**Título:** Random walks in cooling random environments: a journey on the one-dimensional lattice

**Palestrante:** Luca Avena (Leiden University)**Data:** 17/05/2021**Horário:** 15:00h**Local:** Transmissão online.

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**Resumo: **RWRE (Random Walk in Random Environment) is a classical model for particles moving in a non-homogeneous medium presenting impurities. It consists of a random walk on a graph with random transition probabilities determined by an underlying (static) field of random variables. The long term behavior of RWRE is well-understood, at least on the one-dimensional integer lattice, where trapping effects due to the spatial non-homogeneities lead to very different results than for a standard homogeneous random walk (e.g. non-local recurrence criterion, transient sub-ballistic regimes, anomalous diffusions, sub-exponential large deviations, aging).

In this talk we are interested in perturbing the underlying static random environment by repeatedly re-sampling it from a given law along a sequence of prescribed times, the so-called cooling sequence. This perturbation makes the environment dynamic and the resulting model, recently introduced in the literature, is referred to as RWCRE (Random Walk in Cooling Random Environment).

Depending on the choice of the cooling sequence, RWCRE may present strong homogenization as for a homogeneous Random Walk, or can lead to strong trapping effects as for RWRE. A surprisingly rich palette of possible limit scenarios have been explored in a series of recent papers and ongoing works on the one-dimensional lattice. I plan to give an account of these results and related techniques. Particular emphasis will be given to fluctuations and scaling limits where crossovers and mixtures of different laws emerge as a function of the structure of the cooling sequence.

Based on joint works with Yuki Chino (Taiwan), Conrado da Costa (Durham), Frank den Hollander (Leiden) and Jonathan Peterson (Purdue).

**Título:** Energy estimates and convergence of weak solutions of the porous medium equation

**Palestrante:** Adriana Neumann (UFRGS)**Data:** 10/05/2021**Horário:** 15:00h**Local:** Transmissão online.

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**Resumo: **In this talk, we study the convergence in the strong sense, with respect to the L^2-norm, of the weak solution of the porous medium equation (for short PME) with a type of Robin boundary conditions, by tuning a parameter either to zero or to infinity. The limiting function solves the same equation with Neumann (resp. Dirichlet) boundary conditions when the parameter is taken to zero (resp. infinity).

The keystone to prove this convergence result is a sufficiently strong energy estimate to the weak solution of the PME with a type of Robin boundary conditions.

Our approach to obtaining it is to consider an underlying microscopic dynamics, given by an interacting particle system, whose space-time evolution of the density of particles is ruled by the solution of those equations. We called this microscopic dynamic by the porous medium model (PMM) with slow boundary. The relation between the PMM and PME is stated in the **paper**, through the hydrodynamic limit for the PMM with slow boundary.

It is a joint work with Patrícia Gonçalves (IST - Lisbon) and Renato De Paula (IST - Lisbon), see more **HERE**.

All the talks are held in English.

The videos of the online seminars held in 2020 are available at **HERE**.

For the 2021 series, a few days after each meeting the video should be available **HERE**.

**Título**: The environment seen from a geodesic in last-passage percolation, and the TASEP seen from a second-class particle.

**Palestrante: **James Martin**Data: **28/04/2021

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**ID da reunião**: 958 0581 3232

**Resumo**: We study directed last-passage percolation in Z^2 with i.i.d. exponential weights. What does a geodesic path look like locally, and how do the weights on and nearby the geodesic behave? We show convergence of the distribution of the "environment" as seen from a typical point along the geodesic in a given direction, as its length goes to infinity. We describe the limiting distribution, and can calculate various quantities such as the density function of a typical weight, or the proportion of "corners" along the path. The analysis involves a link with the TASEP (totally asymmetric simple exclusion process) seen from an isolated second-class particle, and we obtain some new convergence and ergodicity results for that process. The talk is based on joint work with Allan Sly and Lingfu Zhang

**Título:** Random growth in 1+1 dimensions, KPZ and KP

**Palestrante:** Daniel Remenik**Data:** 28/04/2021**Horário:** 11:00h**Local:** Transmissão online.

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**Resumo: **The KPZ fixed point is a scaling invariant Markov process which arises as the universal scaling limit of all models in the KPZ universality class, a broad collection of models including one-dimensional random growth, directed polymers and particle systems. In particular, it contains all of the rich fluctuation behavior seen in the class, which for some initial data relates to distributions from random matrix theory (RMT). In this talk I'm going to introduce this process and explain how its finite-dimensional distributions are connected to a famous integrable dispersive PDE, the Kadomtsev-Petviashvili (KP) equation (and, for some special initial data, the simpler Korteweg-de Vries equation). I will also describe how this relation provides an explanation for the appearance in the KPZ universality class of the Tracy-Widom distributions from RMT.

**Título: **Soliton decomposition of the Box-Ball System

**Palestrante: **Leonardo T. Rolla (University of Warwick)**Data:** 26/04/2021**Horario: **15:00h**Local:** Transmissão online.

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**Resumo:** The Box-Ball System is a cellular automaton introduced by Takahashi and Satsuma as a discrete counterpart of the Korteweg & de Vries (KdV) differential equation. Both systems exhibit solitons, solitary waves that conserve shape and speed even after collision with other solitons. A configuration is a binary function on the integers representing boxes which may contain one ball or be empty. A carrier visits successively boxes from left to right, picking balls from occupied boxes and depositing one ball, if carried, at each visited empty box. Conservation of solitons suggests that this dynamics has many spatially-ergodic invariant measures besides the i.i.d. distribution. Building on Takahashi-Satsuma identification of solitons, we provide a soliton decomposition of the ball configurations and show that the dynamics reduces to a hierarchical translation of the components, finally obtaining an explicit recipe to construct a rich family of invariant measures. We also consider the a.s. asymptotic speed of solitons of each size. An extended version of this abstract, references, simulations, and the slides, all can be found **HERE**.

This is a joint work with Pablo A. Ferrari, Chi Nguyen, Minmin Wang.