Truncated linear statistics associated with the top eigenvalues of random matrices

DOI10.1007/S10955-017-1755-5zbMATH Open1376.82028arXiv1609.08296OpenAlexW2524500890MaRDI QIDQ2409996

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Publication date: 16 October 2017

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Abstract: Given a certain invariant random matrix ensemble characterised by the joint probability distribution of eigenvalues $P(lambd{a}_1,ldots,lambd{a}_N)$, many important questions have been related to the study of linear statistics of eigenvalues $L=su{m}_{i=1}^{N}f(lambd{a}_i)$, 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: $ildeL=su{m}_{i=1}^{N_1}f(lambd{a}_i)$. 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 $f(lambda)=sqrtlambda$. Using the Coulomb gas technique, we study the $Noinfty$ limit with $N_1/N$ fixed. We show that the constraint that $ildeL=su{m}_{i=1}^{N_1}f(lambd{a}_i)$ 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.

Full work available at URL: https://arxiv.org/abs/1609.08296

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