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

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≤ 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 ∈ N with unit inverse expectation E[1/c(x, x+1)]=1, we prove that the spectral gap, denoted by gapN,k, of the process satisfies gapN, k=(1+o(1))π²/N² and the eigenfunction gN with gN(1)=1 corresponding to the spectral gap of one particle case is well approximated by hN(x) := cos( (x-1/2)π/N ). Under some further assumptions on c(x, x+1)x ∈ N and k, we prove that around time (1+o(1)) (2 π²)⁻¹ N² log k, the total variation distance to equilibrium drops abruptly from $1$ to $0$.