Semi-classical limit for random walks

Abstract:

Let $(G, \mu )$ be a discrete symmetric random walk on a compact Lie group $G$ with step distribution $\mu$ and let $T_{\mu }$ be the associated transition operator on $L^2(G)$. The irreducibles $V_{\rho }$ of the left regular representation of $G$ on $L^2(G)$ are finite dimensional invariant subspaces for $T_{\mu }$ and the spectrum of $T_{\mu }$ is the union of the sub-spectra $\sigma (T_{\mu }\upharpoonleft _{V_{\rho }})$ on the irreducibles, which consist of real eigenvalues $\{ \lambda _{\rho 1},...,\lambda _{\rho \dim V_{\rho }}\}$. Our main result is an asymptotic expansion for the spectral measures \[ m_{\rho }^{\mu }(\lambda ) := \frac {1}{\dim V_{\rho }} \sum _{j=1}^{\dim V_{\rho }} \delta (\lambda - \lambda _{\rho j})\] along rays of representations in a positive Weyl chamber $\mathbf {t}^*_+$, i.e. for sequences of representations $k \rho$, $k\in \mathbb {N}$ with $k\rightarrow \infty$. As a corollary we obtain some estimates on the spectral radius of the random walk. We also analyse the fine structure of the spectrum for certain random walks on $U(n)$ (for which $T_{\mu }$ is essentially a direct sum of Harper operators).