Electronic Communications in Probability

Heat Kernel Asymptotics on the Lamplighter Group

David Revelle

Full-text: Open access

Abstract

We show that, for one generating set, the on-diagonal decay of the heat kernel on the lamplighter group is asymptotic to $c_1 n^{1/6}\exp[-c_2 n^{1/3}]$. We also make off-diagonal estimates which show that there is a sharp threshold for which elements have transition probabilities that are comparable to the return probability. The off-diagonal estimates also give an upper bound for the heat kernel that is uniformly summable in time. The methods used also apply to a one dimensional trapping problem, and we compute the distribution of the walk conditioned on survival as well as a corrected asymptotic for the survival probability. Conditioned on survival, the position of the walker is shown to be concentrated within $\alpha n^{1/3}$ of the origin for a suitable $\alpha$.

Article information

Source
Electron. Commun. Probab., Volume 8 (2003), paper no. 16, 142-154.

Dates
Accepted: 10 November 2003
First available in Project Euclid: 18 May 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1463608900

Digital Object Identifier
doi:10.1214/ECP.v8-1092

Mathematical Reviews number (MathSciNet)
MR2042753

Zentralblatt MATH identifier
1061.60112

Subjects
Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)

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

Citation

Revelle, David. Heat Kernel Asymptotics on the Lamplighter Group. Electron. Commun. Probab. 8 (2003), paper no. 16, 142--154. doi:10.1214/ECP.v8-1092. https://projecteuclid.org/euclid.ecp/1463608900


Export citation

References

  • Anlauf, Joachim K, Asymptotically exact solution of the one-dimensional trapping problem, Phys. Rev. Letters 52 (1984), 1845–1848.
  • Antal, Peter. Enlargement of obstacles for the simple random walk. Ann. Probab. 23 (1995), no. 3, 1061–1101.
  • Avez, André. Limite de quotients pour des marches aléatoires sur des groupes. (French) C. R. Acad. Sci. Paris Sér. A-B 276 (1973), A317–A320.
  • Cartwright, Donald I. On the asymptotic behaviour of convolution powers of probabilities on discrete groups. Monatsh. Math. 107 1989), no. 4, 287–290.
  • Coulhon, T.; Grigor'yan, A.; Pittet, C. A geometric approach to on-diagonal heat kernel lower bounds on groups. Ann. Inst. Fourier (Grenoble) 51 (2001), no. 6, 1763–1827.
  • Dicks, Warren; Schick, Thomas. The spectral measure of certain elements of the complex group ring of a wreath product. Geom. Dedicata 93 (2002), 121–137.
  • Èrshler, A. G. On the asymptotics of the rate of departure to infinity. (Russian) Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 283 (2001), Teor. Predst. Din. Sist. Komb. i Algoritm. Metody. 6, 251–257, 263.
  • Grigorchuk, Rostislav I.;.Zuk, Andrzej. The lamplighter group as a group generated by a 2-state automaton, and its spectrum. Geom. Dedicata 87 (2001), no. 1-3, 209–244.
  • Hebisch, W.; Saloff-Coste, L. Gaussian estimates for Markov chains and random walks on groups. Ann. Probab. 21 (1993), no. 2, 673–709.
  • Hughes, Barry D. Random walks and random e nvironments. Vol. 1. Random walks. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1995. xxii+631 pp. ISBN: 0-19-853788-3.
  • Lawler, Gregory F. Intersections of random walks. Probability and its Applications. Birkhäuser Boston, Inc., Boston, MA, 1991. 219 pp. ISBN: 0-8176-3557-2.
  • Pittet, C.; Saloff-Coste, L. On random walks on wreath products. Ann. Probab. 30 (2002), no. 2, 948–977.
  • Pittet, Ch.; Saloff-Coste, L. On the stability of the behavior of random walks on groups. J. Geom. Anal. 10 (2000), no. 4, 713–737.
  • Pruitt, William E. The rate of escape of random walk. Ann. Probab. 18 (1990), no. 4, 1417–1461.
  • Sznitman, Alain-Sol. Brownian motion, obstacles and random media. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 1998. xvi+353 pp. ISBN: 3-540-64554-3.
  • Unnikrishnan, K.; Prasad, M. A. Random walk on a one-dimensional lattice with a random distribution of traps. Phys. Lett. A 100 (1984), no. 1, 19–20.
  • Varopoulos, Nicholas Th. Random walks on soluble groups. Bull. Sci. Math. (2) 107 (1983), no. 4, 337–344.
  • Woess, Wolfgang. Random walks on infinite graphs and groups–-a survey on selected topics. Bull. London Math. Soc. 26 (1994), no. 1, 1–60.