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 √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.
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BibTeX
@misc{bhowal2026higherorder,
title = {{Higher-Order Fluctuations of the Log-Partition Function for Ising Models on Inhomogeneous Random Graphs}},
author = {Sanchayan Bhowal and Somabha Mukherjee},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.01298},
eprint = {2610.01298},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.01298},
}
RIS
TY - UNPB
TI - Higher-Order Fluctuations of the Log-Partition Function for Ising Models on Inhomogeneous Random Graphs
AU - Sanchayan Bhowal
AU - Somabha Mukherjee
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.01298
UR - https://arxiv.org/abs/2610.01298
AB - 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 √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.
ER -