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

### Florent Jouve

Université Paris-Sud 11, Orsay, France

## 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 $Ω(n,m)(Z)$.

## Cite this article

Florent Jouve, The large sieve and random walks on left cosets of arithmetic groups. Comment. Math. Helv. 85 (2010), no. 3, pp. 647–704

DOI 10.4171/CMH/207