Upper and Lower Bounds on Expected Soft Maxima of Gaussian Processes

Abstract

We obtain upper and lower bounds for "smoothed" versions of the expected supremum of centered Gaussian processes with finite or countable index sets. These so-called soft maxima are computed in terms of expected values of random Gibbs averages at inverse temperature $\beta > 0$ and reduce to expected suprema in the zero-temperature limit $\beta \to \infty$. Our analysis builds on ideas from statistical physics and information theory, and relies crucially on the tensorization technique introduced recently by Liu (2025). The bounds retain the same multiscale structure as in the expressions for the expected supremum derived using the method of generic chaining, with a truncation term governed by the inverse temperature $\beta$. In the zero-temperature limit, we recover the majorizing measure theorem. As an illustrative example, we apply our results to the analysis of the quenched free energy in the Sherrington-Kirkpatrick model and obtain a Parisi formula in the finite system size setting.