Distribution of the spectrum of large non-negative matrices 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 ?
 A: This is a non-centered iid random matrix whose entries have mean one and variance one (and decay exponentially at infinity), and as such, is subject to the circular law with one outlier.  Thus, there will be one eigenvalue roughly near n, and the rest will be more or less uniformly distributed in the complex disk of radius $\sqrt{n}$.  The latter result is due to Chafai (at least for almost all of the eigenvalues), and the former is due to Silverstein.  I discuss some finer aspects of the outlier eigenvalue (and show that all the other eigenvalues are nearly contained in the disk) in a more recent paper.  (See in particular Phillip Wood's Figure 3 in that paper for an example of the eigenvalue distribution of a similar matrix model to the one you propose.)
A: starting from the Gaussian probability measure $\prod_{i,j=1}^{n}e^{-m_{ij}^{2}}$ constrained by $m_{i,j}\geq 0$, I would surmise that the positivity constraint would become irrelevant for eigenvalue correlation functions in the large-$n$ limit; such large-$n$ universality of spectral correlations holds for real symmetric matrices (the Gaussian orthogonal ensemble), and I would expect it to hold for real asymmetric matrices as well.   
without the constraint $m_{i,j}\geq 0$ we have the socalled real Ginibre ensemble of random real asymmetric matrices; the literature on this ensemble is extensive, an entry point could be http://arXiv.org/abs/0706.2020
A: I believe the distribution of eigenvalues of a random symmetric matrix tends towards the Wigner semicircle distribution as $n \rightarrow \infty$.
