Full Current Statistics for a Disordered Open Exclusion Process

TL;DR

This paper derives exact joint current statistics for a disordered open exclusion process, revealing a renormalization property and providing detailed spectral information of the tilted generator.

Contribution

It provides the first exact calculation of the joint current distribution and cumulant generating function for a disordered open exclusion process on a one-dimensional lattice.

Findings

Exact cumulant generating function derived

Conditioned process is a renormalized version of the original

Eigenvalues and eigenvectors of the tilted generator are explicitly determined

Abstract

We consider the nonabelian sandpile model defined on directed trees by Ayyer, Schilling, Steinberg and Thi\'ery (Commun. Math. Phys, 2013) and restrict it to the special case of a one-dimensional lattice of $n$ sites which has open boundaries and disordered hopping rates. We focus on the joint distribution of the integrated currents across each bond simultaneously, and calculate its cumulant generating function exactly. Surprisingly, the process conditioned on seeing specified currents across each bond turns out to be a renormalised version of the same process. We also remark on a duality property of the large deviation function. Lastly, all eigenvalues and both Perron eigenvectors of the tilted generator are determined.