Fluctuations of the Ising free energy on Erdős-Rényi graphs

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