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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Semi-classical limit for random walks


Authors: Ursula Porod and Steve Zelditch
Journal: Trans. Amer. Math. Soc. 352 (2000), 5317-5355
MSC (1991): Primary 60B15, 60J15, 22E30; Secondary 58F06
Published electronically: May 12, 2000
MathSciNet review: 1650038
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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

\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 $U(n)$ (for which $T_{\mu}$ is essentially a direct sum of Harper operators).


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Additional Information

Ursula Porod
Affiliation: Department of Mathematics, University of California, Berkeley, California 94720

Steve Zelditch
Affiliation: Department of Mathematics, Johns Hopkins University, Baltimore, Maryland 21218

DOI: http://dx.doi.org/10.1090/S0002-9947-00-02453-3
PII: S 0002-9947(00)02453-3
Received by editor(s): December 12, 1997
Received by editor(s) in revised form: August 25, 1998
Published electronically: May 12, 2000
Additional Notes: Supported by the Miller Institute for Basic Research in Science and partially by NSF grant #DMS-9404637.
Article copyright: © Copyright 2000 American Mathematical Society