Spectra of sparse non-hermitian random matrices: an analytical solution.

We present the exact analytical expression for the spectrum of a sparse non-hermitian random matrix ensemble, generalizing two standard results in random-matrix theory: this analytical expression constitutes a non-hermitian version of the Kesten-McKay measure as well as a sparse realization of Girko's elliptic law. Our exact result opens new perspectives in the study of several physical problems modelled on sparse random graphs, which are locally treelike. In this context, we show analytically that the convergence rate of a transport process on a very sparse graph depends in a nonmonotonic way upon the degree of symmetry of the graph edges.