Electronic Journal of Probability

On cover times for 2D lattices

Jian Ding

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We study the cover time $\tau_{\mathrm{cov}}$ by (continuous-time) random walk on the 2D box of side length $n$ with wired boundary or on the 2D torus,and show that in both cases with probability approaching $1$ as $n$ increases, $\sqrt{\tau_{\mathrm{cov}}}=\sqrt{2n^2} \left[\sqrt{2/\pi} \log n + O(\log\log n)\right]$. This improves a result of Dembo, Peres, Rosen, and Zeitouni (2004) and makes progresstowards a conjecture of Bramson and Zeitouni (2009).

Article information

Electron. J. Probab., Volume 17 (2012), paper no. 45, 18 pp.

Accepted: 16 June 2012
First available in Project Euclid: 4 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 60G60: Random fields 60G15: Gaussian processes

Cover times Gaussian free fields random walks

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Ding, Jian. On cover times for 2D lattices. Electron. J. Probab. 17 (2012), paper no. 45, 18 pp. doi:10.1214/EJP.v17-2089. https://projecteuclid.org/euclid.ejp/1465062367

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  • D. Aldous and J. Fill. Reversible Markov Chains and Random Walks on Graphs. In preparation, available at http://www.stat.berkeley.edu/~aldous/RWG/book.html.
  • Aldous, David J. Random walk covering of some special trees. J. Math. Anal. Appl. 157 (1991), no. 1, 271–283.
  • Bolthausen, Erwin; Deuschel, Jean-Dominique; Giacomin, Giambattista. Entropic repulsion and the maximum of the two-dimensional harmonic crystal. Ann. Probab. 29 (2001), no. 4, 1670–1692.
  • Bramson, Maury; Zeitouni, Ofer. Tightness of the recentered maximum of the two-dimensional discrete Gaussian free field. Comm. Pure Appl. Math. 65 (2012), no. 1, 1–20.
  • Bramson, Maury; Zeitouni, Ofer. Tightness for a family of recursion equations. Ann. Probab. 37 (2009), no. 2, 615–653.
  • Daviaud, Olivier. Extremes of the discrete two-dimensional Gaussian free field. Ann. Probab. 34 (2006), no. 3, 962–986.
  • Dembo, Amir; Peres, Yuval; Rosen, Jay; Zeitouni, Ofer. Cover times for Brownian motion and random walks in two dimensions. Ann. of Math. (2) 160 (2004), no. 2, 433–464.
  • J. Ding. Asymptotics of cover times via gaussian free fields: bounded-degree graphs and general trees. Preprint, availabel at erb|http://arxiv.org/abs/1103.4402|.
  • J. Ding, J. Lee, and Y. Peres. Cover times, blanket times, and majorizing measures. Annals of Mathematics. to appear.
  • Ding, Jian; Zeitouni, Ofer. A sharp estimate for cover times on binary trees. Stochastic Process. Appl. 122 (2012), no. 5, 2117–2133. http://arxiv.org/abs/1104.0434
  • Dynkin, E. B. Gaussian and non-Gaussian random fields associated with Markov processes. J. Funct. Anal. 55 (1984), no. 3, 344–376.
  • Dynkin, E. B. Local times and quantum fields. Seminar on stochastic processes, 1983 (Gainesville, Fla., 1983), 69–83, Progr. Probab. Statist., 7, Birkhäuser Boston, Boston, MA, 1984.
  • Eisenbaum, Nathalie. Une version sans conditionnement du théorème d'isomorphisms de Dynkin. (French) [An unconditioned version of Dynkin's isomorphism theorem] Séminaire de Probabilités, XXIX, 266–289, Lecture Notes in Math., 1613, Springer, Berlin, 1995.
  • Eisenbaum, Nathalie; Kaspi, Haya; Marcus, Michael B.; Rosen, Jay; Shi, Zhan. A Ray-Knight theorem for symmetric Markov processes. Ann. Probab. 28 (2000), no. 4, 1781–1796.
  • Fernique, X. Regularité des trajectoires des fonctions aléatoires gaussiennes. (French) École d'Été de Probabilités de Saint-Flour, IV-1974, pp. 1–96. Lecture Notes in Math., Vol. 480, Springer, Berlin, 1975.
  • Janson, Svante. Gaussian Hilbert spaces. Cambridge Tracts in Mathematics, 129. Cambridge University Press, Cambridge, 1997. x+340 pp. ISBN: 0-521-56128-0
  • Lawler, Gregory F.; Limic, Vlada. Random walk: a modern introduction. Cambridge Studies in Advanced Mathematics, 123. Cambridge University Press, Cambridge, 2010. xii+364 pp. ISBN: 978-0-521-51918-2
  • Levin, David A.; Peres, Yuval; Wilmer, Elizabeth L. Markov chains and mixing times. With a chapter by James G. Propp and David B. Wilson. American Mathematical Society, Providence, RI, 2009. xviii+371 pp. ISBN: 978-0-8218-4739-8
  • R. Lyons, with Y. Peres. Probability on Trees and Networks. In preparation. Current version available at texttt http://mypage.iu.edu/~rdlyons/prbtree/book.pdf, 2009.
  • Marcus, Michael B.; Rosen, Jay. Sample path properties of the local times of strongly symmetric Markov processes via Gaussian processes. Ann. Probab. 20 (1992), no. 4, 1603–1684.
  • Marcus, Michael B.; Rosen, Jay. Gaussian processes and local times of symmetric Lévy processes. Lévy processes, 67–88, Birkhäuser Boston, Boston, MA, 2001.
  • Marcus, Michael B.; Rosen, Jay. Markov processes, Gaussian processes, and local times. Cambridge Studies in Advanced Mathematics, 100. Cambridge University Press, Cambridge, 2006. x+620 pp. ISBN: 978-0-521-86300-1; 0-521-86300-7
  • Matthews, Peter. Covering problems for Markov chains. Ann. Probab. 16 (1988), no. 3, 1215–1228.
  • Slepian, David. The one-sided barrier problem for Gaussian noise. Bell System Tech. J. 41 1962 463–501.