Higher-Order Fluctuations of the Log-Partition Function for Ising Models on Inhomogeneous Random Graphs

Abstract

Classical central limit theorems (CLTs) for log-partition functions of Ising models on Erdős-Rényi random graphs lead to degenerate limits in the high-temperature regime when self-loops are absent. In this paper, we fill this surprising gap in the more general context of Ising models on inhomogeneous random graphs. In particular, we show that if $G(N,W)$ is an inhomogeneous random graph without self-loops generated by a graphon $W$, then, in the high-temperature regime, the log-partition function of the corresponding Ising model satisfies a CLT on the $N$-scale, with a limiting variance that incorporates contributions from cycles of all orders. This behavior contrasts sharply with that of the corresponding model in which self-loops are allowed, where the analogous Gaussian fluctuations occur on the $\sqrt{N}$-scale, with their asymptotic variance determined entirely by the diagonal profile of $W$. Our analysis yields, as a byproduct of independent interest, a CLT for the log-determinant of the associated resolvent matrix, established by combining the Lindeberg CLT for triangular arrays with suitable bounds on higher-order trace terms.