The Annals of Probability

Cycle density in infinite Ramanujan graphs

Russell Lyons and Yuval Peres

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Abstract

We introduce a technique using nonbacktracking random walk for estimating the spectral radius of simple random walk. This technique relates the density of nontrivial cycles in simple random walk to that in nonbacktracking random walk. We apply this to infinite Ramanujan graphs, which are regular graphs whose spectral radius equals that of the tree of the same degree. Kesten showed that the only infinite Ramanujan graphs that are Cayley graphs are trees. This result was extended to unimodular random rooted regular graphs by Abért, Glasner and Virág. We show that an analogous result holds for all regular graphs: the frequency of times spent by simple random walk in a nontrivial cycle is a.s. 0 on every infinite Ramanujan graph. We also give quantitative versions of that result, which we apply to answer another question of Abért, Glasner and Virág, showing that on an infinite Ramanujan graph, the probability that simple random walk encounters a short cycle tends to 0 a.s. as the time tends to infinity.

Article information

Source
Ann. Probab., Volume 43, Number 6 (2015), 3337-3358.

Dates
Received: October 2013
Revised: August 2014
First available in Project Euclid: 11 December 2015

Permanent link to this document
https://projecteuclid.org/euclid.aop/1449843632

Digital Object Identifier
doi:10.1214/14-AOP961

Mathematical Reviews number (MathSciNet)
MR3433583

Zentralblatt MATH identifier
1346.60061

Subjects
Primary: 60G50: Sums of independent random variables; random walks 82C41: Dynamics of random walks, random surfaces, lattice animals, etc. [See also 60G50]
Secondary: 05C81: Random walks on graphs

Keywords
Nonbacktracking random walks spectral radius regular graphs

Citation

Lyons, Russell; Peres, Yuval. Cycle density in infinite Ramanujan graphs. Ann. Probab. 43 (2015), no. 6, 3337--3358. doi:10.1214/14-AOP961. https://projecteuclid.org/euclid.aop/1449843632


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References

  • Abért, M., Glasner, Y. and Virág, B. (2015). The measurable Kesten theorem. Ann. Probab. To appear.
  • Alon, N. (1986). Eigenvalues and expanders. Combinatorica 6 83–96. Theory of computing (Singer Island, Fla., 1984).
  • Alon, N., Benjamini, I., Lubetzky, E. and Sodin, S. (2007). Non-backtracking random walks mix faster. Commun. Contemp. Math. 9 585–603.
  • Ancona, A. (1988). Positive harmonic functions and hyperbolicity. In Potential Theory—Surveys and Problems (Prague, 1987) (J. Král, J. Lukeš, I. Netuka and J. Veselý, eds.). Lecture Notes in Math. 1344 1–23. Springer, Berlin.
  • Biggs, N. L., Mohar, B. and Shawe-Taylor, J. (1988). The spectral radius of infinite graphs. Bull. Lond. Math. Soc. 20 116–120.
  • Cheeger, J. (1970). A lower bound for the smallest eigenvalue of the Laplacian. In Problems in Analysis (Papers Dedicated to Salomon Bochner, 1969) (R. C. Gunning, ed.) 195–199. Princeton Univ. Press, Princeton, NJ.
  • Dodziuk, J. (1984). Difference equations, isoperimetric inequality and transience of certain random walks. Trans. Amer. Math. Soc. 284 787–794.
  • Dodziuk, J. and Kendall, W. S. (1986). Combinatorial Laplacians and isoperimetric inequality. In From Local Times to Global Geometry, Control and Physics (Coventry, 1984/85) (K. D. Elworthy, ed.). Pitman Res. Notes Math. Ser. 150 68–74. Longman, Harlow.
  • Gerl, P. (1988). Random walks on graphs with a strong isoperimetric property. J. Theoret. Probab. 1 171–187.
  • Grigorchuk, R. I. (1980). Symmetrical random walks on discrete groups. In Multicomponent Random Systems (R. L. Dobrushin, Ya. G. Sinaĭ and D. Griffeath, eds.). Adv. Probab. Related Topics 6 285–325. Dekker, New York.
  • Kaimanovich, V. A. (1992). Dirichlet norms, capacities and generalized isoperimetric inequalities for Markov operators. Potential Anal. 1 61–82.
  • Kesten, H. (1959a). Full Banach mean values on countable groups. Math. Scand. 7 146–156.
  • Kesten, H. (1959b). Symmetric random walks on groups. Trans. Amer. Math. Soc. 92 336–354.
  • Li, W.-C. W. (2007). Ramanujan graphs and Ramanujan hypergraphs. In Automorphic Forms and Applications (P. Sarnak and F. Shahidi, eds.). IAS/Park City Math. Ser. 12 401–427. Amer. Math. Soc., Providence, RI.
  • Lubotzky, A., Phillips, R. and Sarnak, P. (1988). Ramanujan graphs. Combinatorica 8 261–277.
  • Lyons, R. and Peres, Y. (2015). Probability on Trees and Networks. Preprint, Cambridge Univ. Press. Available at http://pages.iu.edu/~rdlyons/.
  • Margulis, G. A. (1988). Explicit group-theoretic constructions of combinatorial schemes and their applications in the construction of expanders and concentrators. Problemy Peredachi Informatsii 24 51–60.
  • Murty, M. R. (2003). Ramanujan graphs. J. Ramanujan Math. Soc. 18 33–52.
  • Nilli, A. (1991). On the second eigenvalue of a graph. Discrete Math. 91 207–210.
  • Northshield, S. (1992). Cogrowth of regular graphs. Proc. Amer. Math. Soc. 116 203–205.
  • Varopoulos, N. Th. (1985). Isoperimetric inequalities and Markov chains. J. Funct. Anal. 63 215–239.