4 research outputs found
On the spectral distribution of large weighted random regular graphs
McKay proved that the limiting spectral measures of the ensembles of
-regular graphs with vertices converge to Kesten's measure as
. In this paper we explore the case of weighted graphs. More
precisely, given a large -regular graph we assign random weights, drawn from
some distribution , to its edges. We study the relationship
between and the associated limiting spectral distribution
obtained by averaging over the weighted graphs. Among other results, we
establish the existence of a unique `eigendistribution', i.e., a weight
distribution such that the associated limiting spectral
distribution is a rescaling of . Initial investigations suggested
that the eigendistribution was the semi-circle distribution, which by Wigner's
Law is the limiting spectral measure for real symmetric matrices. We prove this
is not the case, though the deviation between the eigendistribution and the
semi-circular density is small (the first seven moments agree, and the
difference in each higher moment is ). Our analysis uses
combinatorial results about closed acyclic walks in large trees, which may be
of independent interest.Comment: Version 1.0, 19 page
A unitary test of the Ratios Conjecture
The Ratios Conjecture of Conrey, Farmer and Zirnbauer predicts the answers to
numerous questions in number theory, ranging from n-level densities and
correlations to mollifiers to moments and vanishing at the central point. The
conjecture gives a recipe to generate these answers, which are believed to be
correct up to square-root cancelation. These predictions have been verified,
for suitably restricted test functions, for the 1-level density of orthogonal
and symplectic families of L-functions. In this paper we verify the
conjecture's predictions for the unitary family of all Dirichlet -functions
with prime conductor; we show square-root agreement between prediction and
number theory if the support of the Fourier transform of the test function is
in (-1,1), and for support up to (-2,2) we show agreement up to a power savings
in the family's cardinality.Comment: Version 2: 24 pages, provided additional details, fixed some small
mistakes and expanded the exposition in place