## The Annals of Probability

### Cycles and eigenvalues of sequentially growing random regular graphs

#### 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

#### 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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