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An invariant ensemble of $N\times N$ random matrices can be characterised bya joint distribution for eigenvalues $P(\lambda_1,\cdots,\lambda_N)$. The studyof the distribution of linear statistics, i.e. of quantities of the form$L=(1/N)\sum_if(\lambda_i)$ where $f(x)$ is a given function, appears in manyphysical 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 $P_N(s=N^{-\eta}\,L)\simeqA_{\beta,N}(s)\,\exp\big\{-(\beta N^2/2)\,\Phi(s)\big\}$, where$\beta\in\{1,\,2,\,4\}$ is the Dyson index. The Coulomb gas technique naturallyprovides the large deviation function $\Phi(s)$, which can be efficientlyobtained thanks to a "thermodynamic identity" introduced earlier. We conjecturethe pre-exponential function $A_{\beta,N}(s)$. We check our conjecture onseveral well controlled cases within the Laguerre and the Jacobi ensembles.Then we apply our main result to a situation where the large deviation functionhas no minimum (and $L$ has infinite moments)~: this arises in the statisticalanalysis of the Wigner time delay for semi-infinite multichannel disorderedwires (Laguerre ensemble). The statistical analysis of the Wigner time delaythen crucially depends on the pre-exponential function $A_{\beta,N}(s)$, whichensures the decay of the distribution for large argument.

journal of physics. a, mathematical and theoretical

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