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Semi-classical limit for random walks

Authors:	Ursula Porod Steve Zelditch

Institution:	Department of Mathematics, University of California, Berkeley, California 94720 Steve Zelditch ; Department of Mathematics, Johns Hopkins University, Baltimore, Maryland 21218

Abstract:	Let $(G, \mu)$ be a discrete symmetric random walk on a compact Lie group with step distribution $\mu$ and let $T_{\mu}$ be the associated transition operator on . The irreducibles $V_{\rho}$ of the left regular representation of on 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 $\begin{displaymath}m_{\rho}^{\mu}(\lambda) := \frac{1}{\dim V_{\rho}} \sum_{j=1}^{\dim V_{\rho}} \delta(\lambda - \lambda_{\rho j})\end{displaymath}$ 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 (for which $T_{\mu}$ is essentially a direct sum of Harper operators).

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