### Abstract

Can one count the number of critical points for random smooth functions of many variables? How complex is a typical random smooth function? How complex is the topology of its level sets? We study here the simplest case of smooth Gaussian random functions defined on the sphere in high dimensions. We show that such a randomly chosen smooth function is very complex, i.e. that its number of critical points of given index is exponentially large. We also study the topology of the level sets of these functions, and give sharp estimates of their Euler characteristic. This study, which is a joint work with Tuca Auffinger (Chicago) and partly with Jiri Cerny (Vienna), relies rather surprisingly on Random Matrix Theory, through the use of the classical Kac-Rice formula. The main motivation comes from the study of energy landscapes for general spherical spin-glasses. I will detail the interesting picture we get for the complexity of these random Hamiltonians, for the bottom of the energy landscape, and in particular a strong correlation between the index and the critical value. We also propose a new invariant for the possible transition between the 1-step replica symmetry breaking and a Full Replica symmetry breaking scheme.

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

Title of host publication | XVIIth International Congress on Mathematical Physics: Aalborg, Denmark, 6-11 August 2012 |

Publisher | World Scientific Publishing Co. |

Pages | 403 |

Number of pages | 1 |

ISBN (Electronic) | 9789814449243 |

ISBN (Print) | 9789814449236 |

DOIs | |

State | Published - Jan 1 2013 |

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### ASJC Scopus subject areas

- Physics and Astronomy(all)

### Cite this

*XVIIth International Congress on Mathematical Physics: Aalborg, Denmark, 6-11 August 2012*(pp. 403). World Scientific Publishing Co.. https://doi.org/10.1142/9789814449243_0033

**Complexity of random energy landscapes.** / Ben Arous, Gerard.

Research output: Chapter in Book/Report/Conference proceeding › Chapter

*XVIIth International Congress on Mathematical Physics: Aalborg, Denmark, 6-11 August 2012.*World Scientific Publishing Co., pp. 403. https://doi.org/10.1142/9789814449243_0033

}

TY - CHAP

T1 - Complexity of random energy landscapes

AU - Ben Arous, Gerard

PY - 2013/1/1

Y1 - 2013/1/1

N2 - Can one count the number of critical points for random smooth functions of many variables? How complex is a typical random smooth function? How complex is the topology of its level sets? We study here the simplest case of smooth Gaussian random functions defined on the sphere in high dimensions. We show that such a randomly chosen smooth function is very complex, i.e. that its number of critical points of given index is exponentially large. We also study the topology of the level sets of these functions, and give sharp estimates of their Euler characteristic. This study, which is a joint work with Tuca Auffinger (Chicago) and partly with Jiri Cerny (Vienna), relies rather surprisingly on Random Matrix Theory, through the use of the classical Kac-Rice formula. The main motivation comes from the study of energy landscapes for general spherical spin-glasses. I will detail the interesting picture we get for the complexity of these random Hamiltonians, for the bottom of the energy landscape, and in particular a strong correlation between the index and the critical value. We also propose a new invariant for the possible transition between the 1-step replica symmetry breaking and a Full Replica symmetry breaking scheme.

AB - Can one count the number of critical points for random smooth functions of many variables? How complex is a typical random smooth function? How complex is the topology of its level sets? We study here the simplest case of smooth Gaussian random functions defined on the sphere in high dimensions. We show that such a randomly chosen smooth function is very complex, i.e. that its number of critical points of given index is exponentially large. We also study the topology of the level sets of these functions, and give sharp estimates of their Euler characteristic. This study, which is a joint work with Tuca Auffinger (Chicago) and partly with Jiri Cerny (Vienna), relies rather surprisingly on Random Matrix Theory, through the use of the classical Kac-Rice formula. The main motivation comes from the study of energy landscapes for general spherical spin-glasses. I will detail the interesting picture we get for the complexity of these random Hamiltonians, for the bottom of the energy landscape, and in particular a strong correlation between the index and the critical value. We also propose a new invariant for the possible transition between the 1-step replica symmetry breaking and a Full Replica symmetry breaking scheme.

UR - http://www.scopus.com/inward/record.url?scp=84974823903&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84974823903&partnerID=8YFLogxK

U2 - 10.1142/9789814449243_0033

DO - 10.1142/9789814449243_0033

M3 - Chapter

AN - SCOPUS:84974823903

SN - 9789814449236

SP - 403

BT - XVIIth International Congress on Mathematical Physics: Aalborg, Denmark, 6-11 August 2012

PB - World Scientific Publishing Co.

ER -