The Infinite Block Spin Ising Model

Abstract We study a block mean-field Ising model with N spins split into $$s_N$$ 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 $$N\rightarrow \infty $$ N → ∞ and $$s_N\rightarrow \infty $$ s N → ∞ . The model interpolates between Curie–Weiss model for $$s_N=1$$ s N = 1 , multi-species mean field for fixed $$s_N=s$$ s N = s , and the 1D Ising model for each spin in its own block at $$s_N=N$$ s N = N . Under mild growth conditions on $$s_N$$ 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 $$s_N=o(N/(\log N)^c)$$ 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 $$s_N \rightarrow \infty $$ s N → ∞ .