Electronic Communications in Probability

Cutoff for Ramanujan graphs via degree inflation

Jonathan Hermon

Full-text: Open access

Abstract

Recently Lubetzky and Peres showed that simple random walks on a sequence of $d$-regular Ramanujan graphs $G_n=(V_n,E_n)$ of increasing sizes exhibit cutoff in total variation around the diameter lower bound $\frac{d} {d-2}\log _{d-1}|V_n| $. We provide a different argument under the assumption that for some $r(n) \gg 1$ the maximal number of simple cycles in a ball of radius $r(n)$ in $G_n$ is uniformly bounded in $n$.

Article information

Source
Electron. Commun. Probab., Volume 22 (2017), paper no. 45, 10 pp.

Dates
Received: 26 February 2017
Accepted: 5 July 2017
First available in Project Euclid: 15 August 2017

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1502762751

Digital Object Identifier
doi:10.1214/17-ECP72

Mathematical Reviews number (MathSciNet)
MR3693771

Zentralblatt MATH identifier
1378.05188

Subjects
Primary: 60B10: Convergence of probability measures 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 05C12: Distance in graphs 05C81: Random walks on graphs

Keywords
cutoff Ramanujan graphs degree inflation

Rights
Creative Commons Attribution 4.0 International License.

Citation

Hermon, Jonathan. Cutoff for Ramanujan graphs via degree inflation. Electron. Commun. Probab. 22 (2017), paper no. 45, 10 pp. doi:10.1214/17-ECP72. https://projecteuclid.org/euclid.ecp/1502762751


Export citation

References

  • [1] Abért, M., Glasner, Y. and Virág, B.: The measurable Kesten theorem. Ann. Probab., 44(3):1601–1646, 2016.
  • [2] Riddhipratim. B., Hermon, J. and Peres Y.: Characterization of cutoff for reversible Markov chains. Ann. Probab., 45(3):1448–1487, 2017.
  • [3] Friedman, J.: A proof of Alon’s second eigenvalue conjecture. In Proceedings of the Thirty-Fifth Annual ACM Symposium on Theory of Computing, pages 720–724. ACM, New York, 2003.
  • [4] Lubetzky, E. and Peres, Y.: Cutoff on all Ramanujan graphs. Geom. Funct. Anal., 26(4):1190–1216, 2016.
  • [5] Lubetzky, E. and Sly, A: Cutoff phenomena for random walks on random regular graphs. Duke Math. J., 153(3):475–510, 2010.
  • [6] Lubotzky, A., Phillips, R. and Sarnak, P.: Ramanujan graphs. Combinatorica, 8(3):261–277, 1988.
  • [7] Marcus, A. W., Spielman, D. A., and Srivastava, N.: Interlacing families I: Bipartite Ramanujan graphs of all degrees. Ann. of Math. (2), 182(1):307–325, 2015.
  • [8] Margulis, G. A.: Explicit group-theoretic constructions of combinatorial schemes and their applications in the construction of expanders and concentrators. Problemy Peredachi Informatsii, 24(1):51–60, 1988.
  • [9] Morgenstern, M.: Existence and explicit constructions of $q+1$ regular Ramanujan graphs for every prime power $q$. J. Combin. Theory Ser. B, 62(1):44–62, 1994.
  • [10] Nilli, A.: On the second eigenvalue of a graph. Discrete Math., 91(2):207–210, 1991.