The Infinite Block Spin Ising Model

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