### Abstract

We analyze the landscape of general smooth Gaussian functions on the sphere in dimension N, when N is large. We give an explicit formula for the asymptotic complexity of the mean number of critical points of finite and diverging index at any level of energy and for the mean Euler characteristic of level sets. We then find two possible scenarios for the bottom landscape, one that has a layered structure of critical values and a strong correlation between indexes and critical values and another where even at levels below the limiting ground state energy the mean number of local minima is exponentially large. We end the paper by discussing how these results can be interpreted in the language of spin glasses models.

Original language | English (US) |
---|---|

Pages (from-to) | 4214-4247 |

Number of pages | 34 |

Journal | Annals of Probability |

Volume | 41 |

Issue number | 6 |

DOIs | |

State | Published - Nov 2013 |

### Fingerprint

### Keywords

- Critical points
- Parisi formula
- Random matrices
- Sample
- Spin glasses

### ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Cite this

*Annals of Probability*,

*41*(6), 4214-4247. https://doi.org/10.1214/13-AOP862

**Complexity of random smooth functions on the high-dimensional sphere.** / Auffinger, Antonio; Arous, Gerard Ben.

Research output: Contribution to journal › Article

*Annals of Probability*, vol. 41, no. 6, pp. 4214-4247. https://doi.org/10.1214/13-AOP862

}

TY - JOUR

T1 - Complexity of random smooth functions on the high-dimensional sphere

AU - Auffinger, Antonio

AU - Arous, Gerard Ben

PY - 2013/11

Y1 - 2013/11

N2 - We analyze the landscape of general smooth Gaussian functions on the sphere in dimension N, when N is large. We give an explicit formula for the asymptotic complexity of the mean number of critical points of finite and diverging index at any level of energy and for the mean Euler characteristic of level sets. We then find two possible scenarios for the bottom landscape, one that has a layered structure of critical values and a strong correlation between indexes and critical values and another where even at levels below the limiting ground state energy the mean number of local minima is exponentially large. We end the paper by discussing how these results can be interpreted in the language of spin glasses models.

AB - We analyze the landscape of general smooth Gaussian functions on the sphere in dimension N, when N is large. We give an explicit formula for the asymptotic complexity of the mean number of critical points of finite and diverging index at any level of energy and for the mean Euler characteristic of level sets. We then find two possible scenarios for the bottom landscape, one that has a layered structure of critical values and a strong correlation between indexes and critical values and another where even at levels below the limiting ground state energy the mean number of local minima is exponentially large. We end the paper by discussing how these results can be interpreted in the language of spin glasses models.

KW - Critical points

KW - Parisi formula

KW - Random matrices

KW - Sample

KW - Spin glasses

UR - http://www.scopus.com/inward/record.url?scp=84888417867&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84888417867&partnerID=8YFLogxK

U2 - 10.1214/13-AOP862

DO - 10.1214/13-AOP862

M3 - Article

AN - SCOPUS:84888417867

VL - 41

SP - 4214

EP - 4247

JO - Annals of Probability

JF - Annals of Probability

SN - 0091-1798

IS - 6

ER -