The Lifshitz Tail and Relaxation to Equilibrium in the One-Dimensional Disordered Ising Model

Abstract

We study spectral properties of the generator of the Glauber dynamics for a 1D disordered stochastic Ising model with random bounded couplings. By an explicit representation for the upper branch of the generator we get an asymptotic formula for the integrated density of states of the generator near the upper edge of the spectrum. This asymptotic behavior appears to have the form of the Lifshitz tail, which is typical for random operators near fluctuation boundaries. As a consequence we find the asymptotics for the average over the disorder of the time-autocorrelation function to be

$$\langle \langle \sigma _{\text{0}}^\omega (t),\sigma _0 (0)\rangle _{P(\omega ) = } {\text{exp\{ }} - gt - kt^{1/3} {\text{(1 + }}o(1){\text{)\} as }}t \to \infty $$

with constants g, k depending on the distribution of the random couplings.