Random matrices

A Markov chain arising from Gaussian elimination

Consider a Markov chain on the set of n by n invertible matrices mod 2, where in each step, two distinct rows are chosen uniformly at random, and the first row is added to the second mod 2. How does the mixing time grow with n? And how does it change if only test functions that are computable in polynomial time are allowed?

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Monotonicity of Lyapunov exponents

Consider a random matrix product where each factor is a matrix A or its transpose, chosen independently with probabilities p and 1-p, respectively. How does the growth rate of the product depend on p?

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