Truncated linear statistics associated with the eigenvalues of random matrices II. Partial sums over proper time delays for chaotic quantum dots

Abstract

Invariant ensembles of random matrices are characterized by the distribution of their eigenvalues $\{{\lambda }_1,\cdots ,{\lambda }_N\}$. We study the distribution of truncated linear statistics of the form $\tilde{L}={\sum }_{i=1}^{p}f({\lambda }_i)$ with $p<N$. This problem has been considered by us in a previous paper when the $p$ eigenvalues are further constrained to be the largest ones (or the smallest). In this second paper we consider the same problem without this restriction which leads to a rather different analysis. We introduce a new ensemble which is related, but not equivalent, to the "thinned ensembles" introduced by Bohigas and Pato. This question is motivated by the study of partial sums of proper time delays in chaotic quantum dots, which are characteristic times of the scattering process. Using the Coulomb gas technique, we derive the large deviation function for $\tilde{L}$. Large deviations of linear statistics $L={\sum }_{i=1}^{N}f({\lambda }_i)$ are usually dominated by the energy of the Coulomb gas, which scales as $\sim N^2$, implying that the relative fluctuations are of order $1/N$. For the truncated linear statistics considered here, there is a whole region (including the typical fluctuations region), where the energy of the Coulomb gas is frozen and the large deviation function is purely controlled by an entropic effect. Because the entropy scales as $\sim N$, the relative fluctuations are of order $1/\sqrt{N}$. Our analysis relies on the mapping on a problem of $p$ fictitious non-interacting fermions in $N$ energy levels, which can exhibit both positive and negative effective (absolute) temperatures. We determine the large deviation function characterizing the distribution of the truncated linear statistics, and show that, for the case considered here ($f(\lambda )=1/\lambda$), the corresponding phase diagram is separated in three different phases.

LaTeX, 42 pages, 14 pdf figures , v2: references added

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