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 $d=2m\mathrm{/}n$. 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 $B\ge 0$. If either $B>0$, or $B=0$, $d>1$ and $\beta >\mathrm{ath}(1\mathrm{/}d)$ the limiting distribution is a Gaussian whose variance is of order $\mathrm{\Theta }(n)$ 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 $d\le 1$ or $\beta <\mathrm{ath}(1\mathrm{/}d)$ the limiting distribution is an infinite sum of independent random variables and has bounded variance.