The spectrum of an asymmetric annihilation process

TL;DR

This paper proves a conjecture about the eigenvalues of a Markovian exclusion model called the asymmetric annihilation process, by generalizing the model and explicitly calculating its eigenvalues, thus advancing understanding in nonequilibrium statistical physics.

Contribution

The paper proves the eigenvalue conjecture for the asymmetric annihilation process by generalizing the Markov matrix and explicitly computing its eigenvalues.

Findings

Eigenvalues of the generalized Markov matrix are explicitly calculated.

The generalized model reduces to the original model via specialization.

Partition function derivation for the generalized model is outlined.

Abstract

In recent work on nonequilibrium statistical physics, a certain Markovian exclusion model called an asymmetric annihilation process was studied by Ayyer and Mallick. In it they gave a precise conjecture for the eigenvalues (along with the multiplicities) of the transition matrix. They further conjectured that to each eigenvalue, there corresponds only one eigenvector. We prove the first of these conjectures by generalizing the original Markov matrix by introducing extra parameters, explicitly calculating its eigenvalues, and showing that the new matrix reduces to the original one by a suitable specialization. In addition, we outline a derivation of the partition function in the generalized model, which also reduces to the one obtained by Ayyer and Mallick in the original model.