Truncated linear statistics of eigenvalues

Aurélien Grabsch

Mon, Nov. 27th 2017, 11:00-12:00

Salle Claude Itzykson, Bât. 774, Orme des Merisiers

Since the pioneer work of Wigner, random matrix theory have been applied to many fields in physics. Invariant ensembles have played a prominent role in physical applications, when eigenvalues and eigenvectors are uncorrelated. 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 \textit{truncated} linear statistics $\tilde{L}=\sum_{i=1}^K f(\lambda_i)$, where the summation is restricted to number $K < N$ of the eigenvalues. In this talk, I will analyse two cases: first the case where the linear statistics is restricted to the largest (or smallest) $K$ eigenvalues. Then, I will discuss the case where any eigenvalue can contribute to the truncated linear statistics, without restriction on the ordering. I will show that, in a certain regime, this last problem can be mapped onto a system of non-interacting fermions which can have either positive or negative temperature. \\ \\ Refs: \\ - Truncated linear statistics associated with the top eigenvalues of random matrices. Aurélien Grabsch, Satya N. Majumdar and Christophe Texier. J. Stat. Phys. 167(2), 234-259 (2017) \\ - Truncated linear statistics associated with the eigenvalues of random matrices II. Partial sums over proper time delays for chaotic quantum dots. Aurélien Grabsch, Satya N. Majumdar and Christophe Texier. J. Stat. Phys. 167(2) 1452-1488 (2017)

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