Abstract:
Eric Rains proved surprising results about the distribution of Xp where X is a Haar-distributed random element of a compact Lie group and p is a natural number. For example, when X is a random unitary matrix of size n, the eigenvalues of Xp are distributed like the eigenvalues of p independent random unitary matrices of sizes ≈ n/p. We use these results to prove bounds on the Fourier coefficients of the distribution of Xp. As an application, we prove a dimension-independent mixing result. Namely, we show that for every p there is some natural t, such that the product of t independent random p-powers in every compact connected Lie group is approximately uniform (measured in the L∞-distance).
