Disordered Gibbs measures and Gaussian conditioning
Amir Dembo, Eliran Subag
TL;DR
This work develops a general principle for understanding observables $f_N(\boldsymbol{\sigma})$ evaluated at Gibbs samples from Gaussian mean-field spin-glass Hamiltonians. In the high-temperature regime, concentration under the Gibbs measure is deduced from conditional Gaussian laws given $H_N(\boldsymbol{\sigma})/N$ near the typical energy $F'(\beta)$, establishing a precise exponential decay criterion. In the challenging low-temperature regime, the analysis is restricted to spherical $k$-RSB spin glasses and requires a multi-level conditioning guided by the Parisi measure, including energies and gradients at a hierarchical set of points with prescribed overlaps. The framework is applied to compute the Franz-Parisi potential at any temperature and to relate Langevin dynamics with Gibbs initial data to dynamics with conditioned disorder, enabling rigorous asymptotics and reducing complex disordered dynamics to tractable deterministic-conditional problems. The backbone of the method combines pure-states decompositions, multi-level Kac-Rice formulas, and conditional Gaussian laws, yielding new rigorous results in both static (Parisi-type) and dynamic settings for disordered mean-field systems.
Abstract
We study the law of a random field $f_N(\boldsymbolσ)$ evaluated at a random sample from the Gibbs measure associated to a Gaussian field $H_N(\boldsymbolσ)$. In the high-temperature regime, we show that bounds on the probability that $f_N(\boldsymbolσ)\in A$ for $\boldsymbolσ$ randomly sampled from the Gibbs measure can be deduced from similar bounds for deterministic $\boldsymbolσ$ under the conditional Gaussian law given that $H_N(\boldsymbolσ)/N=E$ for $E$ close to the derivative $F'(β)$ of the free energy (which is the typical value of $H_N(\boldsymbolσ)/N$ 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 $q_i$ denoting the locations of the non-zero atoms of the Parisi measure, in addition to specifying that $H_N(\boldsymbolσ)/N=E$, here one needs to also condition on the energy and its gradient at points $\mathbf{x}_1,\ldots,\mathbf{x}_k$ such that $\langle \mathbf{x}_i,\mathbf{x}_j\rangle/N=q_{i\wedge j}$ and $\langle \mathbf{x}_i,\boldsymbolσ\rangle/N\approx q_{i}$. 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.