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arXiv · 2601.08590

Fluctuations of the Ising free energy on Erd\H{o}s-R\'enyi graphs

Amin Coja-Oghlan·Dominik Kaaser·Maurice Rolvien·Pavel Zakharov·Kostas Zampetakis
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Abstract

We investigate the ferromagnetic Ising model on the Erd\H{o}s-R\'enyi random graph $\mathbb{G}(n,m)$ with bounded average degree $d=2m/n$. Specifically, we determine the limiting distribution of $\log Z_{\mathbb{G}(n,m)}(\beta,B)$, where $Z_{\mathbb{G}(n,m)}(\beta,B)$ is the partition function at inverse temperature $\beta>0$ and external field $B\geq0$. If either $B>0$, or $B=0$, $d>1$ and $\beta>\operatorname{ath}(1/d)$ the limiting distribution is a Gaussian whose variance is of order $\Theta(n)$ 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 $\beta<\operatorname{ath}(1/d)$ the limiting distribution is an infinite sum of independent random variables and has bounded variance.

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Amin Coja-Oghlan, Dominik Kaaser, Maurice Rolvien, Pavel Zakharov, Kostas Zampetakis. 2026-01-13. Fluctuations of the Ising free energy on Erd\H{o}s-R\'enyi graphs. https://arxiv.org/abs/2601.08590

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