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July 2014 Cycles and eigenvalues of sequentially growing random regular graphs
Tobias Johnson, Soumik Pal
Ann. Probab. 42(4): 1396-1437 (July 2014). DOI: 10.1214/13-AOP864


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.


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Tobias Johnson. Soumik Pal. "Cycles and eigenvalues of sequentially growing random regular graphs." Ann. Probab. 42 (4) 1396 - 1437, July 2014.


Published: July 2014
First available in Project Euclid: 3 July 2014

zbMATH: 1355.60012
MathSciNet: MR3262482
Digital Object Identifier: 10.1214/13-AOP864

Primary: 05C80 , 60B20

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

Rights: Copyright © 2014 Institute of Mathematical Statistics


Vol.42 • No. 4 • July 2014
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