Given a Gaussian energy function H(x) and another random process O(x) (an observable) both defined on the same configuration space, what is the law of O(y) for y sampled from the Gibbs measure associated to H(x)? We will see the answer to this question in the high-temperature phase in a general setting, and in the more challenging low-temperature phase for spherical spin glasses. For both, the answer will be given in terms of the law of O(x) for deterministic x, conditional on an appropriate event. In the former case the conditioning will only involve the value of the energy H(x) at the same point x. In the latter, we additionally need to specify the energy and its derivatives over a sequence of critical points with a certain geometry.

This is based on joint work with Amir Dembo.

Abstracts from: https://statistics.stanford.edu/events/probability-seminar