Cycles and eigenvalues of sequentially growing random regular graphs

The Annals of Probability (Impact Factor: 1.42). 03/2012; 42(4). DOI: 10.1214/13-AOP864
Source: arXiv


Consider the sum of d many iid 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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