A second-moment proof of quenched equals annealed for the Potts model on random regular graphs
Abstract
Ferromagnetic spin models on random regular graphs, such as the Ising and Potts models, have been studied extensively. Several equivalent variational and recursive representations of their limiting pressure are known, and these representations can be used to establish agreement between the quenched and annealed pressures. We give a concise overview of these representations and highlight the connections among the different formulations. We then derive analogous representations for the exponential growth rate of the second moment of the partition function, and prove that it has the same order as the square of the annealed pressure, up to a subexponential error. Combined with concentration of the quenched pressure, this identity yields a transparent second-moment proof of quenched--annealed agreement on random regular graphs. By bringing these representations and arguments together in a unified presentation, we aim to provide an accessible entry point to the literature on ferromagnetic spin models on random regular graphs.
Related
- Annealed correlation and quenched noise sensitivity in biased deep neural networks
- Defining the Crick-Franklin-Watson genes of any robot (equals a finite state machine) and defining the Shannon genetic code attached to each gene. We also define the Krohn-Rhodes complexity of any regular maximal prefix code
- Moment estimates for chaoses with regular moment growth
- Second moment of incomplete Eisenstein series revisited
- Hard-Edge Determinant Fields and Second-Moment Universality for Random Matrix Ratios
BibTeX
@misc{hofstad2026secondmoment,
title = {{A second-moment proof of quenched equals annealed for the Potts model on random regular graphs}},
author = {Remco van der Hofstad and Kai-Chun Huang and Noela Müller and Wioletta M. Ruszel},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10034},
eprint = {2610.10034},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.10034},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - A second-moment proof of quenched equals annealed for the Potts model on random regular graphs
AU - Remco van der Hofstad
AU - Kai-Chun Huang
AU - Noela Müller
AU - Wioletta M. Ruszel
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.10034
UR - https://arxiv.org/abs/2610.10034
AB - Ferromagnetic spin models on random regular graphs, such as the Ising and Potts models, have been studied extensively. Several equivalent variational and recursive representations of their limiting pressure are known, and these representations can be used to establish agreement between the quenched and annealed pressures. We give a concise overview of these representations and highlight the connections among the different formulations. We then derive analogous representations for the exponential growth rate of the second moment of the partition function, and prove that it has the same order as the square of the annealed pressure, up to a subexponential error. Combined with concentration of the quenched pressure, this identity yields a transparent second-moment proof of quenched--annealed agreement on random regular graphs. By bringing these representations and arguments together in a unified presentation, we aim to provide an accessible entry point to the literature on ferromagnetic spin models on random regular graphs.
ER -