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!