Semi-classical limit for random walks
HTML articles powered by AMS MathViewer
- 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
- PDF | Request permission
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
- Cédric Béguin, Alain Valette, and Andrzej 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), no. 4, 337–356. MR 1436310, DOI 10.1016/S0393-0440(96)00024-1
- Jean Bellissard, Noncommutative methods in semiclassical analysis, Transition to chaos in classical and quantum mechanics (Montecatini Terme, 1991) Lecture Notes in Math., vol. 1589, Springer, Berlin, 1994, pp. 1–64. MR 1323220, DOI 10.1007/BFb0074074
- Theodor Bröcker and Tammo tom Dieck, Representations of compact Lie groups, Graduate Texts in Mathematics, vol. 98, Springer-Verlag, New York, 1985. MR 781344, DOI 10.1007/978-3-662-12918-0
- L. Boutet de Monvel and V. Guillemin, The spectral theory of Toeplitz operators, Annals of Mathematics Studies, vol. 99, Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1981. MR 620794, DOI 10.1515/9781400881444
- Man Duen Choi, George A. Elliott, and Noriko Yui, Gauss polynomials and the rotation algebra, Invent. Math. 99 (1990), no. 2, 225–246. MR 1031901, DOI 10.1007/BF01234419
- Yves Colin de Verdière, Distribution de points sur une sphère (d’après Lubotzky, Phillips et Sarnak), Astérisque 177-178 (1989), Exp. No. 703, 83–93 (French). Séminaire Bourbaki, Vol. 1988/89. MR 1040569
- J. J. Duistermaat, Fourier integral operators, Courant Institute of Mathematical Sciences, New York University, New York, 1973. Translated from Dutch notes of a course given at Nijmegen University, February 1970 to December 1971. MR 0451313
- J. J. Duistermaat and V. W. Guillemin, The spectrum of positive elliptic operators and periodic bicharacteristics, Invent. Math. 29 (1975), no. 1, 39–79. MR 405514, DOI 10.1007/BF01405172
- William Fulton and Joe Harris, Representation theory, Graduate Texts in Mathematics, vol. 129, Springer-Verlag, New York, 1991. A first course; Readings in Mathematics. MR 1153249, DOI 10.1007/978-1-4612-0979-9
- Alain Grigis and Johannes Sjöstrand, Microlocal analysis for differential operators, London Mathematical Society Lecture Note Series, vol. 196, Cambridge University Press, Cambridge, 1994. An introduction. MR 1269107, DOI 10.1017/CBO9780511721441
- Victor Guillemin, Lectures on spectral theory of elliptic operators, Duke Math. J. 44 (1977), no. 3, 485–517. MR 448452
- V. Guillemin and S. Sternberg, Homogeneous quantization and multiplicities of group representations, J. Functional Analysis 47 (1982), no. 3, 344–380. MR 665022, DOI 10.1016/0022-1236(82)90111-2
- V. Guillemin and S. Sternberg, Geometric quantization and multiplicities of group representations, Invent. Math. 67 (1982), no. 3, 515–538. MR 664118, DOI 10.1007/BF01398934
- Victor Guillemin and Shlomo Sternberg, Geometric asymptotics, Mathematical Surveys, No. 14, American Mathematical Society, Providence, R.I., 1977. MR 0516965, DOI 10.1090/surv/014
- Pierre de la Harpe, A. Guyan Robertson, and Alain Valette, On the spectrum of the sum of generators for a finitely generated group, Israel J. Math. 81 (1993), no. 1-2, 65–96. MR 1231179, DOI 10.1007/BF02761298
- Alfred Rosenblatt, Sur les points singuliers des équations différentielles, C. R. Acad. Sci. Paris 209 (1939), 10–11 (French). MR 85
- V. A. Kaĭmanovich and A. M. Vershik, Random walks on discrete groups: boundary and entropy, Ann. Probab. 11 (1983), no. 3, 457–490. MR 704539, DOI 10.1214/aop/1176993497
- Harry Kesten, Symmetric random walks on groups, Trans. Amer. Math. Soc. 92 (1959), 336–354. MR 109367, DOI 10.1090/S0002-9947-1959-0109367-6
- A. A. Kirillov, Elements of the theory of representations, Grundlehren der Mathematischen Wissenschaften, Band 220, Springer-Verlag, Berlin-New York, 1976. Translated from the Russian by Edwin Hewitt. MR 0412321, DOI 10.1007/978-3-642-66243-0
- A. Lubotzky, R. Phillips, and P. Sarnak, Hecke operators and distributing points on the sphere. I, Comm. Pure Appl. Math. 39 (1986), no. S, suppl., S149–S186. Frontiers of the mathematical sciences: 1985 (New York, 1985). MR 861487, DOI 10.1002/cpa.3160390710
- Cahit Arf, Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper, J. Reine Angew. Math. 181 (1939), 1–44 (German). MR 18, DOI 10.1515/crll.1940.181.1
- Ursula Porod, The cut-off phenomenon for random reflections, Ann. Probab. 24 (1996), no. 1, 74–96. MR 1387627, DOI 10.1214/aop/1042644708
- U. Porod and S. Zelditch, Semiclassical limit of random walks. II, Asymptotic Analysis 18, 215–261 (1998).
- Jeffrey S. Rosenthal, Random rotations: characters and random walks on $\textrm {SO}(N)$, Ann. Probab. 22 (1994), no. 1, 398–423. MR 1258882
- Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR 1232192
- Toshikazu Sunada, A discrete analogue of periodic magnetic Schrödinger operators, Geometry of the spectrum (Seattle, WA, 1993) Contemp. Math., vol. 173, Amer. Math. Soc., Providence, RI, 1994, pp. 283–299. MR 1298211, DOI 10.1090/conm/173/01831
- Ming Liao, Stochastic flows on the boundaries of Lie groups, Stochastics Stochastics Rep. 39 (1992), no. 4, 213–237. MR 1275123, DOI 10.1080/17442509208833776
- François Trèves, Introduction to pseudodifferential and Fourier integral operators. Vol. 1, University Series in Mathematics, Plenum Press, New York-London, 1980. Pseudodifferential operators. MR 597144, DOI 10.1007/978-1-4684-8780-0
- J. Zak, Magnetic translation group. II. Irreducible representations, Phys. Rev. (2) 134 (1964), A1607–A1611. MR 177770
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