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

An invariant ensemble of N× N random matrices can be characterised by a joint distribution for eigenvalues P(λ₁,⋯,λN). The study of the distribution of linear statistics, i.e. of quantities of the form L=(1/N)∑ᵢf(λᵢ) where $f(x)$ is a given function, appears in many physical problems. In the N→∞ limit, L scales as L~ N^η, where the scaling exponent η depends on the ensemble and the function f. Its distribution can be written under the form PN(s=N-η\,L) Aβ,N(s)\,exp\-(β N²/2)\,Φ(s)\, where β∈\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 Aβ,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 Aβ,N(s), which ensures the decay of the distribution for large argument.