Critical Free-Energy Variance in the Sherrington-Kirkpatrick Model
Abstract
We study the variance of the free energy in the Gaussian Sherrington-Kirkpatrick model at zero external field. For every system size and inverse temperature β>0, we prove 2N2Var FNoff(β) ≤ logχ+(β2−1)(χ−1), χ=N𝔼⟨ R1,22⟩β, where R1,2 is the replica overlap and FNoff=N−1log ZNoff omits diagonal disorder. The proof uses an exact identity for the edges incident to one spin, Schur complements, and Gaussian disorder interpolation. At criticality the bound reduces to 2N2Var FNoff(1)≤logχ. An explicit nonnegative integral measures its gap. A second decomposition relates this gap to a mixed cubic overlap functional. Existing moment and cavity estimates imply that its required integral is bounded. Critical susceptibility asymptotics give the sharp logarithmic upper coefficient and show that the gap converges exactly when the variance has a constant term. We extend an entropy bound throughout bounded critical temperature windows and establish overlap moment stability on a smaller interval.
Related
- Fluctuations of the Sherrington-Kirkpatrick free energy at critical temperature
- Free energy and fluctuations of generalized bipartite Sherrington-Kirkpatrick models and generalized Hopfield models
- Free energy fluctuation of soft spherical Sherrington-Kirkpatrick model
- Upper deviations of the Sherrington-Kirkpatrick free energy: the fifth-order law and the onset of constrained Parisi measures
- Critical-Window Fluctuations and Disorder Universality for the Sherrington--Kirkpatrick Model
BibTeX
@misc{tierz2026critical,
title = {{Critical Free-Energy Variance in the Sherrington-Kirkpatrick Model}},
author = {Miguel Tierz},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00011},
eprint = {2610.00011},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00011},
}
RIS
TY - UNPB
TI - Critical Free-Energy Variance in the Sherrington-Kirkpatrick Model
AU - Miguel Tierz
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.00011
UR - https://arxiv.org/abs/2610.00011
AB - We study the variance of the free energy in the Gaussian Sherrington-Kirkpatrick model at zero external field. For every system size and inverse temperature β>0, we prove 2N^2Var F_N^off(β) ≤ logχ+(β^2−1)(χ−1), χ=N𝔼⟨ R_1,2^2⟩_β, where R_1,2 is the replica overlap and F_N^off=N^−1log Z_N^off omits diagonal disorder. The proof uses an exact identity for the edges incident to one spin, Schur complements, and Gaussian disorder interpolation. At criticality the bound reduces to 2N^2Var F_N^off(1)≤logχ. An explicit nonnegative integral measures its gap. A second decomposition relates this gap to a mixed cubic overlap functional. Existing moment and cavity estimates imply that its required integral is bounded. Critical susceptibility asymptotics give the sharp logarithmic upper coefficient and show that the gap converges exactly when the variance has a constant term. We extend an entropy bound throughout bounded critical temperature windows and establish overlap moment stability on a smaller interval.
ER -