Disordered Gibbs measures and Gaussian conditioning

We study the law of a random field fN(σ ) f N ( σ ) evaluated at a random sample from the Gibbs measure associated to a Gaussian field HN(σ ) H N ( σ ) . In the high-temperature regime, we show that bounds on the probability that fN(σ )∈ A f N ( σ ) ∈ A for σ σ randomly sampled from the Gibbs measure can be deduced from similar bounds for deterministic σ σ under the conditional Gaussian law given that HN(σ )/N=E H N ( σ ) / N = E for E close to the derivative F'(β ) F ′ ( β ) of the free energy (which is the typical value of HN(σ )/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 qᵢ q i denoting the locations of the non-zero atoms of the Parisi measure, in addition to specifying that HN(σ )/N=E H N ( σ ) / N = E , here one needs to also condition on the energy and its gradient at points x₁,… ,xₖ