Abstract
We show that on every Ramanujan graph \({G}\), the simple random walk exhibits cutoff: when \({G}\) has \({n}\) vertices and degree \({d}\), the total-variation distance of the walk from the uniform distribution at time \({t=\frac{d}{d-2} \log_{d-1} n + s\sqrt{\log n}}\) is asymptotically \({{\mathbb{P}}(Z > c \, s)}\) where \({Z}\) is a standard normal variable and \({c=c(d)}\) is an explicit constant. Furthermore, for all \({1 \leq p \leq \infty}\), \({d}\)-regular Ramanujan graphs minimize the asymptotic \({L^p}\)-mixing time for SRW among all \({d}\)-regular graphs. Our proof also shows that, for every vertex \({x}\) in \({G}\) as above, its distance from \({n-o(n)}\) of the vertices is asymptotically \({\log_{d-1} n}\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
D. Aldous. Random walks on finite groups and rapidly mixing Markov chains. In Seminar on probability, XVII. In: Lecture Notes in Math., Vol. 986. Springer, Berlin, (1983), pp. 243–297.
Aldous D., Diaconis P.: Shuffling cards and stopping times. Am. Math. Monthly 93(5), 333–348 (1986)
D. Aldous and J. A. Fill. Reversible markov chains and random walks on graphs (2002). http://www.stat.berkeley.edu/~aldous/RWG/book.html.
Alon N.: Eigenvalues and expanders. Combinatorica 6(2), 83–96 (1986)
Alon N., Benjamini I., Lubetzky E., Sodin S.: Non-backtracking random walks mix faster. Commun. Contemp. Math. 9(4), 585–603 (2007)
Alon N., Milman V. D.: \({\lambda_1,}\) isoperimetric inequalities for graphs, and superconcentrators. J. Combin. Theory Ser. B 38(1), 73–88 (1985)
Angel O., Friedman J., Hoory S.: The non-backtracking spectrum of the universal cover of a graph. Trans. Am. Math. Soc. 367(6), 4287–4318 (2015)
Bass H.: The Ihara–Selberg zeta function of a tree lattice. Internat. J. Math. 3(6), 717–797 (1992)
C. Bordenave. A new proof of Friedman’s second eigenvalue Theorem and its extension to random lifts (2015). arXiv:1502.04482.
Chen G.-Y., Saloff-Coste L.: The cutoff phenomenon for ergodic Markov processes. Electron. J. Probab. 13(3), 26–78 (2008)
Chung F. R. K.: Diameters and eigenvalues. J. Am. Math. Soc. 2(2), 187–196 (1989)
Chung F. R. K., Faber V., Manteuffel T. A.: An upper bound on the diameter of a graph from eigenvalues associated with its Laplacian. SIAM J. Discrete Math. 7(3), 443–457 (1994)
G. Davidoff, P. Sarnak, and A. Valette. In: Elementary number theory, group theory, and Ramanujan graphs. London Mathematical Society Student Texts, Vol. 55. Cambridge University Press, Cambridge (2003).
Diaconis P., Shahshahani M.: Generating a random permutation with random transpositions. Z. Wahrsch. Verw. Gebiete 57(2), 159–179 (1981)
R. Durrett. Random graph dynamics. In: Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge (2010).
Feller W.: An Introduction to Probability Theory and its Applications, Vol. I. 3rd ed. Wiley, New York (1968)
Friedman J.: A proof of Alon’s second eigenvalue conjecture and related problems. Mem. Amer. Math. Soc. 195(910), viii+100 (2008)
J. Friedman and D. Kohler. The relativized second eigenvalue conjecture of Alon (2014). arXiv:1403.3462.
S. Hoory, N. Linial, and A. Wigderson. Expander graphs and their applications. Bull. Am. Math. Soc. (N.S.), (4)43 (2006), 439–561 (electronic).
Kotani M., Sunada T.: Zeta functions of finite graphs. J. Math. Sci. Univ. Tokyo 7(1), 7–25 (2000)
Lalley S. P.: Finite range random walk on free groups and homogeneous trees. Ann. Probab. 21(4), 2087–2130 (1993)
Lubetzky E., Sly A.: Cutoff phenomena for random walks on random regular graphs. Duke Math. J. 153(3), 475–510 (2010)
Lubetzky E., Sly A.: Explicit expanders with cutoff phenomena. Electron. J. Probab. 16(15), 419–435 (2011)
A. Lubotzky. Discrete groups, expanding graphs and invariant measures. In: Modern Birkhäuser Classics. Birkhäuser, Basel (2010).
Lubotzky A., Phillips R., Sarnak P.: Ramanujan graphs. Combinatorica 8(3), 261–277 (1988)
R. Lyons and Y. Peres. Probability on Trees and Networks. Cambridge University Press, Cambridge (2016) (In preparation). http://pages.iu.edu/~rdlyons/.
Marcus A., Spielman D. A., Srivastava N.: Interlacing families I: bipartite Ramanujan graphs of all degrees. Ann. Math. 182(1), 307–325 (2015)
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)
Nilli A.: On the second eigenvalue of a graph. Discrete Math. 91(2), 207–210 (1991)
Y. Peres. American Institute of Mathematics (AIM) research workshop “Sharp Thresholds for Mixing Times”, Palo Alto, December (2004). http://www.aimath.org/WWN/mixingtimes.
N. T. Sardari. Diameter of Ramanujan graphs and random Cayley graphs with numerics (2015) (Preprint). arXiv:1511.09340.
P. Sarnak. Letter to Scott Aaronson and Andrew Pollington on the Solovay–Kitaev Theorem and Golden Gates (with an appendix on optimal lifting of integral points). February (2015). http://publications.ias.edu/sarnak/paper/2637.
Serre J.-P.: Répartition asymptotique des valeurs propres de l’opérateur de Hecke \({T_p}\). J. Am. Math. Soc. 10(1), 75–102 (1997)
Stein E. M., Weiss G.: Introduction to Fourier Analysis on Euclidean spaces. Princeton University Press, Princeton (1971)
W. Woess. Denumerable Markov chains. Generating Functions, Boundary Theory, Random Walks on Trees. European Mathematical Society (EMS), Zürich (2009).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Lubetzky, E., Peres, Y. Cutoff on all Ramanujan graphs. Geom. Funct. Anal. 26, 1190–1216 (2016). https://doi.org/10.1007/s00039-016-0382-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00039-016-0382-7