The Annals of Probability

Cycles and eigenvalues of sequentially growing random regular graphs

Tobias Johnson and Soumik Pal

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Abstract

Consider the sum of $d$ many i.i.d. random permutation matrices on $n$ labels along with their transposes. The resulting matrix is the adjacency matrix of a random regular (multi)-graph of degree $2d$ on $n$ vertices. It is known that the distribution of smooth linear eigenvalue statistics of this matrix is given asymptotically by sums of Poisson random variables. This is in contrast with Gaussian fluctuation of similar quantities in the case of Wigner matrices. It is also known that for Wigner matrices the joint fluctuation of linear eigenvalue statistics across minors of growing sizes can be expressed in terms of the Gaussian Free Field (GFF). In this article, we explore joint asymptotic (in $n$) fluctuation for a coupling of all random regular graphs of various degrees obtained by growing each component permutation according to the Chinese Restaurant Process. Our primary result is that the corresponding eigenvalue statistics can be expressed in terms of a family of independent Yule processes with immigration. These processes track the evolution of short cycles in the graph. If we now take $d$ to infinity, certain GFF-like properties emerge.

Article information

Source
Ann. Probab., Volume 42, Number 4 (2014), 1396-1437.

Dates
First available in Project Euclid: 3 July 2014

Permanent link to this document
https://projecteuclid.org/euclid.aop/1404394068

Digital Object Identifier
doi:10.1214/13-AOP864

Mathematical Reviews number (MathSciNet)
MR3262482

Zentralblatt MATH identifier
1355.60012

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

Keywords
Random regular graphs eigenvalue fluctuations Chinese restaurant process minors of random matrices

Citation

Johnson, Tobias; Pal, Soumik. Cycles and eigenvalues of sequentially growing random regular graphs. Ann. Probab. 42 (2014), no. 4, 1396--1437. doi:10.1214/13-AOP864. https://projecteuclid.org/euclid.aop/1404394068


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References

  • [1] Adler, M., Nordenstam, E. and van Moerbeke, P. (2011). The Dyson Brownian minor process. Preprint. Available at arXiv:1006.2956.
  • [2] Arratia, R. and Tavaré, S. (1992). The cycle structure of random permutations. Ann. Probab. 20 1567–1591.
  • [3] Barbour, A. D., Holst, L. and Janson, S. (1992). Poisson Approximation. Oxford Studies in Probability 2. Oxford Univ. Press, New York.
  • [4] Ben Arous, G. and Dang, K. (2011). On fluctuations of eigenvalues of random permutation matrices. Preprint. Available at arXiv:1106.2108.
  • [5] Billingsley, P. (1999). Convergence of Probability Measures, 2nd ed. Wiley, New York.
  • [6] Borodin, A. (2010). CLT for spectra of submatrices of Wigner random matrices. Preprint. Available at arXiv:1010.0898.
  • [7] Borodin, A. (2010). CLT for spectra of submatrices of Wigner random matrices II. Stochastic evolution. Preprint. Available at arXiv:1011.3544.
  • [8] Borodin, A. and Ferrari, P. L. (2008). Anisotropic growth of random surfaces in $2+1$ dimensions. Preprint. Available at arXiv:0804.3035.
  • [9] Bourgade, P., Najnudel, J. and Nikeghbali, A. (2013). A unitary extension of virtual permutations. Int. Math. Res. Not. IMRN 2013 4101–4134.
  • [10] Carmona, P., Petit, F. and Yor, M. (1998). Beta-gamma random variables and intertwining relations between certain Markov processes. Rev. Mat. Iberoam. 14 311–367.
  • [11] Diaconis, P. and Fill, J. A. (1990). Strong stationary times via a new form of duality. Ann. Probab. 18 1483–1522.
  • [12] Dumitriu, I., Johnson, T., Pal, S. and Paquette, E. (2013). Functional limit theorems for random regular graphs. Probab. Theory Related Fields 156 921–975.
  • [13] Dumitriu, I. and Pal, S. (2012). Sparse regular random graphs: Spectral density and eigenvectors. Ann. Probab. 40 2197–2235.
  • [14] Elon, Y. (2008). Eigenvectors of the discrete Laplacian on regular graphs—A statistical approach. J. Phys. A 41 435203, 17.
  • [15] Elon, Y. (2010). Gaussian waves on the regular tree. Preprint. Available at arXiv:0907.5065.
  • [16] Elon, Y. and Smilansky, U. (2010). Percolating level sets of the adjacency eigenvectors of $d$-regular graphs. J. Phys. A 43 455209, 13.
  • [17] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley, New York.
  • [18] Ferrari, P. L. (2010). From interacting particle systems to random matrices. J. Stat. Mech. Theory Exp. 10 P10016, 15.
  • [19] Ferrari, P. L. and Frings, R. (2010). On the partial connection between random matrices and interacting particle systems. J. Stat. Phys. 141 613–637.
  • [20] Jakobson, D., Miller, S. D., Rivin, I. and Rudnick, Z. (1999). Eigenvalue spacings for regular graphs. In Emerging Applications of Number Theory (Minneapolis, MN, 1996). IMA Vol. Math. Appl. 109 317–327. Springer, New York.
  • [21] Johansson, K. and Nordenstam, E. (2006). Eigenvalues of GUE minors. Electron. J. Probab. 11 1342–1371.
  • [22] Johnson, T. (2014). Eigenvalue fluctuations for random regular graphs. Ph.D. thesis, Univ. Washington.
  • [23] Kerov, S., Olshanski, G. and Vershik, A. (2004). Harmonic analysis on the infinite symmetric group. Invent. Math. 158 551–642.
  • [24] Linial, N. and Puder, D. (2010). Word maps and spectra of random graph lifts. Random Structures Algorithms 37 100–135.
  • [25] Miller, S. J. and Novikoff, T. (2008). The distribution of the largest nontrivial eigenvalues in families of random regular graphs. Experiment. Math. 17 231–244.
  • [26] Oren, I., Godel, A. and Smilansky, U. (2009). Trace formulae and spectral statistics for discrete Laplacians on regular graphs. I. J. Phys. A 42 415101, 20.
  • [27] Oren, I. and Smilansky, U. (2010). Trace formulas and spectral statistics for discrete Laplacians on regular graphs (II). J. Phys. A 43 225205, 13.
  • [28] Pitman, J. (2006). Combinatorial Stochastic Processes. Lecture Notes in Math. 1875. Springer, Berlin.
  • [29] Sheffield, S. (2007). Gaussian free fields for mathematicians. Probab. Theory Related Fields 139 521–541.
  • [30] Smilansky, U. (2010). Discrete graphs—A paradigm model for quantum chaos. Séminaire Poincaré XIV 89–114. Available at http://www.bourbaphy.fr/smilansky.pdf.
  • [31] Spohn, H. (1998). Dyson’s model of interacting Brownian motions at arbitrary coupling strength. Markov Process. Related Fields 4 649–661.
  • [32] Tran, L. V., Vu, V. H. and Wang, K. (2013). Sparse random graphs: Eigenvalues and eigenvectors. Random Structures Algorithms 42 110–134.
  • [33] Wieand, K. (2000). Eigenvalue distributions of random permutation matrices. Ann. Probab. 28 1563–1587.
  • [34] 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.