2024/09/28 by Amir Dembo, Eliran Subag, Dembo, Amir +1
Physics and Astronomy · #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #Statistical Mechanics and Entropy
paper · pdf · doi:10.48550/arxiv.2409.19453
openalex publication_date 2024/09/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the law of a random field fN(\boldsymbolσ) evaluated at a random sample from the Gibbs measure associated to a Gaussian field HN(\boldsymbolσ). In the high-temperature regime, we show that bounds on the probability that fN(\boldsymbolσ)∈ A for \boldsymbolσ randomly sampled from the Gibbs measure can be deduced from similar bounds for deterministic \boldsymbolσ under the conditional Gaussian law given that HN(\boldsymbolσ)/N=E for E close to the derivative F'(β) of the free energy (which is the typical value of HN(\boldsymbolσ)/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(\boldsymbolσ)/N=E, here one needs to also condition on the energy and its gradient at points x1,…,xk such that ⟨ xi,xj⟩/N=qi\wedge j and ⟨ xi,\boldsymbolσ⟩/N≈ 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.