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: 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.
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: Random walk on the simple symmetric exclusion process
Palestrante: Daniel Kious
Data: : 31/03/2021
Horario: 13:00h
Local: Transmissão online
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Resumo: In a joint work with Marcelo R. Hilário and Augusto Teixeira, we investigate the long-term behavior of a random walker evolving on top of the simple symmetric exclusion process (SSEP) at equilibrium. At each jump, the random walker is subject to a drift that depends on whether it is sitting on top of a particle or a hole. The asymptotic behavior is expected to depend on the density ρ in [0, 1] of the underlying SSEP. Our first result is a law of large numbers (LLN) for the random walker for all densities ρ except for at most two values ρ− and ρ+ in [0, 1], where the speed (as a function fo the density) possibly jumps from, or to, 0. Second, we prove that, for any density corresponding to a non-zero speed regime, the fluctuations are diffusive and a Central Limit Theorem holds. Our main results extend to environments given by a family of independent simple symmetric random walks in equilibrium
ID da reunião: 958 0581 3232
