Disordered Gibbs measures and Gaussian conditioning

• 成果类型: Article; Early Access
• 署名作者: Dembo, Amir; Subag, Eliran
• 署名单位: Stanford University; Weizmann Institute of Science
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051; 1432-2064
• DOI: 10.1007/s00440-026-01526-5
• 发表日期: 2026-08-22
• 关键词: spin-glass model; metastable states; free-energy; complexity; DYNAMICS; ultrametricity; limit

摘要

We study the law of a random field fN(sigma) evaluated at a random sample from the Gibbs measure associated to a Gaussian field HN(sigma) . In the high-temperature regime, we show that bounds on the probability that fN(sigma)is an element of A for sigma randomly sampled from the Gibbs measure can be deduced from similar bounds for deterministic sigma under the conditional Gaussian law given that HN(sigma)/N=E for E close to the derivative F '(beta) of the free energy (which is the typical value of HN(sigma)/N under the Gibbs measure). In the more challenging low-temperature regime we restrict to k -rsb spherical spin glasses, proving a similar result, now with a more elaborate conditioning. Namely, with qi denoting the locations of the non-zero atoms of the Parisi measure, in addition to specifying that HN(sigma)/N=E , here one needs to also condition on the energy and its gradient at points x1,& mldr;,xk such that < xi,xj >/N=qi boolean AND j and < xi,sigma >/N approximate to qi . Like in the high-temperature phase, the energy and gradient values on which one conditions are also specified by the model's Parisi measure. We apply our general results to two important problems from statistical physics. That is, computing the Franz-Parisi potential at any temperature and, reducing certain asymptotics of Langevin dynamics with initial conditions distributed according to the Gibbs measure, to the more manageable problem of studying dynamics with non-random initial conditions and conditional disorder.