The infinite block spin Ising model

TL;DR

This paper analyzes a generalized block spin Ising model that interpolates between classical models, proving laws of large numbers and CLTs as both the number of spins and blocks grow, revealing new phenomena in high-dimensional limits.

Contribution

It introduces a novel infinite block spin Ising model and establishes its probabilistic behavior in the simultaneous limit of large spins and blocks, extending classical results.

Findings

Law of large numbers for the model

Multivariate CLT with Green's function covariance

High-temperature CLT valid up to s_N=o(N/( ext{log} N)^c)

Abstract

We study a block mean-field Ising model with $N$ spins split into $s_N$ blocks, with Curie-Weiss interaction within blocks and nearest-neighbor coupling between blocks. While previous models deal with the block magnetization for a fixed number of blocks, we study the the simultaneous limit $N\to\infty$ and $s_N\to\infty$. The model interpolates between Curie-Weiss model for $s_N=1$, multi-species mean field for fixed $s_N=s$, and the 1D Ising model for each spin in its own block at $s_N=N$. Under mild growth conditions on $s_N$, we prove a law of large numbers and a multivariate CLT with covariance given by the lattice Green's function. For instance, the high temperature CLT essentially covers the optimal range up to $s_N=o(N/(\log N)^c)$ and the low temperature regime is new even for fixed number of blocks $s > 2$. In addition to the standard competition between entropy and energy, a new obstacle in the proofs is a curse of dimensionality as $s_N \to \infty$.