Abstract
We prove a quenched functional central limit theorem for a one-dimensional random walk driven by a simple symmetric exclusion process. This model can be viewed as a special case of the random walk in a balanced random environment, for which the weak quenched limit is constructed as a function of the invariant measure of the environment viewed from the walk. We bypass the need to show the existence of this invariant measure. Instead, we find the limit of the quadratic variation of the walk and give an explicit formula for it.
Highlights
We prove a quenched functional central limit theorem for a one-dimensional random walk driven by a simple symmetric exclusion process
The model belongs to the class of random walks in dynamical random environments
To define a random walk driven by the simple symmetric exclusion process (SSEP), one fixes parameters p1, p0, ρ ∈ [0, 1], λ0, λ1 > 0 and makes the random walk jump from x ∈ Z to x + 1 at time t at rate λ1p1ηt(x) + λ0p0(1 − ηt(x)), where ηt(x) is the state of the exclusion process at site x and time t, started from equilibrium at density ρ
Summary
We prove a quenched functional central limit theorem for a one-dimensional random walk driven by a simple symmetric exclusion process. Since the underlying dynamic environment in our model has only two types of sites (particles/holes), the key to analyzing the growth rate of the quadratic variation is to compute the asymptotic fraction of time, limt→∞ t−1 t 0 We accomplish this by providing an explicit function φ and explicit constants a and b such that Lφ ≈ aξ0 + b, where ξx(t) := ηt(x + Xt) and L denotes the generator of the process (ξ(t))t≥0, the environment as seen by the walk. In the context of random walks in random environments, it has been used in [1], [20] and [23], among other works
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