Metastability of the Ising model on random regular graphs at zero temperature

TL;DR

This paper analyzes the metastability of the ferromagnetic Ising model on random regular graphs at zero temperature, showing how transition times grow exponentially with system size and parameters.

Contribution

It provides a rigorous proof of the exponential scaling of transition times in the zero-temperature limit for the Ising model on random regular graphs.

Findings

Transition time scales as exp(β(r/2 + O(√r)) n) as β→∞

Bounds on isoperimetric number of random regular graphs are used

Pathwise approach is employed for the proof

Abstract

We study the metastability of the ferromagnetic Ising model on a random $r$-regular graph in the zero temperature limit. We prove that in the presence of a small positive external field the time that it takes to go from the all minus state to the all plus state behaves like ${\exp(\beta (r/2+ \mathcal{O}(\sqrt{r}))n)}$ when the inverse temperature $\beta\rightarrow\infty$ and the number of vertices $n$ is large enough but fixed. The proof is based on the so-called pathwise approach and bounds on the isoperimetric number of random regular graphs.