We investigate the long-term behavior of a random walker evolving on top of the simple symmetric exclusion process (SSEP) at equilibrium, in dimension one. 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, so that its asymptotic behavior is expected to depend on the density $\rho \in [0, 1]$ of the underlying SSEP. Our first result is a law of large numbers (LLN) for the random walker for all densities $\rho$ except for at most two values $\rho_-, \rho_+ \in [0, 1]$. The asymptotic speed we obtain in our LLN is a monotone function of $\rho$. Also, $\rho_-$ and $\rho_+$ are characterized as the two points at which the speed may jump to (or from) zero. Furthermore, for all the values of densities where the random walk experiences a non-zero speed, we can prove that it satisfies a functional central limit theorem (CLT). For the special case in which the density is $1/2$ and the jump distribution on an empty site and on an occupied site are symmetric to each other, we prove a LLN with zero limiting speed. Finally, we prove similar LLN and CLT results for a different environment, given by a family of independent simple symmetric random walks in equilibrium.

中文翻译：

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简单对称排除过程的随机游走

我们研究了随机游走器在平衡的简单对称排除过程 (SSEP) 之上进化的长期行为，在维度一。在每次跳跃时，随机游走者都会受到漂移的影响，这取决于它是坐在一个粒子还是一个洞的顶部，因此它的渐近行为预计取决于密度 $\rho \in [0, 1] $ 基础SSEP。我们的第一个结果是对于所有密度 $\rho$ 的随机游走者的大数定律 (LLN)，除了最多两个值 $\rho_-, \rho_+ \in [0, 1]$。我们在 LLN 中获得的渐近速度是 $\rho$ 的单调函数。此外，$\rho_-$ 和 $\rho_+$ 被表征为速度可能跳到（或从）零的两个点。此外，对于随机游走经历非零速度的所有密度值，我们可以证明它满足泛函中心极限定理（CLT）。对于密度为$1/2$并且空站点和占用站点上的跳跃分布彼此对称的特殊情况，我们证明了具有零极限速度的LLN。最后，我们证明了不同环境下类似的 LLN 和 CLT 结果，由平衡中的独立简单对称随机游走族给出。