$ there is an infinite stack of "cookies". Upon each visit to a site the walker eats a cookie from the bottom of the stack at that site and chooses the probability to jump to the right according to the ``flavor'' of the cookie eaten. We assume that the cookie stacks are i.i.d. and that the cookie ``flavors'' at each stack, $\omega_x(i),\, i\in\mathbb{N}$, follow a finite state Markov chain in $i$. Thus, the environment at each site is dynamic: it evolves according to the local time of the walk at each site rather than according to the random walk time. This talk will give an overview of some of the models of self-interacting random walks and then will discuss functional limit theorems for excited random walks with Markovian cookie stacks in the recurrent regime.

Joint work with Thomas Mountford (École polytechnique fédérale de Lausanne, Switzerland) and Jonathon Peterson (Purdue University, USA).

Tuesday, July 13, 14:00 ~ 14:35 UTC-3

## Random walk in a field of weakly killing exclusion particles

### Dirk Erhard

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

Consider a simple random walk, and independently of it the simple symmetric exclusion process on $\mathbb{Z}^d$. The random walk gets killed at rate epsilon when it shares a site with an exclusion particle. Using the relation to the parabolic Anderson model and the theory of regularity structures, I will present exact asymptotics for the survival probability of the random walk as epsilon tends to zero in dimension $d=3$. To establish these, precise bounds on the joint cumulants of the exclusion process are needed, which hold as soon as $d\geq 3$. I will also discuss what is missing to establish the corresponding result in $d=2$.

Joint work with Martin Hairer (Imperial College London).

Tuesday, July 13, 14:50 ~ 15:25 UTC-3

## Stochastic recursions on directed random graphs

### Mariana Olvera-Cravioto

#### University of North Carolina at Chapel Hill, United States - This email address is being protected from spambots. You need JavaScript enabled to view it.

We study a family of Markov processes on directed graphs where the values at each vertex are influenced by the values of its inbound neighbors and by independent fluctuations either on the vertices themselves or on the edges connecting them to their inbound neighbors. Typical examples include PageRank, the generalized deGroot model, and other information propagation processes on directed graphs. Assuming a stationary distribution exists for this Markov chain, our goal is to characterize the marginal distribution of a uniformly chosen vertex in the graph. In order to obtain a meaningful characterization, we assume that the underlying graph converges in the local weak sense to a marked Galton-Watson process, e.g., a directed configuration model or any rank-1 inhomogeneous random digraph. We then prove that the stationary distribution we study on the graph converges in a Wasserstein metric to a distribution characterized through a branching distributional fixed-point equation and its endogenous solution.

Joint work with Tzu-Chi Lin (University of North Carolina at Chapel Hill) and Nicolás Fraiman (University of North Carolina at Chapel Hill).

Tuesday, July 13, 15:25 ~ 16:00 UTC-3

## Mean Field behavior in Coalescing Random Walk

### Jonathan Hermon

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

We study Coalescing Random Walks on general graphs. We determine the asymptotic of the probability that the origin is occupied at time t for transient vertex-transitive graphs. Previously this probability was only known for $\mathbb{Z}^d$. We prove analogous results for finite graphs (vertex transitive or the configuration model with minimal degree 3) under certain finitary notions of transience. In particular, it follows that the number of remaining particles evolves like Kingman's coalescence up to a scaling of the time by a constant with a certain probabilistic interpretation.

Joint work with Shuangping Li,, Dong Yao, and Lingfu Zhang.

Wednesday, July 14, 12:00 ~ 12:35 UTC-3

## Random interfaces beyond $\mathbb{Z}^d$

### Alessandra Cipriani

#### TU Delft, The Netherlands - This email address is being protected from spambots. You need JavaScript enabled to view it.

The discrete membrane model (MM) is a random interface which is sampled from a Gaussian distribution indexed over the square lattice $\mathbb{Z}^d$. It can be described as a Gaussian perturbation of biharmonic functions. It is a close relative of the discrete Gaussian free field (DGFF), which is also a Gaussian perturbation, but of harmonic functions. Working with the two models presents some key differences. In particular, a lot of tools (electrical networks, random walk representation of the covariance) are available for the DGFF and lack in the MM. In this talk we will investigate a random walk representation for the covariances of the MM to study the model beyond the square lattice $\mathbb{Z}^d$.

Joint work with Biltu Dan (IISc Bangalore, India), Rajat Subhra Hazra (University of Leiden, The Netherlands) and Rounak Ray (TU Eindhoven, The Netherlands).

Wednesday, July 14, 12:35 ~ 13:10 UTC-3

## Random Schrödinger operators, integrated density of states and localization.

### Constanza Rojas-Molina

#### CY Cergy Paris Universite, France - This email address is being protected from spambots. You need JavaScript enabled to view it.

The theory of random Schrödinger operators originated in the late 70s to give a rigorous drescription to the absence of electron propagation in materials with impurities, a phenomenon known as Anderson localization. Since then, this theory has been developed and refined to be applied to different settings and iffernt types of disorder. Despite the fact that Anderson localization as consequence of the disorder present in the medium is well understood by known, some of the original questions remain open.

In this talk, after a brief introduction to the subject of random Schrödinger operators, we will review some recent results and discuss how this theory finds new applications today, motivated by connections to random walks in different settings, as for example, random walks with long jumps or quantum random walks.

Wednesday, July 14, 13:25 ~ 14:00 UTC-3

## Hydrodynamic large deviations of TASEP

### Li-Cheng Tsai

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

We consider the large deviations from the hydrodynamic limit of the Totally Asymmetric Simple Exclusion Process (TASEP), which is related to the entropy production in the inviscid Burgers equation. Here we prove the full large deviation principle. Our method relies on the explicit formula of Matetski, Quastel, and Remenik (2016) for the transition probabilities of the TASEP.

Joint work with Jeremy Quastel (University of Toronto, Canada).

Wednesday, July 14, 14:00 ~ 14:35 UTC-3

## Hydrodynamics of Porous Medium with Slow Boundary

### Adriana Neumann

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

We analyze the hydrodynamic behavior of the porous medium model (PMM) in a discrete uni-dimensional lattice. Our strategy relies on the entropy method of Guo, Papanicolau and Varadhan. However, this method cannot be straightforwardly applied, since there are configurations that do not evolve according to the dynamics (blocked configurations). To avoid this problem, we slightly perturbed the dynamics in such a way that the macroscopic behavior of the system keeps following the porous medium equation (PME), but with boundary conditions that depend on the boundary strength. These boundary conditions are Robin, Neumann, and Dirichlet. We also prove the convergence of the weak solution of the porous medium equation with a type of Robin boundary conditions, by tuning a parameter either to zero or to infinity. The convergence is in the strong sense, concerning the $L^2-$norm, and the limiting function solves the same equation with Neumann (resp. Dirichlet) boundary conditions when the parameter is taken to zero (resp. infinity).

Joint work with Patrícia Gonçalves (Instituto Superior Técnico, Portugal), Renato De Paula (Instituto Superior Técnico, Portugal) and Leonardo Bonorino (Federal University of Rio Grande do Sul).

Wednesday, July 14, 14:50 ~ 15:25 UTC-3

## Scaling limit for heavy-tailed ballistic deposition with $p$-sticking

### Santiago Saglietti

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

Ballistic deposition is a model for interface growth in which unit blocks fall vertically at random over the different sites of $\mathbb{Z}$ and stick to the interface at the first point of contact, causing it to grow. We consider a variant of this model in which the blocks have random heights, which are i.i.d. with some common distribution having a heavy right tail, and each block can only stick to the interface at the first point of contact with probability $p$ (otherwise, it continues to fall vertically until it lands on some previously deposited block). We study scaling limits of the resulting interface for the different values of $p$ and show that there is a phase transition as $p$ goes from $1$ to $0$.

Joint work with Francis Comets (Université de Paris, France) and Joseba Dalmau (NYU-Shanghai, China).

Wednesday, July 14, 15:25 ~ 16:00 UTC-3

## The directed landscape

### Duncan Dauvergne

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

The directed landscape is a random directed metric on the spacetime plane that arises as the scaling limit of integrable models of last passage percolation. It is expected to be the universal scaling limit for all models in the KPZ universality class for random growth. In this talk, I will describe its construction in terms of the Airy line ensemble via an isometric property of the Robinson-Schensted-Knuth correspondence.

Joint work with Janosch Ortmann (Université du Québec à Montréal) and Bálint Virág (University of Toronto).