Exact eigenspectrum of the symmetric simple exclusion process on the complete, complete bipartite, and related graphs

TL;DR

This paper derives the exact eigenspectrum of the symmetric simple exclusion process on specific graphs by mapping it to a quantum spin model, providing detailed degeneracy analysis and connecting to broader spectral conjectures.

Contribution

It presents an elementary solution for the eigenspectrum of the process on complete and bipartite graphs, linking to higher spin models and spectral gap conjectures.

Findings

Exact eigenspectrum formulas derived for specific graphs

Degeneracy structures of the spectra analyzed in detail

Connections made to spectral gap conjectures and spin models

Abstract

We show that the infinitesimal generator of the symmetric simple exclusion process, recast as a quantum spin-1/2 ferromagnetic Heisenberg model, can be solved by elementary techniques on the complete, complete bipartite, and related multipartite graphs. Some of the resulting infinitesimal generators are formally identical to homogeneous as well as mixed higher spins models. The degeneracies of the eigenspectra are described in detail, and the Clebsch-Gordan machinery needed to deal with arbitrary spin-s representations of the SU(2) is briefly developed. We mention in passing how our results fit within the related questions of a ferromagnetic ordering of energy levels and a conjecture according to which the spectral gaps of the random walk and the interchange process on finite simple graphs must be equal.