Zero-temperature dynamics in the dilute Curie-Weiss model

TL;DR

This paper studies zero-temperature dynamics in a disordered Curie-Weiss Ising model on dense random graphs, showing it typically reaches a uniform ground state and relating this to the local MINCUT problem, with differences observed in heavy-tailed coupling cases.

Contribution

It proves that zero-temperature dynamics in the disordered Curie-Weiss model typically avoids local minima and reaches a uniform ground state, linking this to the local MINCUT problem.

Findings

Dynamics avoids local minima in typical cases.

Greedy local MINCUT search fails to find non-trivial cuts.

Heavy-tailed couplings can lead to different local minima.

Abstract

We consider the Ising model on a dense Erd\H{o}s--R\'enyi random graph, $\mathcal G(N,p)$, with $p>0$ fixed---equivalently, a disordered Curie--Weiss Ising model with $\mbox{Ber}(p)$ couplings---at zero temperature. The disorder may induce local energy minima in addition to the two uniform ground states. In this paper we prove that, starting from a typical initial configuration, the zero-temperature dynamics avoids all such local minima and absorbs into a predetermined one of the two uniform ground states. We relate this to the local MINCUT problem on dense random graphs; namely with high probability, the greedy search for a local MINCUT of $\mathcal G(N,p)$ with $p>0$ fixed, started from a uniform random partition, fails to find a non-trivial cut. In contrast, in the disordered Curie--Weiss model with heavy-tailed couplings, we demonstrate that zero-temperature dynamics has positive probability of absorbing in a random local minimum different from the two homogenous ground states.