**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: **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.

**Título**: Condensation of SIP particles and sticky Brownian motio

**Palestrante**: Gioia Carinci, University of Modena and R. Emilia.**Data: **14/04/2021**Horario: **14:00h**Local**: Transmissão online.

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**ID da reunião**: 958 0581 3232

**Resumo**: The symmetric inclusion process (SIP) is a particle system with attractive interaction. We study its behavior in the condensation regime attained for large values of the attraction intensity. Using Mosco convergence of Dirichlet forms, we prove convergence to sticky Brownian motion for the distance of two SIP particles. We use this result to obtain, via duality, an explicit scaling for the variance of the density field in this regime, for the SIP initially started from a homogeneous product measure. This provides relevant new information on the coarsening dynamics of condensing particle systems on the infinite lattice. Joint work with M. Ayala and F. Redig.

**Título**: A tale of two balloons

**Palestrante**: Yinon Spinka**Data: **21/04/2021**Horario: **13:00h**Local**: Transmissão online.

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**ID da reunião**: 958 0581 3232

**Resumo**: From each point of a Poisson point process start growing a balloon at rate 1. When two balloons touch, they pop and disappear. Will balloons reach the origin infinitely often or not? We answer this question for various underlying spaces. En route we find a new(ish) 0-1 law, and generalize bounds on independent sets that are factors of IID on trees. Joint work with Omer Angel and Gourab Ray.

**Título**: Gravitational allocation of uniform points on the sphere

**Palestrante**: Yuval Peres (Kent State University)**Data**: 12/04/2021**Horário**: 15:00h**Local**: Transmissão online

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**Resumo**: Given uniform points on the surface of a two-dimensional sphere, how can we partition the sphere fairly among them? "Fairly" means that each region has the same area. It turns out that if the given points apply a two-dimensional gravity force to the rest of the sphere, then the basins of attraction for the resulting gradient flow yield such a partition-with exactly equal areas, no matter how the points are distributed. This is related to work of Nazarov-Sodin-Volberg on Gaussian analytic functions. (See the cover of the AMS Notices at http://www.ams.org/publications/journals/notices/201705/rnoti-cvr1.pdf.) Our main result is that this partition minimizes, up to a bounded factor, the average distance between points in the same cell. I will also present an application to almost optimal matching on the sphere, connecting to a classical result of Ajtai, Komlos and Tusnady (Combinatorica 1984).

Joint work with Nina Holden and Alex Zhai.