Disordered Gibbs measures and Gaussian conditioning

Abstract: We study the law of a random field $f_N(\boldsymbol{\sigma})$ evaluated at a random sample from the Gibbs measure associated to a Gaussian field $H_N(\boldsymbol{\sigma})$. In the high-temperature regime, we show that bounds on the probability that $f_N(\boldsymbol{\sigma})\in A$ for $\boldsymbol{\sigma}$ randomly sampled from the Gibbs measure can be deduced from similar bounds for deterministic $\boldsymbol{\sigma}$ under the conditional Gaussian law given that $H_N(\boldsymbol{\sigma})/N=E$ for $E$ close to the derivative $F'(\beta)$ of the free energy (which is the typical value of $H_N(\boldsymbol{\sigma})/N$ 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 $q_i$ denoting the locations of the non-zero atoms of the Parisi measure, in addition to specifying that $H_N(\boldsymbol{\sigma})/N=E$, 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 $\langle \mathbf{x}i,\boldsymbol{\sigma}\rangle/N\approx q{i}$. Like in the high-temperature phase, the energy and gradient values on which one conditions are also specified by the model's Parisi measure. As an application, we compute the Franz-Parisi potential at any temperature. Our results are also relevant to the study of Langevin dynamics with initial conditions distributed according to the Gibbs measure, which is one of the main motivations of this work.