The infinite block spin Ising model

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 $s_N \to \infty$.

💡 Research Summary

The paper introduces and rigorously analyzes a novel “block mean‑field Ising” model in which the total number of spins N and the number of blocks sN both tend to infinity. The spins are Ising variables σi∈{−1,+1}. The index set {1,…,N} is partitioned into sN blocks of equal size N/sN. Within each block every pair of spins interacts with strength β>0 (Curie‑Weiss interaction), while spins in neighboring blocks interact with strength α>0 (nearest‑neighbour coupling on a one‑dimensional periodic lattice). The Hamiltonian can be written in terms of the block magnetization vector m=(m1,…,msN) as
 H_N(m)= (N/2sN) mᵀ A m,
where A is a circulant matrix with diagonal entries β and off‑diagonal entries α on the first sub‑ and super‑diagonals (periodic). Its eigenvalues are λj=β+2α cos(2πj/sN). The model interpolates between the Curie‑Weiss model (sN=1), bipartite or multi‑species mean‑field models (fixed sN), and the one‑dimensional nearest‑neighbour Ising chain (sN=N).

The main contributions are two theorems describing the macroscopic behaviour of the block magnetization under the Gibbs measure μN,β,α.

Theorem 2.1 (Uniform Law of Large Numbers).
Assume sN=o(N log N).
High‑temperature regime (β+2α≤1): for any ε>0,
 μN,β,α( max_{k≤sN}|mk|>ε ) → 0 as N→∞.
Thus every block magnetization converges to zero in probability.
Low‑temperature regime (β+2α>1): let m* be the largest solution of x=tanh((β+2α)x). Then for any ε>0,
 μN,β,α( min{ max_k|mk−m*|, max_k|mk+m*| } > ε ) → 0.
Hence the whole vector m concentrates on the two symmetry‑broken states ±(m*,…,m*). The result holds uniformly over all blocks, despite the growing dimension.

Theorem 2.3 (Multivariate Central Limit Theorem).
High‑temperature CLT: If sN=o(N (log N)^{‑5/2}) and β+2α<1, the rescaled magnetization √(N/sN) m converges in finite‑dimensional distributions to a centered Gaussian with covariance Σ=(I−A)^{‑1}. For fixed sN this reduces to the classical Curie‑Weiss CLT; when sN→∞ the entries of Σ decay geometrically and equal the lattice Green’s function
 Σ_{ij}= ((1−β)−√{(1−β)²−4α²})^{|i−j|} / (2α √{(1−β)²−4α²}).
Low‑temperature CLT: If sN=o(√N (log N)^{‑1}) and β+2α>1, then for any δ>0 the fluctuations around each broken‑symmetry state satisfy
 √(N/sN)(m∓1 m*) ⇒ N(0, Σ*),
where Σ*=(1−(m*)²)