Algebraic Properties of a Disordered Asymmetric Glauber Model

TL;DR

This paper analyzes a disordered asymmetric Glauber model for Ising spins on a 1D lattice, providing explicit eigenvalues, conjecturing normalization formulas, and exploring ferromagnetic and antiferromagnetic limits.

Contribution

It explicitly calculates all eigenvalues of the transition matrix for the model and proposes a formula for the normalization factor, advancing understanding of disordered Glauber dynamics.

Findings

Eigenvalues of the transition matrix are explicitly calculated.

A conjectured formula for the normalization factor is proposed.

Normalization formulas are proved in ferromagnetic and antiferromagnetic limits.

Abstract

We consider a variant of Glauber dynamics of Ising spins on a one-dimensional lattice, where each spin flips according to the relative state of the spin to its left. Moreover, each bond allows for two rates; flips which equalize nearest neighbor spins, and flips which "unequalize" them. In addition, the leftmost spin flips depending on the spin at that site. We explicitly calculate all eigenvalues of the transition matrix for all system sizes and conjecture a formula for the normalization factor of the model. We then analyze two limits of this model, which are analogous to ferromagnetic and antiferromagnetic behavior in the Ising model for which we are able to prove an analogous formula for the normalization factor.