Publication Date:
2012-09-03
Description:
Consider d uniformly random permutation matrices on n labels. Consider the sum of these matrices along with their transposes. The total can be interpreted as the adjacency matrix of a random regular graph of degree 2 d on n vertices. We consider limit theorems for various combinatorial and analytical properties of this graph (or the matrix) as n grows to infinity, either when d is kept fixed or grows slowly with n . In a suitable weak convergence framework, we prove that the (finite but growing in length) sequences of the number of short cycles and of cyclically non-backtracking walks converge to distributional limits. We estimate the total variation distance from the limit using Stein’s method. As an application of these results we derive limits of linear functionals of the eigenvalues of the adjacency matrix. A key step in this latter derivation is an extension of the Kahn–Szemerédi argument for estimating the second largest eigenvalue for all values of d and n . Content Type Journal Article Pages 1-55 DOI 10.1007/s00440-012-0447-y Authors Ioana Dumitriu, Department of Mathematics, University of Washington, Seattle, WA 98195, USA Tobias Johnson, Department of Mathematics, University of Washington, Seattle, WA 98195, USA Soumik Pal, Department of Mathematics, University of Washington, Seattle, WA 98195, USA Elliot Paquette, Department of Mathematics, University of Washington, Seattle, WA 98195, USA Journal Probability Theory and Related Fields Online ISSN 1432-2064 Print ISSN 0178-8051
Print ISSN:
0178-8051
Electronic ISSN:
1432-2064
Topics:
Mathematics
