Truncated linear statistics associated with the top eigenvalues of random matrices

Abstract

Given a certain invariant random matrix ensemble characterised by the joint probability distribution of eigenvalues $P({\lambda }_1,\dots ,{\lambda }_N)$, many important questions have been related to the study of linear statistics of eigenvalues $L={\sum }_{i=1}^{N}f({\lambda }_i)$, where $f(\lambda )$ is a known function. We study here truncated linear statistics where the sum is restricted to the $N_1<N$ largest eigenvalues: $\tilde{L}={\sum }_{i=1}^{N_1}f({\lambda }_i)$. Motivated by the analysis of the statistical physics of fluctuating one-dimensional interfaces, we consider the case of the Laguerre ensemble of random matrices with $f(\lambda )=\sqrt{\lambda }$. Using the Coulomb gas technique, we study the $N\to \infty$ limit with $N_1/N$ fixed. We show that the constraint that $\tilde{L}={\sum }_{i=1}^{N_1}f({\lambda }_i)$ 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. Assuming that $f(\lambda )$ is monotonous, we show that these features arise for any random matrix ensemble and truncated linear statitics, which makes the scenario described here robust and universal.

LaTeX, 30 pages, 20 pdf figures. Updated version: a typo has been corrected in Eq. (3.30) and more details are provided in the Appendix

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