This question is related to that of Thurston. However, I am not interested in algebraic integers, and I wish to focus on random matrices instead of random polynomials.

When considering (entrywise) non-negative matrices $M$, a natural probability measure seems to be $$\prod_{i,j=1}^ne^{-m_{ij}}dm_{ij}.$$ By Perron-Frobenius theorem, the spectral radius $\rho(M)$ is an eigenvalue, associated to a non-negative eigenvector. Almost surely, $M$ is positive and therefore this eigenvalue is simple and its eigenvector is positive.

What is the distribution of the eigenvalues of $M$ as the size $n$ goes to infinity ? What is the relevant normalization ? Should we consider $\lambda/\rho(M)$ or $\lambda/\sqrt{\rho(M)}$ or something else ? Is it the same asymptotics as in the case of the conjugates of the algebraic Perron numbers considered by Thurston ?

Note that because of the constraint $m_{ij}\ge0$, an exponential law looked more natural to me than a Gaussian. Has anyone an other suggestion of probability over non-negative matrices ?