Complexity of Gaussian random fields with isotropic increments

2020/07/15 by Antonio Auffinger, Qiang Zeng, Auffinger, Antonio +1 · 1 citation
Mathematics · #FOS: Mathematics #FOS: Physical sciences #Geometry and complex manifolds #Mathematical Physics (math-ph) #Probability (math.PR) #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2007.07668

openalex publication_date 2020/07/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the energy landscape of a model of a single particle on a random potential, that is, we investigate the topology of level sets of smooth random fields on \mathbb R^N of the form X_N(x) +\fracμ2 ‖x‖², where X_N is a Gaussian process with isotropic increments. We derive asymptotic formulas for the mean number of critical points with critical values in an open set as the dimension N goes to infinity. In a companion paper, we provide the same analysis for the number of critical points with a given index.

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