The infinite block spin Ising model
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866917306043465728 |
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| author | Jalowy, Jonas Lammers, Isabel Löwe, Matthias |
| author_facet | Jalowy, Jonas Lammers, Isabel Löwe, Matthias |
| contents | 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 $s_N \to \infty$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_01994 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The infinite block spin Ising model Jalowy, Jonas Lammers, Isabel Löwe, Matthias Probability 60F05, 82B20 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 $s_N \to \infty$. |
| title | The infinite block spin Ising model |
| topic | Probability 60F05, 82B20 |
| url | https://arxiv.org/abs/2603.01994 |