3,166 research outputs found
Trace Formulae and Spectral Statistics for Discrete Laplacians on Regular Graphs (I)
Trace formulae for d-regular graphs are derived and used to express the
spectral density in terms of the periodic walks on the graphs under
consideration. The trace formulae depend on a parameter w which can be tuned
continuously to assign different weights to different periodic orbit
contributions. At the special value w=1, the only periodic orbits which
contribute are the non back- scattering orbits, and the smooth part in the
trace formula coincides with the Kesten-McKay expression. As w deviates from
unity, non vanishing weights are assigned to the periodic walks with
back-scatter, and the smooth part is modified in a consistent way. The trace
formulae presented here are the tools to be used in the second paper in this
sequence, for showing the connection between the spectral properties of
d-regular graphs and the theory of random matrices.Comment: 22 pages, 3 figure
The Relativized Second Eigenvalue Conjecture of Alon
We prove a relativization of the Alon Second Eigenvalue Conjecture for all
-regular base graphs, , with : for any , we show that
a random covering map of degree to has a new eigenvalue greater than
in absolute value with probability .
Furthermore, if is a Ramanujan graph, we show that this probability is
proportional to , where
is an integer depending on , which can be computed by a finite algorithm for
any fixed . For any -regular graph, , is
greater than .
Our proof introduces a number of ideas that simplify and strengthen the
methods of Friedman's proof of the original conjecture of Alon. The most
significant new idea is that of a ``certified trace,'' which is not only
greatly simplifies our trace methods, but is the reason we can obtain the
estimate above. This estimate represents an
improvement over Friedman's results of the original Alon conjecture for random
-regular graphs, for certain values of
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