Point process convergence of extremes in $K$-symmetric exclusion

TL;DR

This paper studies the extreme particle behavior in $K$-symmetric exclusion processes, showing convergence to a Poisson process and extending known results for the case $K=1$ to larger $K$, with new techniques for controlling correlations.

Contribution

It establishes point process convergence of extremes in $K$-symmetric exclusion and extends superdiffusive scaling limits from $K=1$ to $K extgreater 1$, using more general methods.

Findings

Convergence of extremal particle point processes to Poisson random measures.

Asymptotic distributions for extreme particles and spacings.

Robustness of superdiffusive scaling limits for $K extgreater 1$.

Abstract

We consider the behavior of extremal particles in $K$-symmetric exclusion on $\mathbb{Z}$ when the process starts from certain infinite-particle step configurations where there are no particles to the right of a maximal one. In such a system, the occupancy of a site is limited to at most $K \geq 1$. Let $X^{(0)}_t\geq X^{(1)}_t\geq \cdots$ denote the order statistics of the particles in the system. We show that the point process $\sum_{m=0}^\infty \delta_{v_t(X_{t/K}^{(m)})}$ converges in distribution as $t \to \infty$ to a Poisson random measure on $\mathbb{R}$ with intensity proportional to $e^{-x}\,dx$, where $v_t(x) = (\sigma b_t)^{-1}x - a_t$, $a_t = \log(t/ (\sqrt{2\pi} \log t))$, $b_t = (t/\log t)^{1/2}$, and $\sigma$ is the standard deviation of the random walk jump probabilities. From this limit, we further deduce the asymptotic joint distributions for the extreme statistics and the spacings between them. Moreover, to probe effects of the number of particles on the behavior of the extremes, we consider an array of truncated step profiles supported on blocks of $L(t)$ sites at times $t\geq 0$. Letting $L(t) \to \infty$ with $t \to \infty$, we obtain Poisson random measure limits in different scaling regimes determined by $L(t)$. These results show robustness of both previously known and newly introduced superdiffusive scaling limits for the extremes in the symmetric exclusion process ($K=1$) by extending them to the larger class of $K\geq 2$ exclusion. Furthermore, proofs are more general than previously known techniques, relying on moment bounds and a semigroup monotonicity estimate to control particle correlations.