Fluctuations of the Ising free energy on Erdős-Rényi graphs
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arXiv
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| Auteurs principaux: | , , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866915737558319104 |
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| author | Coja-Oghlan, Amin Kaaser, Dominik Rolvien, Maurice Zakharov, Pavel Zampetakis, Kostas |
| author_facet | Coja-Oghlan, Amin Kaaser, Dominik Rolvien, Maurice Zakharov, Pavel Zampetakis, Kostas |
| contents | We investigate the ferromagnetic Ising model on the Erdős-Rényi random graph $\mathbb{G}(n,m)$ with bounded average degree $d=2m/n$. Specifically, we determine the limiting distribution of $\log Z_{\mathbb{G}(n,m)}(β,B)$, where $Z_{\mathbb{G}(n,m)}(β,B)$ is the partition function at inverse temperature $β>0$ and external field $B\geq0$.
If either $B>0$, or $B=0$, $d>1$ and $β>\operatorname{ath}(1/d)$ the limiting distribution is a Gaussian whose variance is of order $Θ(n)$ and is described by a family of stochastic fixed point problems that encode the root magnetisation of two correlated Galton-Watson trees. By contrast, if $B=0$ and either $d\leq1$ or $β<\operatorname{ath}(1/d)$ the limiting distribution is an infinite sum of independent random variables and has bounded variance. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_08590 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Fluctuations of the Ising free energy on Erdős-Rényi graphs Coja-Oghlan, Amin Kaaser, Dominik Rolvien, Maurice Zakharov, Pavel Zampetakis, Kostas Combinatorics Mathematical Physics Probability 05C80, 82B44, 82B20 We investigate the ferromagnetic Ising model on the Erdős-Rényi random graph $\mathbb{G}(n,m)$ with bounded average degree $d=2m/n$. Specifically, we determine the limiting distribution of $\log Z_{\mathbb{G}(n,m)}(β,B)$, where $Z_{\mathbb{G}(n,m)}(β,B)$ is the partition function at inverse temperature $β>0$ and external field $B\geq0$. If either $B>0$, or $B=0$, $d>1$ and $β>\operatorname{ath}(1/d)$ the limiting distribution is a Gaussian whose variance is of order $Θ(n)$ and is described by a family of stochastic fixed point problems that encode the root magnetisation of two correlated Galton-Watson trees. By contrast, if $B=0$ and either $d\leq1$ or $β<\operatorname{ath}(1/d)$ the limiting distribution is an infinite sum of independent random variables and has bounded variance. |
| title | Fluctuations of the Ising free energy on Erdős-Rényi graphs |
| topic | Combinatorics Mathematical Physics Probability 05C80, 82B44, 82B20 |
| url | https://arxiv.org/abs/2601.08590 |