Cutoff and Dynamical Phase Transition for the General Multi-component Ising Model

Abstract

We study the multi-component Ising model, which is also known as the block Ising model. In this model, the particles are partitioned into a fixed number of groups with a fixed proportion, and the interaction strength is determined by the group to which each particle belongs. We demonstrate that the Glauber dynamics on our model exhibits the cutoff\(\text{-- }\)metastability phase transition as passing the critical inverse-temperature \(\beta _{cr}\), which is determined by the proportion of the groups and their interaction strengths, regardless of the total number of particles. For \(\beta <\beta _{cr}\), the dynamics shows a cutoff at \(\alpha n\log n\) with a window size O(n), where \(\alpha \) is a constant independent of n. For \(\beta =\beta _{cr}\), we prove that the mixing time is of order \(n^{3/2}\). In particular, we deduce the so-called non-central limit theorem for the block magnetizations to validate the optimal bound at \(\beta =\beta _{cr}\). For \(\beta >\beta _{cr}\), we examine the metastability, which refers to the exponential mixing time. Our results, based on the position of the employed Ising model on the complete multipartite graph, generalize the results of previous versions of the model.