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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
DOI: https://doi.org/10.1090/S0002-9947-00-02453-3
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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  • [BVZ] C. Beguin, A. Valette, and A. Zuk, On the spectrum of a random walk on the discrete Heisenberg group and the norm of Harper's operator, J. Geom. Phys. 21 (1997), 337-356. MR 99e:60025
  • [B] J. Bellissard, Noncommutative methods in semiclassical analysis, Springer Lecture Notes in Mathematics 1589, Transition to Chaos (S. Graffi, ed.), (1994). MR 96e:81050
  • [BtD] T. Bröcker and T. tom Dieck, Representations of Compact Lie Groups; Graduate Texts in Math., Springer-Verlag New York, 1985. MR 86i:22023
  • [BdMG] L. Boutet de Monvel and V. Guillemin, The Spectral Theory of Toeplitz Operators; Princeton University Press, Princeton, NJ, 1981. MR 85j:58141
  • [CEY] M-D. Choi, G. A. Elliott, and N. Yui, Gauss polynomials and the rotation algebra, Inventiones Math. 99, 225-246 (1990). MR 91b:46067
  • [CV] Y. Colin de Verdiere, Distribution de points sur une sphère, Sém. Bourbaki 41 (1988-89) no. 703, in Asterisque 177-78, 83-93 (1989). MR 91k:11089
  • [D] J. J. Duistermaat, Fourier Integral Operators; Courant Institute of Mathematical Sciences Lecture Notes, New York, 1973. MR 56:9600
  • [DG] J. J. Duistermaat and V. Guillemin, The spectrum of positive elliptic operators and periodic bicharacteristics; Inventiones Math. 29, 39-79 (1975). MR 53:9307
  • [FH] W. Fulton and J. Harris, Representation Theory, A First Course; Graduate Texts in Mathematics 129, Springer-Verlag, New York, 1991. MR 93a:20069
  • [GSj] A. Grigis and J. Sjöstrand, Microlocal Analysis for Differential Operators; London Math. Society Lecture Note Series 196, Cambridge University Press, 1994. MR 95d:35009
  • [G] V. Guillemin, Lectures on spectral theory of elliptic operators; Duke Math. J. 44 No. 3, 485-517 (1977). MR 56:6758
  • [GS1] V. Guillemin and S. Sternberg, Homogeneous quantization and multiplicities of group representations; J. Funct. Anal. 47 No. 3, 344-380 (1982). MR 84d:58034
  • [GS2] V. Guillemin and S. Sternberg, Geometric quantization and multiplicities of group representations; Inventiones Math. 67 515-538 (1982). MR 83m:58040
  • [GS3] V. Guillemin and S. Sternberg, Geometric Asymptotics; Amer. Math. Soc., Providence, 1977. MR 58:24404
  • [HRV] P. de la Harpe, A. G. Robertson, and A. Valette, On the spectrum of the sum of generators of a finitely generated group, Israel J.Math. 81 (1993), 65-96. MR 94j:22007
  • [Ho] L. Hormander, The Analysis of Linear Partial Differential Operators I-IV; Springer-Verlag, Berlin, Heidelberg, 1983-1985. MR 85g:35002a; MR 85g:35002b; MR 87d:35002a; MR 87d:35002b
  • [KV] V. A. Kaimanovich and A. M. Vershik, Random walks on discrete groups: boundary and entropy; Annals Prob. 11 No. 3, 457-490 (1983). MR 85d:60024
  • [Ke] H. Kesten, Symmetric random walks on groups; Trans. Amer. Math. Soc. 92, 336-354 (1959). MR 22:253
  • [Ki] A. A. Kirillov, Elements of the theory of representations; Grundlehren der mathematischen Wissenschaften 220, Springer-Verlag, Berlin, New York, 1976. MR 54:447
  • [LPS] A. Lubotzky, R. Phillips, and P. Sarnak, Hecke operators and distributing points on the sphere I; Comm. Pure Appl. Math. 39 No. S, suppl., S149-S186 (1986). MR 88m:11025a
  • [N] K. Nomizu, Lie Groups and Differential Geometry; The Mathematical Society of Japan, Tokyo, 1956. MR 18:821d
  • [P] U. Porod, The cut-off phenomenon for random reflections, Annals of Prob. 24 (1996), 74-96. MR 97e:60012
  • [PZ] U. Porod and S. Zelditch, Semiclassical limit of random walks. II, Asymptotic Analysis 18, 215-261 (1998). CMP 99:08
  • [R] J. Rosenthal, Random rotations: characters and random walks on $SO(N)$; Annals Prob. 22 No. 1, 398-423 (1994). MR 95c:60008
  • [Stein] E. M. Stein, Harmonic Analysis, Princeton Mathematical Series, 43, Princeton University Press, Princeton, 1993. MR 95c:42002
  • [Su] T. Sunada, A discrete analogue of periodic magnetic Schrodinger operators, in Geometry of the Spectrum, Amer. Math. Soc. Contemp. Math., 173, (R. Brooks, C. Gordon, P. Perry, eds.) Amer. Math. Soc. (1994). MR 95i:58185
  • [TU] M. E. Taylor and A. Uribe, Semiclassical spectra of gauge fields, J. Funct. Anal. 110 (1992), 1-46. MR 95d:58143
  • [T] F. Treves, Introduction to Pseudodifferential and Fourier Integral Operators. II; Plenum Press, New York, 1980. MR 82i:35173
  • [Z] J. Zak, Magnetic translation group I, II; Phys. Rev. A 134, 1602-1611 (1964). MR 31:2032

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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: https://doi.org/10.1090/S0002-9947-00-02453-3
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

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