The Annals of Probability

On the almost eigenvectors of random regular graphs

Ágnes Backhausz and Balázs Szegedy

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 $d\geq3$ be fixed and $G$ be a large random $d$-regular graph on $n$ vertices. We show that if $n$ is large enough then the entry distribution of every almost eigenvector of $G$ (with entry sum 0 and normalized to have length $\sqrt{n}$) is close to some Gaussian distribution $N(0,\sigma)$ in the weak topology where $0\leq\sigma\leq1$. Our theorem holds even in the stronger sense when many entries are looked at simultaneously in small random neighborhoods of the graph. Furthermore, we also get the Gaussianity of the joint distribution of several almost eigenvectors if the corresponding eigenvalues are close. Our proof uses graph limits and information theory. Our results have consequences for factor of i.i.d. processes on the infinite regular tree.

In particular, we obtain that if an invariant eigenvector process on the infinite $d$-regular tree is in the weak closure of factor of i.i.d. processes then it has Gaussian distribution.

Article information

Source
Ann. Probab., Volume 47, Number 3 (2019), 1677-1725.

Dates
Received: February 2017
Revised: May 2018
First available in Project Euclid: 2 May 2019

Permanent link to this document
https://projecteuclid.org/euclid.aop/1556784030

Digital Object Identifier
doi:10.1214/18-AOP1294

Mathematical Reviews number (MathSciNet)
MR3945757

Zentralblatt MATH identifier
07067280

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

Keywords
Random regular graphs graph limits group-invariant processes

Citation

Backhausz, Ágnes; Szegedy, Balázs. On the almost eigenvectors of random regular graphs. Ann. Probab. 47 (2019), no. 3, 1677--1725. doi:10.1214/18-AOP1294. https://projecteuclid.org/euclid.aop/1556784030


Export citation

References

  • [1] Alon, N., Benjamini, I., Lubetzky, E. and Sodin, S. (2007). Non-backtracking random walks mix faster. Commun. Contemp. Math. 9 585–603.
    Zentralblatt MATH: 1140.60301
    Digital Object Identifier: doi:10.1142/S0219199707002551
  • [2] Anantharaman, N. and Le Masson, E. (2015). Quantum ergodicity on large regular graphs. Duke Math. J. 164 723–765.
    Zentralblatt MATH: 1386.58015
    Digital Object Identifier: doi:10.1215/00127094-2881592
    Project Euclid: euclid.dmj/1426512106
  • [3] Anantharaman, N. and Sabri, M. (2017). Quantum ergodicity on graphs: From spectral to spatial delocalization. Preprint. Available at arXiv:1704.02766 [math.SP].
    arXiv: 1704.02766
    Zentralblatt MATH: 1376.82091
    Digital Object Identifier: doi:10.1063/1.5000962
  • [4] Backhausz, Á. and Szegedy, B. (2018). On large girth regular graphs and random processes on trees. Random Structures Algorithms. To appear.
    Zentralblatt MATH: 1401.05267
    Digital Object Identifier: doi:10.1002/rsa.20769
  • [5] Backhausz, Á., Szegedy, B. and Virág, B. (2015). Ramanujan graphings and correlation decay in local algorithms. Random Structures Algorithms 47 424–435.
    Zentralblatt MATH: 1325.05159
    Digital Object Identifier: doi:10.1002/rsa.20562
  • [6] Backhausz, Á. and Virág, B. (2017). Spectral measures of factor of i.i.d. processes on vertex-transitive graphs. Ann. Inst. Henri Poincaré Probab. Stat. 53 2260–2278.
    Zentralblatt MATH: 1387.60063
    Digital Object Identifier: doi:10.1214/16-AIHP790
    Project Euclid: euclid.aihp/1511773745
  • [7] Bauerschmidt, R., Huang, J., Knowles, A. and Yau, H.-T. (2016). Local Kesten–McKay law for random regular graphs. Preprint. Available at arXiv:1609.09052 [math.PR].
    arXiv: 1609.09052
  • [8] Bauerschmidt, R., Huang, J., Knowles, A. and Yau, H.-T. (2017). Bulk eigenvalue statistics for random regular graphs. Ann. Probab. 45 3626–3663.
    Zentralblatt MATH: 1379.05098
    Digital Object Identifier: doi:10.1214/16-AOP1145
    Project Euclid: euclid.aop/1511773660
  • [9] Bauerschmidt, R., Knowles, A. and Yau, H.-T. (2017). Local semicircle law for random regular graphs. Comm. Pure Appl. Math. 70 1898–1960.
    Zentralblatt MATH: 1372.05194
    Digital Object Identifier: doi:10.1002/cpa.21709
  • [10] Benaych-Georges, F., Knowles, A. and Yau, H.-T. (2017). Lectures on the local semicircle law for Wigner matrices. In Advanced Topics in Random Matrices. Panoramas et Synthèses 53. Société Mathématique de France, Paris.
  • [11] Bloemendal, A., Knowles, A., Yau, H.-T. and Yin, J. (2016). On the principal components of sample covariance matrices. Probab. Theory Related Fields 164 459–552.
    Zentralblatt MATH: 1339.15023
    Digital Object Identifier: doi:10.1007/s00440-015-0616-x
  • [12] Bollobás, B. (2001). Random Graphs, 2nd ed. Cambridge Studies in Advanced Mathematics 73. Cambridge Univ. Press, Cambridge.
  • [13] Bollobás, B. and Riordan, O. (2011). Sparse graphs: Metrics and random models. Random Structures Algorithms 39 1–38.
  • [14] Bordenave, C. (2015). A new proof of Friedman’s second eigenvalue theorem and its extension to random lifts. Preprint. Available at arXiv:1502.04482 [math.CO].
    arXiv: 1502.04482
  • [15] Bourgade, P., Huang, J. and Yau, H.-T. (2017). Eigenvector statistics of sparse random matrices. Electron. J. Probab. 22 Paper No. 64, 38.
    Zentralblatt MATH: 1372.05195
    Digital Object Identifier: doi:10.1214/17-EJP81
  • [16] Bourgade, P. and Yau, H.-T. (2017). The eigenvector moment flow and local quantum unique ergodicity. Comm. Math. Phys. 350 231–278.
    Zentralblatt MATH: 1379.58014
    Digital Object Identifier: doi:10.1007/s00220-016-2627-6
  • [17] Bowen, L. (2010). The ergodic theory of free group actions: Entropy and the $f$-invariant. Groups Geom. Dyn. 4 419–432.
    Zentralblatt MATH: 1201.37006
    Digital Object Identifier: doi:10.4171/GGD/89
  • [18] Brooks, S. and Lindenstrauss, E. (2013). Non-localization of eigenfunctions on large regular graphs. Israel J. Math. 193 1–14.
    Zentralblatt MATH: 1317.05110
    Digital Object Identifier: doi:10.1007/s11856-012-0096-y
  • [19] Conley, C. T., Marks, A. S. and Tucker-Drob, R. D. (2016). Brooks’ theorem for measurable colorings. Forum Math. Sigma 4 e16, 23.
    Zentralblatt MATH: 06601294
    Digital Object Identifier: doi:10.1017/fms.2016.14
  • [20] Cover, T. M. and Thomas, J. A. (2006). Elements of Information Theory, 2nd ed. Wiley-Interscience, Hoboken, NJ.
    Zentralblatt MATH: 1140.94001
  • [21] Csóka, E., Gerencsér, B., Harangi, V. and Virág, B. (2015). Invariant Gaussian processes and independent sets on regular graphs of large girth. Random Structures Algorithms 47 284–303.
  • [22] Dumitriu, I., Johnson, T., Pal, S. and Paquette, E. (2013). Functional limit theorems for random regular graphs. Probab. Theory Related Fields 156 921–975.
    Zentralblatt MATH: 1271.05088
    Digital Object Identifier: doi:10.1007/s00440-012-0447-y
  • [23] Dumitriu, I. and Pal, S. (2012). Sparse regular random graphs: Spectral density and eigenvectors. Ann. Probab. 40 2197–2235.
    Zentralblatt MATH: 1255.05173
    Digital Object Identifier: doi:10.1214/11-AOP673
    Project Euclid: euclid.aop/1349703320
  • [24] Elon, Y. (2009). Gaussian waves on the regular tree. Preprint. Available at arXiv:0907.5065 [math-ph].
    arXiv: 0907.5065
  • [25] Friedman, J. (2008). A proof of Alon’s second eigenvalue conjecture and related problems. Mem. Amer. Math. Soc. 195 viii $+$ 100.
  • [26] Friedman, N. A. and Ornstein, D. S. (1970). On isomorphism of weak Bernoulli transformations. Adv. Math. 5 365–394.
    Zentralblatt MATH: 0203.05801
    Digital Object Identifier: doi:10.1016/0001-8708(70)90010-1
  • [27] Gaboriau, D. and Lyons, R. (2009). A measurable-group-theoretic solution to von Neumann’s problem. Invent. Math. 177 533–540.
    Zentralblatt MATH: 1182.43002
    Digital Object Identifier: doi:10.1007/s00222-009-0187-5
  • [28] Gamarnik, D. and Sudan, M. (2014). Limits of local algorithms over sparse random graphs [extended abstract]. In ITCS’14—Proceedings of the 2014 Conference on Innovations in Theoretical Computer Science 369–375. ACM, New York.
    Zentralblatt MATH: 1365.05277
  • [29] Geisinger, L. (2015). Convergence of the density of states and delocalization of eigenvectors on random regular graphs. J. Spectr. Theory 5 783–827.
    Zentralblatt MATH: 1384.60024
    Digital Object Identifier: doi:10.4171/JST/114
  • [30] Harangi, V. and Virág, B. (2015). Independence ratio and random eigenvectors in transitive graphs. Ann. Probab. 43 2810–2840.
    Zentralblatt MATH: 1323.05101
    Digital Object Identifier: doi:10.1214/14-AOP952
    Project Euclid: euclid.aop/1441792299
  • [31] Hatami, H., Lovász, L. and Szegedy, B. (2014). Limits of locally-globally convergent graph sequences. Geom. Funct. Anal. 24 269–296.
    Zentralblatt MATH: 1294.05109
    Digital Object Identifier: doi:10.1007/s00039-014-0258-7
  • [32] Huang, J., Landon, B. and Yau, H.-T. (2015). Bulk universality of sparse random matrices. J. Math. Phys. 56 123301, 19.
    Zentralblatt MATH: 1329.05262
    Digital Object Identifier: doi:10.1063/1.4936139
  • [33] Knowles, A. and Yin, J. (2013). Eigenvector distribution of Wigner matrices. Probab. Theory Related Fields 155 543–582.
    Zentralblatt MATH: 1268.15033
    Digital Object Identifier: doi:10.1007/s00440-011-0407-y
  • [34] Lubotzky, A., Phillips, R. and Sarnak, P. (1988). Ramanujan graphs. Combinatorica 8 261–277.
    Zentralblatt MATH: 0661.05035
    Digital Object Identifier: doi:10.1007/BF02126799
  • [35] Lyons, R. (2017). Factors of IID on trees. Combin. Probab. Comput. 26 285–300.
    Zentralblatt MATH: 1371.05129
    Digital Object Identifier: doi:10.1017/S096354831600033X
  • [36] Lyons, R. and Nazarov, F. (2011). Perfect matchings as IID factors on non-amenable groups. European J. Combin. 32 1115–1125.
    Zentralblatt MATH: 1229.05115
    Digital Object Identifier: doi:10.1016/j.ejc.2011.03.008
  • [37] O’Rourke, S., Vu, V. and Wang, K. (2016). Eigenvectors of random matrices: A survey. J. Combin. Theory Ser. A 144 361–442.
    Zentralblatt MATH: 1347.15044
    Digital Object Identifier: doi:10.1016/j.jcta.2016.06.008
  • [38] Puder, D. (2015). Expansion of random graphs: New proofs, new results. Invent. Math. 201 845–908.
    Zentralblatt MATH: 1320.05115
    Digital Object Identifier: doi:10.1007/s00222-014-0560-x
  • [39] Rahman, M. (2016). Factor of IID percolation on trees. SIAM J. Discrete Math. 30 2217–2242.
    Zentralblatt MATH: 1351.05206
    Digital Object Identifier: doi:10.1137/15M1021362
  • [40] Rahman, M. and Virág, B. (2017). Local algorithms for independent sets are half-optimal. Ann. Probab. 45 1543–1577.
    Zentralblatt MATH: 1377.60049
    Digital Object Identifier: doi:10.1214/16-AOP1094
    Project Euclid: euclid.aop/1494835225
  • [41] Tao, T. and Vu, V. (2012). Random matrices: Universal properties of eigenvectors. Random Matrices Theory Appl. 1 1150001, 27.
    Zentralblatt MATH: 1248.15031
    Digital Object Identifier: doi:10.1142/S2010326311500018
  • [42] 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.
    Zentralblatt MATH: 0935.05080

The Institute of Mathematical Statistics

The Annals of Probability

New content alerts

Email RSS ToC RSS Article
  • You have access to this content.
  • You have partial access to this content.
  • You do not have access to this content.
Turn Off MathJax

What is MathJax?

More like this

+ See more