Distribution of spectral linear statistics on random matrices beyond the large deviation function -- Wigner time delay in multichannel disordered wires

Abstract

An invariant ensemble of $N\times N$ random matrices can be characterised by a joint distribution for eigenvalues $P({\lambda }_1,\cdots ,{\lambda }_N)$. The study of the distribution of linear statistics, i.e. of quantities of the form $L=(1/N){\sum }_{i}f({\lambda }_i)$ where $f(x)$ is a given function, appears in many physical problems. In the $N\to \infty$ limit, $L$ scales as $L\sim N^{\eta }$, where the scaling exponent $\eta$ depends on the ensemble and the function $f$. Its distribution can be written under the form $P_N(s=N^{-\eta }L)\simeq A_{\beta ,N}(s)\exp \{-(\beta N^2/2)\Phi (s)\}$, where $\beta \in \{1,2,4\}$ is the Dyson index. The Coulomb gas technique naturally provides the large deviation function $\Phi (s)$, which can be efficiently obtained thanks to a "thermodynamic identity" introduced earlier. We conjecture the pre-exponential function $A_{\beta ,N}(s)$. We check our conjecture on several well controlled cases within the Laguerre and the Jacobi ensembles. Then we apply our main result to a situation where the large deviation function has no minimum (and $L$ has infinite moments)~: this arises in the statistical analysis of the Wigner time delay for semi-infinite multichannel disordered wires (Laguerre ensemble). The statistical analysis of the Wigner time delay then crucially depends on the pre-exponential function $A_{\beta ,N}(s)$, which ensures the decay of the distribution for large argument.

LaTeX , 30 pages , 12 pdf figures ; v2: paper reorganised, conclusion extended and refs. added

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