seminar: Central limit theorem for a tagged particle in asymmetric simple exclusion
speaker: Ana Patricia Gonçalves (IMPA)abstract: Consider the Asymmetric Simple Exclusion in $\mathbb{Z}$. In this process, particles evolve on $\mathbb{Z}$ according to interacting random walks with an exclusion rule which prevents more than one particle per site. The dynamics can be informally described as follows. Fix a probability $p(.)$ on $\mathbb{Z}{d}$. Each particle, independently from the others, waits a mean one exponential time, at the end of which being at $x$ it jumps to $x+y$ at rate $p(y)$. If the site is occupied the jump is suppressed to respect the exclusion rule. In this case, $p(1)=p$ and $p(-1)=1-p$. For $0\leq{\alpha}\leq{1}$, denote by $\nu{\alpha}$ the Bernoulli product measure on $\{0,1\}{\mathbb{Z}}$ with density $\alpha$. It is known that $\nu{\alpha}$ is an invariant state for the exclusion process. Consider then, the asymmetric exclusion in equilibrium and in the hyperbolic time scale - $tN$. The results we are going to present can be summarized as follows: We prove a Central Limit theorem for the empirical measure associated to this process and for the current over a bond. Taking as initial measure the Bernoulli product measure with parameter $\alpha$, conditioned to have a particle at the origin, and denoting by $X{t}$, the position at time $t$ of the particle initially at the origin, we prove that $X{t}$ properly centered and rescaled, converges in law to the Brownian Motion. The proof relies on a simple relation between the position of the Tagged particle with the current and the density of particles. We also prove that in a higher time scale and for $\alpha=12$, the Central Limit Theorem for the empirical measure and for the Current still hold.
timetable:
Thu 8 Jun, 14:50 - 15:20, Sala Stemmi
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