Distribution Of Linear Statistics Research Articles

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 . The distribution of linear statistics, i.e. of quantities of the form where f(x) is a given function, appears in many physical problems. In the limit, L scales as , where the scaling exponent η depends on the ensemble and the function f(x). Its distribution can be written in the form , where is the Dyson index. The Coulomb gas technique naturally provides the large deviation function , which can be efficiently obtained thanks to a ‘thermodynamic identity’ introduced earlier. We conjecture the pre-exponential function . 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 , which ensures the decay of the distribution for large argument.

Reconstruction of the General Distribution by the Distribution of Some Statistics

We investigate the problem of characterizing the distribution of independent identically distributed random variables $X_1,\ldots,X_m$ (general distribution) by the distribution of linear statistics and statistics of maximum with positive coefficients. Necessary and sufficient conditions are found under which such a characterization takes place.