Random walks veering left

Raoul Normand (Academia Sinica, Taipei, Taiwan)
Bálint Virág (University of Toronto)


We study coupled random walks in the plane such that, at each step, the walks change direction by a uniform random angle plus an extra deterministic angle $\theta$. We compute the Hausdorff dimension of the $\theta$ for which the walk has an unusual behavior. This model is related to a study of the spectral measure of some random matrices. The same techniques allow to study the boundary behavior of some Gaussian analytic functions.

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Pages: 1-25

Publication Date: October 21, 2013

DOI: 10.1214/EJP.v18-2523


  • Benjamini, Itai; Häggström, Olle; Peres, Yuval; Steif, Jeffrey E. Which properties of a random sequence are dynamically sensitive? Ann. Probab. 31 (2003), no. 1, 1--34. MR1959784
  • George Bennett, Probability inequalities for the sum of independent random variables, Journal of the American Statistical Association 57 (1962), no. 297, pp. 33--45.
  • Chatterjee, Sourav. A generalization of the Lindeberg principle. Ann. Probab. 34 (2006), no. 6, 2061--2076. MR2294976
  • Jitomirskaya, Svetlana; Last, Yoram. Power-law subordinacy and singular spectra. I. Half-line operators. Acta Math. 183 (1999), no. 2, 171--189. MR1738043
  • Kahane, Jean-Pierre. Some random series of functions. Second edition. Cambridge Studies in Advanced Mathematics, 5. Cambridge University Press, Cambridge, 1985. xiv+305 pp. ISBN: 0-521-24966-X; 0-521-45602-9 MR0833073
  • Killip, Rowan; Nenciu, Irina. Matrix models for circular ensembles. Int. Math. Res. Not. 2004, no. 50, 2665--2701. MR2127367
  • Peter Mörters and Yuval Peres, phBrownian motion, Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge University Press, Cambridge, 2010, With an appendix by Oded Schramm and Wendelin Werner. MR2604525 (2011i:60152)
  • Peres, Yuval; Virág, Bálint. Zeros of the i.i.d. Gaussian power series: a conformally invariant determinantal process. Acta Math. 194 (2005), no. 1, 1--35. MR2231337
  • Arnab Sen and Bálint Virág, phThe top eigenvalue of the random Toeplitz matrix and the Sine kernel, arXiv:1109.5494.
  • Simon, Barry. OPUC on one foot. Bull. Amer. Math. Soc. (N.S.) 42 (2005), no. 4, 431--460 (electronic). MR2163705
  • Simon, Barry. Orthogonal polynomials on the unit circle. Part 1. Classical theory. American Mathematical Society Colloquium Publications, 54, Part 1. American Mathematical Society, Providence, RI, 2005. xxvi+466 pp. ISBN: 0-8218-3446-0 MR2105088
  • Simon, Barry. Orthogonal polynomials on the unit circle. Part 2. Spectral theory. American Mathematical Society Colloquium Publications, 54, Part 2. American Mathematical Society, Providence, RI, 2005. pp. i--xxii and 467--1044. ISBN: 0-8218-3675-7 MR2105089
  • Takács, Lajos. Combinatorial methods in the theory of stochastic processes. Reprint of the 1967 original. Robert E. Krieger Publishing Co., Huntington, N. Y., 1977. xi+262 pp. MR0431313

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