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Accueil du site > Seminars > Séminaires théorie > Theory Club Thursday February 27 2020 at 13:30 in room 412B. Aurélien Grabsch: "Truncated linear statistics associated with the top eigenvalues of random matrices and application to an interface model".

## Theory Club Thursday February 27 2020 at 13:30 in room 412B. Aurélien Grabsch: "Truncated linear statistics associated with the top eigenvalues of random matrices and application to an interface model"

#### Unless otherwise stated, seminars and defences take place at 11:30 in room 454A of Condorcet building.

Truncated linear statistics associated with the top eigenvalues of random matrices and application to an interface model

Aurélien Grabsch

Abstract: Since the pioneer work of Wigner, random matrix theory has been applied to many fields. Invariant ensembles, in which eigenvalues and eigenvectors are uncorrelated, have played a prominent role in physical applications. Many important physical observables take the form of linear statistics of eigenvalues $\\lambda_i\_i=1,\ldots,N$, i.e. $L = \sum_i=1^N f(\lambda_i)$, where $N$ is the total number of eigenvalues and $f$ is any given function depending on the physical situation under consideration. We have recently introduced a new type of problem: motivated by the analysis of the statistical physics of fluctuating one-dimensional interfaces, we have studied the distribution of truncated linear statistics $\tildeL=\sum_i=1^K f(\lambda_i)$, where the summation is restricted to the $K < N$ largest eigenvalues. In this talk I will analyse the distribution of these truncated linear statistics, in the limit $N \to \infty$ with $K/N$ fixed, using the Coulomb gas technique. I will show that the constraint that $\tildeL=\sum_i=1^K f(\lambda_i)$ is fixed drives an infinite order phase transition in the underlying Coulomb gas. This transition corresponds to a change in the density of the gas, from a density defined on two disjoint intervals to a single interval. In this latter case the density presents a logarithmic divergence inside the bulk.

Thursday February 27 at 13:30 in room 412B

Contact : Équipe séminaires / Seminar team - Published on / Publié le 24 February 2020