Orthogonal polynomials, random matrices and the numerical inversion of Laplace transform of positive functions

A method for the numerical inversion of the Laplace transform of a continuous positive function $f(t)$ is proposed. Random matrices distributed according to a Gibbs law whose energy $V(x)$ is a function of $f(t)$ are considered as well as random polynomials orthogonal with respect to $w(x)=e^{-V(x)}$. The equation relating $w(x)$ to the reproducing kernel and to the condensed density of the roots of the random orthogonal polynomials is exploited. Basic results from the theories of orthogonal polynomials, random matrices and random polynomials are revisited in order to provide a unified and almost self--contained context. The qualitative behavior of the solutions provided by the proposed method is illustrated by numerical examples and discussed by using logarithmic potentials with external fields that give insight into the asymptotic behavior of the condensed density when the number of data points goes to infinity.