## 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 R^{N} of the form XN(x)+μ2‖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.

Original language | English (US) |
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Pages (from-to) | 951-993 |

Number of pages | 43 |

Journal | Communications in Mathematical Physics |

Volume | 402 |

Issue number | 1 |

DOIs | |

State | Published - Aug 2023 |

### Funding

Antonio Auffinger: research partially supported by NSF Grant CAREER DMS-1653552 and NSF Grant DMS-1517894. Qiang Zeng: research partially supported by SRG 2020-00029-FST and FDCT 0132/2020/A3.

## ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics