The large sieve and random walks on left cosets of arithmetic groups

Abstract

Building on Kowalski’s work on random walks on the groups SL(n,ℤ) and Sp(2g,ℤ), we consider similar problems (we try to estimate the probability with which, after k steps, the matrix obtained has a characteristic polynomial with maximal Galois group or has no nonzero squares among its entries) for more general classes of sets: in GL(n,A), where A is a subring of ℚ containing ℤ that we specify, we perform a random walk on the set of matrices with fixed determinant D ∈ A×. We also investigate the case where the set involved is any of the two left cosets of the special orthogonal group SO(n,m)(ℤ) with respect to the spinorial kernel Ω(n,m)(ℤ).