The Annals of Probability

Size biased couplings and the spectral gap for random regular graphs

Nicholas Cook, Larry Goldstein, and Tobias Johnson

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Let $\lambda$ be the second largest eigenvalue in absolute value of a uniform random $d$-regular graph on $n$ vertices. It was famously conjectured by Alon and proved by Friedman that if $d$ is fixed independent of $n$, then $\lambda=2\sqrt{d-1}+o(1)$ with high probability. In the present work, we show that $\lambda=O(\sqrt{d})$ continues to hold with high probability as long as $d=O(n^{2/3})$, making progress toward a conjecture of Vu that the bound holds for all $1\le d\le n/2$. Prior to this work the best result was obtained by Broder, Frieze, Suen and Upfal (1999) using the configuration model, which hits a barrier at $d=o(n^{1/2})$. We are able to go beyond this barrier by proving concentration of measure results directly for the uniform distribution on $d$-regular graphs. These come as consequences of advances we make in the theory of concentration by size biased couplings. Specifically, we obtain Bennett-type tail estimates for random variables admitting certain unbounded size biased couplings.

Article information

Source
Ann. Probab., Volume 46, Number 1 (2018), 72-125.

Dates
Received: January 2016
Revised: February 2017
First available in Project Euclid: 5 February 2018

Permanent link to this document
https://projecteuclid.org/euclid.aop/1517821219

Digital Object Identifier
doi:10.1214/17-AOP1180

Mathematical Reviews number (MathSciNet)
MR3758727

Zentralblatt MATH identifier
06865119

Subjects
Primary: 05C80: Random graphs [See also 60B20]
Secondary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52) 60E15: Inequalities; stochastic orderings

Keywords
Second eigenvalue random regular graph Alon’s conjecture size biased coupling Stein’s method concentration

Citation

Cook, Nicholas; Goldstein, Larry; Johnson, Tobias. Size biased couplings and the spectral gap for random regular graphs. Ann. Probab. 46 (2018), no. 1, 72--125. doi:10.1214/17-AOP1180. https://projecteuclid.org/euclid.aop/1517821219


Export citation

References

  • [1] Alon, N. (1986). Eigenvalues and expanders. Combinatorica 6 83–96. Theory of computing (Singer Island, Fla., 1984).
  • [2] Alon, N. and Milman, V. D. (1985). $\lambda_{1}$, isoperimetric inequalities for graphs, and superconcentrators. J. Combin. Theory Ser. B 38 73–88.
  • [3] Arratia, R. and Baxendale, P. (2015). Bounded size bias coupling: A Gamma function bound, and universal Dickman-function behavior. Probab. Theory Related Fields 162 411–429.
  • [4] Arratia, R., Goldstein, L. and Kochman, F. (2015). Size bias for one and all. ArXiv preprint. Available at arXiv:1308.2729.
  • [5] Baldi, P., Rinott, Y. and Stein, C. (1989). A normal approximation for the number of local maxima of a random function on a graph. In Probability, Statistics, and Mathematics 59–81. Academic Press, Boston, MA.
  • [6] Barbour, A. D., Holst, L. and Janson, S. (1992). Poisson Approximation. Oxford Studies in Probability 2. Oxford Univ. Press, New York.
  • [7] Bartroff, J., Goldstein, L. and Işlak, Ü. (2014). Bounded size biased couplings, log concave distributions and concentration of measure for occupancy models. ArXiv preprint. Available at arXiv:1402.6769.
  • [8] Bauerschmidt, R., Huang, J., Knowles, A. and Yau, H.-T. (2015). Bulk eigenvalue statistics for random regular graphs. ArXiv preprint. Available at arXiv:1505.06700.
  • [9] Bauerschmidt, R., Knowles, A. and Yau, H.-T. (2015). Local semicircle law for random regular graphs. ArXiv preprint. Available at arXiv:1503.08702.
  • [10] Bennett, G. (1962). Probability inequalities for the sum of independent random variables. J. Amer. Statist. Assoc. 57 33–45.
  • [11] Bernstein, S. (1924). On a modification of Chebyshev’s inequality and of the error formula of Laplace. Ann. Sci. Inst. Sav. Ukraine, Sect. Math. 1 38–49.
  • [12] Bordenave, C. (2015). A new proof of Friedman’s second eigenvalue theorem and its extension to random lifts. ArXiv preprint. Available at arXiv:1502.04482.
  • [13] Boucheron, S., Lugosi, G. and Massart, P. (2013). Concentration Inequalities. A Nonasymptotic Theory of Independence. Oxford Univ. Press, Oxford.
  • [14] Broder, A. and Shamir, E. (1987). On the second eigenvalue of random regular graphs. In 28th Annual Symposium on Foundations of Computer Science (Los Angeles) 286–294. IEEE Comput. Soc. Press, Washington, DC.
  • [15] Broder, A. Z., Frieze, A. M., Suen, S. and Upfal, E. (1999). Optimal construction of edge-disjoint paths in random graphs. SIAM J. Comput. 28 541–573.
  • [16] Chatterjee, S. (2007). Stein’s method for concentration inequalities. Probab. Theory Related Fields 138 305–321.
  • [17] Chatterjee, S. and Dey, P. S. (2010). Applications of Stein’s method for concentration inequalities. Ann. Probab. 38 2443–2485.
  • [18] Chen, L. H. Y., Goldstein, L. and Shao, Q.-M. (2011). Normal Approximation by Stein’s Method. Probability and Its Applications (New York). Springer, Heidelberg.
  • [19] Chung, F., Lu, L. and Vu, V. (2003). Spectra of random graphs with given expected degrees. Proc. Natl. Acad. Sci. USA 100 6313–6318.
  • [20] Coja-Oghlan, A. and Lanka, A. (2009). The spectral gap of random graphs with given expected degrees. Electron. J. Combin. 16 Research Paper 138.
  • [21] Cook, N. A. (2017). Discrepancy properties for random regular digraphs. Random Structures Algorithms 50 23–58.
  • [22] Cook, N. A. (2017). On the singularity of adjacency matrices for random regular digraphs. Probab. Theory Related Fields 167 143–200.
  • [23] Dumitriu, I., Johnson, T., Pal, S. and Paquette, E. (2013). Functional limit theorems for random regular graphs. Probab. Theory Related Fields 156 921–975.
  • [24] Dumitriu, I. and Pal, S. (2012). Sparse regular random graphs: Spectral density and eigenvectors. Ann. Probab. 40 2197–2235.
  • [25] Feige, U. and Ofek, E. (2005). Spectral techniques applied to sparse random graphs. Random Structures Algorithms 27 251–275.
  • [26] Friedman, J. (1991). On the second eigenvalue and random walks in random $d$-regular graphs. Combinatorica 11 331–362.
  • [27] Friedman, J. (2008). A proof of Alon’s second eigenvalue conjecture and related problems. Mem. Amer. Math. Soc. 195 viii+100.
  • [28] Friedman, J., Kahn, J. and Szemerédi, E. (1989). On the second eigenvalue of random regular graphs. In Proceedings of the Twenty-First Annual ACM Symposium on Theory of Computing, STOC’89 587–598. ACM, New York.
  • [29] Friedman, J. and Wigderson, A. (1995). On the second eigenvalue of hypergraphs. Combinatorica 15 43–65.
  • [30] Ghosh, S. and Goldstein, L. (2011). Concentration of measures via size-biased couplings. Probab. Theory Related Fields 149 271–278.
  • [31] Ghosh, S., Goldstein, L. and Raič, M. (2011). Concentration of measure for the number of isolated vertices in the Erdős–Rényi random graph by size bias couplings. Statist. Probab. Lett. 81 1565–1570.
  • [32] Goldstein, L. and Işlak, Ü. (2014). Concentration inequalities via zero bias couplings. Statist. Probab. Lett. 86 17–23.
  • [33] Goldstein, L. and Rinott, Y. (1996). Multivariate normal approximations by Stein’s method and size bias couplings. J. Appl. Probab. 33 1–17.
  • [34] Greenhill, C., Janson, S., Kim, J. H. and Wormald, N. C. (2002). Permutation pseudographs and contiguity. Combin. Probab. Comput. 11 273–298.
  • [35] Hoeffding, W. (1951). A combinatorial central limit theorem. Ann. Math. Stat. 22 558–566.
  • [36] Hoeffding, W. (1963). Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58 13–30.
  • [37] Hoory, S., Linial, N. and Wigderson, A. (2006). Expander graphs and their applications. Bull. Amer. Math. Soc. (N.S.) 43 439–561.
  • [38] Janson, S. (2009). The probability that a random multigraph is simple. Combin. Probab. Comput. 18 205–225.
  • [39] Johnson, T. (2015). Exchangeable pairs, switchings, and random regular graphs. Electron. J. Combin. 22 Paper 1.33.
  • [40] Keshavan, R. H., Montanari, A. and Oh, S. (2010). Matrix completion from a few entries. IEEE Trans. Inform. Theory 56 2980–2998.
  • [41] Krivelevich, M., Sudakov, B., Vu, V. H. and Wormald, N. C. (2001). Random regular graphs of high degree. Random Structures Algorithms 18 346–363.
  • [42] Lubetzky, E., Sudakov, B. and Vu, V. (2011). Spectra of lifted Ramanujan graphs. Adv. Math. 227 1612–1645.
  • [43] Lubotzky, A. (2012). Expander graphs in pure and applied mathematics. Bull. Amer. Math. Soc. (N.S.) 49 113–162.
  • [44] McKay, B. D., Wormald, N. C. and Wysocka, B. (2004). Short cycles in random regular graphs. Electron. J. Combin. 11 Research Paper 66.
  • [45] Miller, S. J., Novikoff, T. and Sabelli, A. (2008). The distribution of the largest nontrivial eigenvalues in families of random regular graphs. Exp. Math. 17 231–244.
  • [46] Nilli, A. (1991). On the second eigenvalue of a graph. Discrete Math. 91 207–210.
  • [47] Raič, M. (2007). CLT-related large deviation bounds based on Stein’s method. Adv. in Appl. Probab. 39 731–752.
  • [48] Ross, N. (2011). Fundamentals of Stein’s method. Probab. Surv. 8 210–293.
  • [49] Stein, C. (1986). Approximate Computation of Expectations. Institute of Mathematical Statistics Lecture Notes—Monograph Series 7. IMS, Hayward, CA.
  • [50] Tikhomirov, K. and Youssef, P. (2016). The spectral gap of dense random regular graphs. ArXiv preprint. Available at arXiv:1610.01765.
  • [51] Tracy, C. A. and Widom, H. (1994). Level-spacing distributions and the Airy kernel. Comm. Math. Phys. 159 151–174.
  • [52] Tran, L. V., Vu, V. H. and Wang, K. (2013). Sparse random graphs: Eigenvalues and eigenvectors. Random Structures Algorithms 42 110–134.
  • [53] Vu, V. (2008). Random discrete matrices. In Horizons of Combinatorics. Bolyai Soc. Math. Stud. 17 257–280. Springer, Berlin.
  • [54] Wormald, N. C. (1999). Models of random regular graphs. In Surveys in Combinatorics, 1999 (Canterbury). London Mathematical Society Lecture Note Series 267 239–298. Cambridge Univ. Press, Cambridge.