Spectral gap and cutoff of the simple exclusion process with IID conductances

Shangjie Yang

Published 2024-08-13Version 1

In this paper, we study the spectral gap and mixing time of the simple exclusion process with $k$ particles in the line segment $[1, N]$ with conductances $c(x, x+1)_{1\le x<N}$ where $c(x, x+1)>0$ is the rate of swapping the contents of the two sites $x$ and $ x+1$. Under the IID assumption on $(c(x, x+1))_{x \in \mathbb{N}}$ with unit inverse expectation $\mathbb{E}[1/c(x, x+1)]=1$, we prove that the spectral gap, denoted by $\mathrm{gap}_{N,k}$, of the process satisfies $\mathrm{gap}_{N, k}=(1+o(1))\pi^2/N^2$ and the eigenfunction $g_N$ with $g_N(1)=1$ corresponding to the spectral gap of one particle case is well approximated by $h_N(x) := \cos\left( (x-1/2)\pi/N \right)$. Under some further assumptions on $c(x, x+1)_{x \in \mathbb{N} }$ and $k$, we prove that around time $(1+o(1)) (2 \pi^2)^{-1} N^2 \log k$, the total variation distance to equilibrium drops abruptly from $1$ to $0$.