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 $P(λ_1,\cdots,λ_N)$. The study of the distribution of linear statistics, i.e. of quantities of the form $L=(1/N)\sum_if(λ_i)$ where $f(x)$ is a given function, appears in many physical problems. In the $N\to\infty$ limit, $L$ scales as $L\sim N^η$, where the scaling exponent $η$ depends on the ensemble and the function $f$. Its distribution can be written under the form $P_N(s=N^{-η}\,L)\simeq A_{β,N}(s)\,\exp\big\{-(βN^2/2)\,Φ(s)\big\}$, where $β\in\{1,\,2,\,4\}$ is the Dyson index. The Coulomb gas technique naturally provides the large deviation function $Φ(s)$, which can be efficiently obtained thanks to a "thermodynamic identity" introduced earlier. We conjecture the pre-exponential function $A_{β,N}(s)$. 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 $A_{β,N}(s)$, which ensures the decay of the distribution for large argument.