Random Walk on the Simple Symmetric Exclusion Process

Abstract

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 $$\rho \in [0, 1]$$ ρ ∈ [ 0 , 1 ] of the underlying SSEP. Our first result is a law of large numbers (LLN) for the random walker for all densities $$\rho $$ ρ except for at most two values $$\rho _-, \rho _+ \in [0, 1]$$ ρ - , ρ + ∈ [ 0 , 1 ] . The asymptotic speed we obtain in our LLN is a monotone function of $$\rho $$ ρ . Also, $$\rho _-$$ ρ - and $$\rho _+$$ ρ + 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. We also prove similar LLN and CLT results for a different environment, given by a family of independent simple symmetric random walks in equilibrium.

Variable speed symmetric random walk driven by the simple symmetric exclusion process

“…Several cases were studied. The results in [17] and [15] that we are about to cite were proven for a discrete-time random walk, but we believe that the continuous-time results we state are true as well. In [17], laws of large numbers and Gaussian fluctuations are proven for λ 0 = λ 1 sufficiently large or sufficiently small and appropriate assumptions on p 0 and p 1 .…”

“…m n, the upper bound in the last equation vanishes as t → ∞. To bound the second term, we apply the Lateral Decoupling Lemma ( [15], Proposition 4.1). To do so, we need the random variable inside the expectation to be a function of the exclusion process only.…”

“…We have P µ (A ) = 1 since it is a intersection of countably many sets while each has probability 1. Choose any η ∈ A , for any n ≥ 1, The next lemma comes from [15]. To get the version stated below, one only needs to change the last line of the original proof, using (4.2).…”

“…To get the version stated below, one only needs to change the last line of the original proof, using (4.2). Proposition 4.4 (Lateral Decoupling, [15] Proposition 4.1). Let f 1 , f 2 : {0, 1} Z × R + → [0, 1] be measurable functions and H, y, α > 0.…”

“…When λ 0 = λ 1 , [10] proves that the limiting speed, if any, is strictly between λ 0 (2p 0 − 1) and λ 1 (2p 1 − 1). In [15] it is proven that, for λ 0 = λ 1 = 1 the law of large numbers holds for all ρ, with only two possible exceptions, and when the speed is not zero a Gaussian central limit theorem holds. Moreover, when p 0 = 1 − p 1 (as in [2] and [17]) and ρ = 1/2 it was shown in [15] that the speed is zero, but it is an interesting open problem to determine the scale of the fluctuations in this case and there are several competing conjectures: in [19] it is conjectured that under the scaling t 3/4 the limiting process is a fractional Brownian motion with Hurst index H = 3/4; in [12], it is conjectured (for a related continuous model) that the fluctuations are either of order t 1/2 (for a fast particle) or t 2/3 (for a slow particle); on the other hand in [16] and [18], it is conjectured that for either fast or slow particle dynamics the fluctuations are always of order t 1/2 for time t sufficiently large.…”