Slow dynamics for the dilute Ising model in the phase coexistence region

TL;DR

This paper proves that in the phase coexistence region of the dilute Ising model, the Glauber dynamics exhibit a power-law decay of spin autocorrelation, confirming long-standing conjectures about slow relaxation.

Contribution

It establishes rigorous bounds on autocorrelation decay in the dilute Ising model, linking phase profile energy barriers to surface tension and phase separation.

Findings

Autocorrelation decays as a negative power of time in all dimensions d ≥ 2.

Upper bound on decay rate established for d=2.

Connection made between phase profiles, surface tension, and energy barriers.

Abstract

In this paper we consider the Glauber dynamics for a disordered ferromagnetic Ising model, in the region of phase coexistence. It was conjectured several decades ago that the spin autocorrelation decays as a negative power of time [Huse and Fisher, Phys. Rev. B, 1987]. We confirm this behavior by establishing a corresponding lower bound in any dimensions $d \geqslant 2$, together with an upper bound when $d=2$. Our approach is deeply connected to the Wulff construction for the dilute Ising model. We consider initial phase profiles with a reduced surface tension on their boundary and prove that, under mild conditions, those profiles are separated from the (equilibrium) pure plus phase by an energy barrier.