Upper deviations of the Sherrington-Kirkpatrick free energy: the fifth-order law and the onset of constrained Parisi measures
Abstract
Let ZN be the partition function of the Sherrington-Kirkpatrick model at inverse temperature β>1. Replica theory (Kondor near the critical temperature, Parisi and Rizzo at all temperatures) predicts that the probability that N−1log ZN exceeds its typical value by δ decays like exp(−Naδ6/5) with an explicit a; Aronow and Lopatto recently proved bounds of this order. The exponent is the Legendre transform of the Parisi problem constrained to carry an atom θ at zero. We show that the constrained minimum exceeds the equilibrium free energy by cβθ5(1+o(1)), where cβ=9/640β3ρβ(0)−3 and ρβ(0)>0 is the density at zero of the Parisi measure. With the fractional-moment formula this gives the exponent 5/6(6cβ)−1/5δ6/5(1+o(1)) as N→∞ and then δ→0. The constrained minimizer has a marginal first positive contact, to leading order at 3θ/(2ρβ(0)) with an atom θ/2. This plateau calculus applies to every mixture with ξ'''(0)=0 whose Parisi measure accumulates at zero; a small field h opens a gap of order |h|2/3, and an atom of the Parisi measure at zero, present in every pure p-spin model with p≥3, makes the rate leave zero linearly. Near criticality the constant tends to Kondor's value 9/5120 and the marginal regime ends at a de Almeida-Thouless threshold. For the Sherrington-Kirkpatrick model at every β>1, a finite-step curvature inequality from an unrefereed preprint shows that every constrained minimizer either has an interval of positive contacts, all marginal, or has exactly two atoms, and that the marginal regime ends at a unique de Almeida-Thouless threshold θc(β). The laws at every β>1 use that the Parisi measure accumulates at zero, a theorem of Lopatto also proved in earlier work of the author.
Related
- Free energy and fluctuations of generalized bipartite Sherrington-Kirkpatrick models and generalized Hopfield models
- Free energy fluctuation of soft spherical Sherrington-Kirkpatrick model
- Fluctuations of the Sherrington-Kirkpatrick free energy at critical temperature
- Critical Free-Energy Variance in the Sherrington-Kirkpatrick Model
- Sampling from the Sherrington-Kirkpatrick Model up to Criticality
BibTeX
@misc{pei2026upper,
title = {{Upper deviations of the Sherrington-Kirkpatrick free energy: the fifth-order law and the onset of constrained Parisi measures}},
author = {Yan Ru Pei},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06480},
eprint = {2610.06480},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.06480},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Upper deviations of the Sherrington-Kirkpatrick free energy: the fifth-order law and the onset of constrained Parisi measures
AU - Yan Ru Pei
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.06480
UR - https://arxiv.org/abs/2610.06480
AB - Let Z_N be the partition function of the Sherrington-Kirkpatrick model at inverse temperature β>1. Replica theory (Kondor near the critical temperature, Parisi and Rizzo at all temperatures) predicts that the probability that N^−1log Z_N exceeds its typical value by δ decays like exp(−Naδ^6/5) with an explicit a; Aronow and Lopatto recently proved bounds of this order. The exponent is the Legendre transform of the Parisi problem constrained to carry an atom θ at zero. We show that the constrained minimum exceeds the equilibrium free energy by c_βθ^5(1+o(1)), where c_β=9/640β^3ρ_β(0)^−3 and ρ_β(0)>0 is the density at zero of the Parisi measure. With the fractional-moment formula this gives the exponent 5/6(6c_β)^−1/5δ^6/5(1+o(1)) as N→∞ and then δ→0. The constrained minimizer has a marginal first positive contact, to leading order at 3θ/(2ρ_β(0)) with an atom θ/2. This plateau calculus applies to every mixture with ξ'''(0)=0 whose Parisi measure accumulates at zero; a small field h opens a gap of order |h|^2/3, and an atom of the Parisi measure at zero, present in every pure p-spin model with p≥3, makes the rate leave zero linearly. Near criticality the constant tends to Kondor's value 9/5120 and the marginal regime ends at a de Almeida-Thouless threshold. For the Sherrington-Kirkpatrick model at every β>1, a finite-step curvature inequality from an unrefereed preprint shows that every constrained minimizer either has an interval of positive contacts, all marginal, or has exactly two atoms, and that the marginal regime ends at a unique de Almeida-Thouless threshold θ_c(β). The laws at every β>1 use that the Parisi measure accumulates at zero, a theorem of Lopatto also proved in earlier work of the author.
ER -