The infinite block spin Ising model

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 $s_N \to \infty$.

Figures (1)

• Figure 1: Three different simulations of the block spin Curie Weiss model for $s_N=15$ blocks with $N/15=2500$ spins in each block (each of which itself is a mean field model). The high temperature regime in the top row ($\beta=0.5, \alpha=0.2$) shows zero magnetization, whereas the low temperature regime in the center does magnetize ($\beta=0.8,\alpha=0.2$). The bottom row shows an unstable (improbable) realization in the low temperature regime, which visualizes the correlation of (cyclically) neighboring block interaction.

Theorems & Definitions (51)

• Theorem 2.1: Uniform Law of Large Numbers
• Proposition 2.2
• Theorem 2.3: CLT in finite dimensional distribution
• Remark 2.4
• Proposition 3.1
• Lemma 3.2
• proof
• proof : Proof of Proposition \ref{['m1']}
• Lemma 4.1
• proof