TITLE: Powers of random matrices
QUESTION [18 upvotes]: Let $M$ be an $n \times n$ matrix whose elements are random reals in [0,1].
Two questions.

 What is the growth rate of the magnitude of the elements of $M^k$ as a function of $k$?  It is definitely
exponential, but maybe the exponent is known?
 Is it the case that eventually one element of $M^k$ dominates, as $k \rightarrow \infty$?
I have some ambiguous experimental evidence that this is the case, but because
of the exponential growth, exact computation is difficult, rendering my "evidence" tenuous
at best and perhaps worthless.

One can ask the same question for matrices whose elements are random reals in [-1,1],
or random 0's and 1's, or random choices among $\lbrace -1, 0, 1\rbrace$, ...
These question have likely been studied.  Thanks for pointers and/or ideas!

REPLY [5 votes]: I don't have enough reputation to leave a comment, so here goes....
As other have observed, the growth rate of $M^k$ is determined by the largest eigenvalue of $M$. I just want to note that for a nonnegative matrix, the largest eigenvalue is always 
between the largest and smallest column sum. 
That, in principle, could allow you to obtain upper and lower bounds by trying to bound how big or small the column sums usually get for the random matrix you described.