Shangjie Yang

UFF

Local: Auditório do Edifício Antonio Galves (NUMEC)

Horário: 14h00

Cutoff of the simple exclusion process with  inhomogeneous conductances

 Resumo:

In this talk, we study the mixing time of the simple exclusion process on a finite line segment with varying edge conductances. In this model, the contents of two adjacent sites are swapped at rates determined by these conductances. Under the assumption that the resistances (the inverse of the conductances) grow linearly, along with certain additional conditions, we prove that the process exhibits the cutoff phenomenon. Specifically, we demonstrate that at a critical time, the total variation distance to equilibrium drops abruptly from 1 to 0.

Guilherme Silva

ICMC-USP

Local: Auditório do Edifício Antonio Galves (NUMEC)

Horário: 14h00

Moderate deviations for random matrices and the stochastic six-vertex model

 Resumo:

The phenomenon of universality is central to random matrix theory and stochastic growth models. In one of its most celebrated forms, it establishes that the fluctuations of extremal eigenvalues in large Hermitian random matrices are described by the Tracy-Widom distribution, entirely irrespective of the original distribution of the matrix entries. Strikingly, across a broad family of 1+1-dimensional stochastic growth models – specifically those sharing the macroscopic symmetries of the Kardar-Parisi-Zhang equation – the long-time fluctuations of the growing interface are governed by this exact same Tracy-Widom distribution, linking the geometry of random growth to the spectrum of random matrices.

At the opposite end of the probability spectrum, large deviations in both stochastic growth and random matrices exhibit a very different behavior: they display universal scaling, but distinctly non-universal rate functions that depend on microscopic details.

A natural question then emerges: how do these systems transition from the universality of typical fluctuations to the non-universal regime of large deviations? In this talk, we will explore this crossover phenomenon, discussing recent moderate deviation results that (try to) bridge the gap between these two distinct regimes.

Luiz Renato Gonçalves Fontes

IME-USP

Local: Auditório do Edifício Antonio Galves (NUMEC)

Horário: 14h00

Random walk in an independent birth-and-death dynamical environment

 Resumo:

We study a continuous time mean zero random walk in Zd in a dynamical random environment given by independent birth-and-death processes (possibly spatially inhomogeneous). When at site x at time t, the walker jumps at a rate given by a suitable (decreasing) function phi of the environment at that time and location. The jump chain The main interest is for the case where phi vanishes at infinity (possibly at infinite speed), suggesting the possibility of subdiffusive behavior. Two main results will be presented: one for the ‘rapidly ergodic’ case, meaning that the bd chain is ergodic and roughly its equilibrium distribution has a finite second moment, where a CLT for the walker position holds. And we get subdiffusive behavior for null recurrent (and transient, with caveats) spatially homogeneous bd chains at high dimensions and phi decreasing not too fast. (Joint with Maicon Pinheiro and Pablo Gomes.)

José Hermenegildo

IME-USP

Local: Auditório do Edifício Antonio Galves (NUMEC)

Horário: 14h00

The Frog Model on ℤ with a Random Survival Parameter 

 Resumo:

In the frog model on ℤ, only particles at the origin are initially active. They perform random walks, awaken sleeping particles at visited sites, and die after finite lifetimes. Each particle receives an independently sampled survival parameter p between zero and one. We study when successive activations can keep the system alive indefinitely.

Starting from the geometric-lifetime model of de Carvalho and Machado with Beta-distributed parameters, we consider general parameter densities behaving near one as (1 − p)^(β − 1) times a slowly varying factor, where β > 0. Assuming independent and identically distributed initial particle counts with positive finite mean, the critical value of β is 1/2 for simple symmetric random walks. We then replace geometric lifetimes with discrete Weibull lifetimes: conditional on p, the probability of completing at least k steps is p^(k^γ), where γ > 0 is the shape parameter. The critical value becomes 1/(2γ) for simple symmetric random walks and 1/γ for biased simple random walks. Nonzero drift changes the time needed to reach distant sites from a quadratic to a linear scale.

Finally, we consider geometric lifetimes with symmetric long jumps, each activating all sites between its endpoints. Finite-variance jumps retain the critical value 1/2. When the probability that a jump has length greater than n is asymptotically proportional to n^(−α), where 1 < α < 2, the critical value becomes 1/α. In these regimes, β below the corresponding critical value gives survival with positive probability, whereas β above it gives almost-sure extinction.

The talk is based on joint work with Gustavo Oshiro de Carvalho and Fábio Prates Machado.

Rodrigo Lambert

UFMG

Local: Auditório do Edifício Antonio Galves (NUMEC)

Horário: 14h00

Asymptotics for an urn model with randomly switching reinforcement 

 Resumo:We introduce and analyze a novel two-player urn model where the reinforcement rule switches randomly between a generalized Pólya scheme and an i.i.d. color-selection mechanism, according to the outcome of a coin toss at each step. This stochastic interference acts as a memory lapse on the process. We establish a strong law of large numbers for the color proportions, functional central limit theorems in the small and critical regimes, and Gaussian fluctuations with a law of the iterated logarithm in the large-urn regime. (Joint work with Manuel González-Navarrete, U. Bío-Bío, Chile.) 

Tertuliano Franco

UFBA

Local: Auditório do Edifício Antonio Galves (NUMEC)

Horário: 14h00

Heat Equation with Wentzel boundary conditions as a limit of random particle systems

 Resumo: In this talk we consider two interacting particle systems in the finite box with N sites, namely, independent random walks and symmetric exclusion processes. Both systems are in contact with a finite reservoir, where the exit rate is proportional to the inverse of $N^\theta$, where  $\theta$ is nonnegative. We prove the hydrodynamic limit for these models, which are given by the heat equation with Wentzell boundary conditions at the boundary at the critical parameter , and exhibit a dynamic phase transition. Moreover, the Wentzell boundary condition is non-linear in the exclusion setting at the critical parameter $\theta=1$. Joint work with Patrícia Gonçalves and Matheus Franco.

Jhon Astoquillca

IME-USP

Local: Auditório do Edifício Antonio Galves (NUMEC)

Horário: 14h00

Collisions of random walks in unimodular random graphs

 Resumo: We look at collisions of two independent random walks on graphs and how they relate to recurrence and transience. We start by recalling what happens in the bounded-degree setting and why things become more subtle when degrees are unbounded. This leads naturally to unimodularity and the Mass-Transport Principle. We then prove a general criterion for having only finitely many collisions and use it in one concrete example: the supercritical Gilbert random geometric graph. In that setting, we recover the expected dimension-dependent picture: infinitely many collisions in dimension two and only finitely many in dimensions three and higher.