Disordered Gibbs measures and Gaussian conditioning

Abstract

Abstract We study the law of a random field $$f_N(\varvec{\sigma })$$ f N ( σ ) evaluated at a random sample from the Gibbs measure associated to a Gaussian field $$H_N(\varvec{\sigma })$$ H N ( σ ) . In the high-temperature regime, we show that bounds on the probability that $$f_N(\varvec{\sigma })\in A$$ f N ( σ ) ∈ A for $$\varvec{\sigma }$$ σ randomly sampled from the Gibbs measure can be deduced from similar bounds for deterministic $$\varvec{\sigma }$$ σ under the conditional Gaussian law given that $$H_N(\varvec{\sigma })/N=E$$ H N ( σ ) / N = E for E close to the derivative $$F'(\beta )$$ F ′ ( β ) of the free energy (which is the typical value of $$H_N(\varvec{\sigma })/N$$ H N ( σ ) / 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 $$q_i$$ q i denoting the locations of the non-zero atoms of the Parisi measure, in addition to specifying that $$H_N(\varvec{\sigma })/N=E$$ H N ( σ ) / N = E , here one needs to also condition on the energy and its gradient at points $$\textbf{x}_1,\ldots ,\textbf{x}_k$$