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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2024 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Semi-classical limit for random walks
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by Ursula Porod and Steve Zelditch
Trans. Amer. Math. Soc. 352 (2000), 5317-5355
DOI: https://doi.org/10.1090/S0002-9947-00-02453-3
Published electronically: May 12, 2000

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).
References
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Bibliographic 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
  • MR Author ID: 186875
  • 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.
  • © Copyright 2000 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 352 (2000), 5317-5355
  • MSC (1991): Primary 60B15, 60J15, 22E30; Secondary 58F06
  • DOI: https://doi.org/10.1090/S0002-9947-00-02453-3
  • MathSciNet review: 1650038