preprint Jonas Jalowy author 1137680000-0001-9624-2685 Isabel Lammers author Matthias Löwe author 94 department 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 $s_N \to \infty$. 2026 eng 2603.01994 @article{Jalowy_Lammers_Löwe_2026, title={The infinite block spin Ising model}, journal={arXiv:2603.01994}, author={Jalowy, Jonas and Lammers, Isabel and Löwe, Matthias}, year={2026} } Jalowy J, Lammers I, Löwe M. The infinite block spin Ising model. <i>arXiv:260301994</i>. Published online 2026. Jalowy, Jonas, et al. “The Infinite Block Spin Ising Model.” <i>ArXiv:2603.01994</i>, 2026. J. Jalowy, I. Lammers, M. Löwe, ArXiv:2603.01994 (2026). Jalowy, Jonas, Isabel Lammers, and Matthias Löwe. “The Infinite Block Spin Ising Model.” <i>ArXiv:2603.01994</i>, 2026. J. Jalowy, I. Lammers, and M. Löwe, “The infinite block spin Ising model,” <i>arXiv:2603.01994</i>. 2026. Jalowy, J., Lammers, I., &#38; Löwe, M. (2026). The infinite block spin Ising model. In <i>arXiv:2603.01994</i>. 648162026-03-03T08:49:16Z2026-03-03T08:49:33Z