The Annals of Probability

The measurable Kesten theorem

Miklós Abért, Yair Glasner, and Bálint Virág

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We give an explicit bound on the spectral radius in terms of the densities of short cycles in finite $d$-regular graphs. It follows that the a finite $d$-regular Ramanujan graph $G$ contains a negligible number of cycles of size less than $c\log\log\vert G\vert$.

We prove that infinite $d$-regular Ramanujan unimodular random graphs are trees. Through Benjamini–Schramm convergence this leads to the following rigidity result. If most eigenvalues of a $d$-regular finite graph $G$ fall in the Alon–Boppana region, then the eigenvalue distribution of $G$ is close to the spectral measure of the $d$-regular tree. In particular, $G$ contains few short cycles.

In contrast, we show that $d$-regular unimodular random graphs with maximal growth are not necessarily trees.

Article information

Ann. Probab. Volume 44, Number 3 (2016), 1601-1646.

Received: February 2012
Revised: May 2014
First available in Project Euclid: 16 May 2016

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Girth spectral radius Ramanujan graphs mass transport principal unimodular random graphs


Abért, Miklós; Glasner, Yair; Virág, Bálint. The measurable Kesten theorem. Ann. Probab. 44 (2016), no. 3, 1601--1646. doi:10.1214/14-AOP937.

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  • [1] Abért, M., Glasner, Y. and Virág, B. (2014). Kesten’s theorem for invariant random subgroups. Duke Math. J. 163 465–488.
  • [2] Aldous, D. and Lyons, R. (2007). Processes on unimodular random networks. Electron. J. Probab. 12 1454–1508.
  • [3] Aldous, D. and Steele, J. M. (2004). The objective method: Probabilistic combinatorial optimization and local weak convergence. In Probability on Discrete Structures. Encyclopaedia Math. Sci. 110 1–72. Springer, Berlin.
  • [4] Antal, P. and Pisztora, A. (1996). On the chemical distance for supercritical Bernoulli percolation. Ann. Probab. 24 1036–1048.
  • [5] Bartholdi, L. (1999). Counting paths in graphs. Enseign. Math. (2) 45 83–131.
  • [6] Benjamini, I. and Schramm, O. (2001). Recurrence of distributional limits of finite planar graphs. Electron. J. Probab. 6 13 pp. (electronic).
  • [7] Bougerol, P. and Jeulin, T. (1999). Brownian bridge on hyperbolic spaces and on homogeneous trees. Probab. Theory Related Fields 115 95–120.
  • [8] Boyd, A. V. (1992). Bounds for the Catalan numbers. Fibonacci Quart. 30 136–138.
  • [9] Friedman, J. (2003). A proof of Alon’s second eigenvalue conjecture. In Proceedings of the Thirty-Fifth Annual ACM Symposium on Theory of Computing 720–724 (electronic). ACM, New York.
  • [10] Glasner, Y. (2003). Ramanujan graphs with small girth. Combinatorica 23 487–502.
  • [11] Grigorchuk, R., Kaimanovich, V. A. and Nagnibeda, T. (2012). Ergodic properties of boundary actions and the Nielsen–Schreier theory. Adv. Math. 230 1340–1380.
  • [12] Kesten, H. (1959). Symmetric random walks on groups. Trans. Amer. Math. Soc. 92 336–354.
  • [13] Levin, D. A., Peres, Y. and Wilmer, E. L. (2009). Markov Chains and Mixing Times. Amer. Math. Soc., Providence, RI.
  • [14] Lubotzky, A. (1994). Discrete Groups, Expanding Graphs and Invariant Measures. Progress in Mathematics 125. Birkhäuser, Basel.
  • [15] Lubotzky, A., Phillips, R. and Sarnak, P. (1988). Ramanujan graphs. Combinatorica 8 261–277.
  • [16] Lyons, R. and Peres, Y. (2015). Cycle density in infinite Ramanujan graphs. Ann. Probab. 43 3337–3358.
  • [17] 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.
  • [18] Massey, W. S. (1977). Algebraic Topology: An Introduction. Springer, New York.
  • [19] McKay, B. D. (1981). The expected eigenvalue distribution of a large regular graph. Linear Algebra Appl. 40 203–216.
  • [20] Morgenstern, M. (1994). Existence and explicit constructions of $q+1$ regular Ramanujan graphs for every prime power $q$. J. Combin. Theory Ser. B 62 44–62.
  • [21] Nilli, A. (1991). On the second eigenvalue of a graph. Discrete Math. 91 207–210.
  • [22] Ortner, R. and Woess, W. (2007). Non-backtracking random walks and cogrowth of graphs. Canad. J. Math. 59 828–844.
  • [23] Paschke, W. L. (1993). Lower bound for the norm of a vertex-transitive graph. Math. Z. 213 225–239.
  • [24] Serre, J.-P. (1997). Répartition asymptotique des valeurs propres de l’opérateur de Hecke $T_{p}$. J. Amer. Math. Soc. 10 75–102.
  • [25] Woess, W. (2000). Random Walks on Infinite Graphs and Groups. Cambridge Tracts in Mathematics 138. Cambridge Univ. Press, Cambridge.