# YEP XVIII: "Spectra of random graphs and related combinatorial problems"

## Mar 27 - Mar 31

#### Summary

Since the first workshop in 2004, the Young European Probabilists (YEP) series has spawned 17 highly successful editions on a wide range of topics.

The theme of its 18th edition, will be spectra of random graphs, random matrices and related combinatorial problems.

The study of matrices with random entries started in the 1950’s and has grown into an immense body of literature until today. While the initial focus was on problems in statistical physics, in recent times random matrices have proven to be an important tool also in a variety of other fields like statistics, network analysis, image processing or machine learning. Moreover, there has been great progess in the study of matrices which arise naturally in random graphs, like the adjacency matrix, the Laplacian matrix, or the transition matrix of the random walk on the graph. The recent theoretical advances in this area are remarkable and one of the key goals of the workshop is to understand the information contained in eigenvalues and eigenvectors of high-dimensional random matrices. A second point of focus are applications of tools stemming from random graphs theory that can be used to study the spectrum of random matrices.

In this workshop, we provide a platform where young and more senior researchers from the area of random matrices, random graphs and related topics can come together and exchange their research, find new collaborations and learn about different perspectives on the topic.

#### Organizers

Alessandra Cipriani | UCL |

Rajat Subhra Hazra | Leiden University |

Matteo Quattropani | Sapienza Università di Roma |

Noela Müller | TU Eindhoven |

**Scientific advisory committee**

Remco van der Hofstad | TU Eindhoven |

Frank den Hollander | Leiden University |

#### Speakers

**Mini-course Speakers**

Alice Guionnet | ENS de Lyon UMPA |

Antti Knowles | University of Geneva |

Justin Salez | Université Paris-Dauphine - CEREMADE |

**Invited Speakers**

Johannes Alt | University of Geneva |

Luisa Andreis | Politecnico di Milano |

Jean Barbier | ICTP |

Simon Coste | University of Paris |

Pierfrancesco Dionigi | Leiden University |

Luca Ganassali | Inria Paris |

Johannes Heiny | Ruhr University Bochum |

Ivan Kryven | Utrecht University |

Camille Male | University of Bordeaux |

Jaron Sanders | TU Eindhoven |

Yizhe Zhu | University of California Irvine |

#### Schedule

The workshop is starting on Monday, March 27, at 9:30 (registration) and is ending on Friday, March 31, at lunchtime. A more detailed schedule is coming soon.

#### Abstracts

**Alice Guionnet, Large deviation for random matrices**In this mini-course I will discuss the theory of large deviations for the spectrum of large random matrices. The eigenvalues of random matrices are complicated functions of their entries, there is not yet a complete theory to estimate the probability of rare events for the spectrum of large random matrices. We will review known results and open problems.

**Antti Knowles, Spectral phases of Erdős-Rényi graphs**

In this minicourse I give an overview of spectral phases of the Erdős-Rényi graph G(N,p). At the critical regime pN \sim\log N and below, the graph is inhomogeneous and presents structures such as hubs and leaves. On the spectral side, this leads to the appearance and coexistence of several phases distinguished by the spatial structure of the eigenvectors. The aim of this lecture is to give an overview of the phase diagram and to go over the main ideas of the proofs in the different phases.

**Justin Salez, *** New applications of local weak convergence
*In the sparse regime, many natural (deterministic or random) graph sequences happen to converge in the

*local weak sense*, a notion first introduced by Benjamini & Schramm and later developped further by Aldous, Lyons & Steele. The limiting objects are probability measures on rooted graphs enjoying a certain form of stationarity known as

*unimodularity*. Those local weak limits are often much more convenient to work with than the finite-graph sequences that they approximate, and they have been shown to capture, in a unified way, the asymptotic behavior of a number of important combinatorial or spectral graph parameters. In this course, I will provide a self-contained introduction to this powerful framework, and illustrate it with a few modern applications. In particular, I will present a recent solution to a classical open question of Milman & Naor on the relation between discrete curvature and spectral expansion.

#### Poster abstracts

Click here for the pdf of poster abstracts.

#### Sponsors

#### Registration

THE REGISTRATION FOR THIS WORKSHOP IS CLOSED FOR PARTICIPANTS

SPEAKERS CAN REGISTER USING THIS: Link