Electronic Journal of Probability

On cover times for 2D lattices

Jian Ding

Full-text: Open access

Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465062367

Digital Object Identifier
doi:10.1214/EJP.v17-2089

Mathematical Reviews number (MathSciNet)
MR2946152

Zentralblatt MATH identifier
1258.60044

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

Keywords
Cover times Gaussian free fields random walks

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

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


Export citation

References

  • 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.