dorsal/arxiv
View SchemaFluctuations of the Ising free energy on Erd\H{o}s-R\'enyi graphs
| Authors | Amin Coja-Oghlan, Dominik Kaaser, Maurice Rolvien, Pavel Zakharov, Kostas Zampetakis |
|---|---|
| Categories | |
| ArXiv ID | 2601.08590vv2 |
| URL | https://arxiv.org/abs/2601.08590 |
| License | http://creativecommons.org/licenses/by/4.0/ |
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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"abstract": "We investigate the ferromagnetic Ising model on the Erd\\H{o}s-R\\\u0027enyi 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 $\\beta\u003e0$ and external field $B\\geq0$.\n If either $B\u003e0$, or $B=0$, $d\u003e1$ and $\\beta\u003e\\operatorname{ath}(1/d)$ 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 $\\beta\u003c\\operatorname{ath}(1/d)$ the limiting distribution is an infinite sum of independent random variables and has bounded variance.",
"arxiv_id": "2601.08590",
"authors": [
"Amin Coja-Oghlan",
"Dominik Kaaser",
"Maurice Rolvien",
"Pavel Zakharov",
"Kostas Zampetakis"
],
"categories": [
"math.CO",
"math-ph",
"math.MP",
"math.PR"
],
"license": "http://creativecommons.org/licenses/by/4.0/",
"title": "Fluctuations of the Ising free energy on Erd\\H{o}s-R\\\u0027enyi graphs",
"url": "https://arxiv.org/abs/2601.08590",
"version": "v2"
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"schema_id": "dorsal/arxiv",
"source": {
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"variant": "snapshot-2026-01-17",
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