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Cutoff on all Ramanujan graphs

Geometric and Functional Analysis volume 26, pages 1190–1216 (2016)Cite this article

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 \({\log_{d-1} n}\).

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References

  1. 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.

  2. Aldous D., Diaconis P.: Shuffling cards and stopping times. Am. Math. Monthly 93(5), 333–348 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  3. D. Aldous and J. A. Fill. Reversible markov chains and random walks on graphs (2002). http://www.stat.berkeley.edu/~aldous/RWG/book.html.

  4. Alon N.: Eigenvalues and expanders. Combinatorica 6(2), 83–96 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  5. Alon N., Benjamini I., Lubetzky E., Sodin S.: Non-backtracking random walks mix faster. Commun. Contemp. Math. 9(4), 585–603 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  6. Alon N., Milman V. D.: \({\lambda_1,}\) isoperimetric inequalities for graphs, and superconcentrators. J. Combin. Theory Ser. B 38(1), 73–88 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  7. 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)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bass H.: The Ihara–Selberg zeta function of a tree lattice. Internat. J. Math. 3(6), 717–797 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  9. C. Bordenave. A new proof of Friedman’s second eigenvalue Theorem and its extension to random lifts (2015). arXiv:1502.04482.

  10. Chen G.-Y., Saloff-Coste L.: The cutoff phenomenon for ergodic Markov processes. Electron. J. Probab. 13(3), 26–78 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chung F. R. K.: Diameters and eigenvalues. J. Am. Math. Soc. 2(2), 187–196 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  12. 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)

    Article  MathSciNet  MATH  Google Scholar 

  13. 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).

  14. Diaconis P., Shahshahani M.: Generating a random permutation with random transpositions. Z. Wahrsch. Verw. Gebiete 57(2), 159–179 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  15. R. Durrett. Random graph dynamics. In: Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge (2010).

  16. Feller W.: An Introduction to Probability Theory and its Applications, Vol. I. 3rd ed. Wiley, New York (1968)

    MATH  Google Scholar 

  17. Friedman J.: A proof of Alon’s second eigenvalue conjecture and related problems. Mem. Amer. Math. Soc. 195(910), viii+100 (2008)

    MATH  Google Scholar 

  18. J. Friedman and D. Kohler. The relativized second eigenvalue conjecture of Alon (2014). arXiv:1403.3462.

  19. S. Hoory, N. Linial, and A. Wigderson. Expander graphs and their applications. Bull. Am. Math. Soc. (N.S.), (4)43 (2006), 439–561 (electronic).

  20. Kotani M., Sunada T.: Zeta functions of finite graphs. J. Math. Sci. Univ. Tokyo 7(1), 7–25 (2000)

    MathSciNet  MATH  Google Scholar 

  21. Lalley S. P.: Finite range random walk on free groups and homogeneous trees. Ann. Probab. 21(4), 2087–2130 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  22. Lubetzky E., Sly A.: Cutoff phenomena for random walks on random regular graphs. Duke Math. J. 153(3), 475–510 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  23. Lubetzky E., Sly A.: Explicit expanders with cutoff phenomena. Electron. J. Probab. 16(15), 419–435 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  24. A. Lubotzky. Discrete groups, expanding graphs and invariant measures. In: Modern Birkhäuser Classics. Birkhäuser, Basel (2010).

  25. Lubotzky A., Phillips R., Sarnak P.: Ramanujan graphs. Combinatorica 8(3), 261–277 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  26. R. Lyons and Y. Peres. Probability on Trees and Networks. Cambridge University Press, Cambridge (2016) (In preparation). http://pages.iu.edu/~rdlyons/.

  27. Marcus A., Spielman D. A., Srivastava N.: Interlacing families I: bipartite Ramanujan graphs of all degrees. Ann. Math. 182(1), 307–325 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  28. 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)

    MathSciNet  Google Scholar 

  29. Nilli A.: On the second eigenvalue of a graph. Discrete Math. 91(2), 207–210 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  30. Y. Peres. American Institute of Mathematics (AIM) research workshop “Sharp Thresholds for Mixing Times”, Palo Alto, December (2004). http://www.aimath.org/WWN/mixingtimes.

  31. N. T. Sardari. Diameter of Ramanujan graphs and random Cayley graphs with numerics (2015) (Preprint). arXiv:1511.09340.

  32. 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.

  33. Serre J.-P.: Répartition asymptotique des valeurs propres de l’opérateur de Hecke \({T_p}\). J. Am. Math. Soc. 10(1), 75–102 (1997)

    Article  Google Scholar 

  34. Stein E. M., Weiss G.: Introduction to Fourier Analysis on Euclidean spaces. Princeton University Press, Princeton (1971)

    MATH  Google Scholar 

  35. W. Woess. Denumerable Markov chains. Generating Functions, Boundary Theory, Random Walks on Trees. European Mathematical Society (EMS), Zürich (2009).

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Authors and Affiliations

  1. Courant Institute, New York University, 251 Mercer St., New York, NY, 10012, USA

    Eyal Lubetzky

  2. Microsoft Research, One Microsoft Way, Redmond, WA, 98052, USA

    Yuval Peres

Authors
  1. Eyal Lubetzky
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  2. Yuval Peres
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Correspondence to Yuval Peres.

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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

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  • DOI: https://doi.org/10.1007/s00039-016-0382-7