910 research outputs found
Spectra of lifted Ramanujan graphs
A random -lift of a base graph is its cover graph on the vertices
, where for each edge in there is an independent
uniform bijection , and has all edges of the form .
A main motivation for studying lifts is understanding Ramanujan graphs, and
namely whether typical covers of such a graph are also Ramanujan.
Let be a graph with largest eigenvalue and let be the
spectral radius of its universal cover. Friedman (2003) proved that every "new"
eigenvalue of a random lift of is with high
probability, and conjectured a bound of , which would be tight by
results of Lubotzky and Greenberg (1995). Linial and Puder (2008) improved
Friedman's bound to . For -regular graphs,
where and , this translates to a bound of
, compared to the conjectured .
Here we analyze the spectrum of a random -lift of a -regular graph
whose nontrivial eigenvalues are all at most in absolute value. We
show that with high probability the absolute value of every nontrivial
eigenvalue of the lift is . This result is
tight up to a logarithmic factor, and for it
substantially improves the above upper bounds of Friedman and of Linial and
Puder. In particular, it implies that a typical -lift of a Ramanujan graph
is nearly Ramanujan.Comment: 34 pages, 4 figure
On the Expansion of Group-Based Lifts
A -lift of an -vertex base graph is a graph on
vertices, where each vertex of is replaced by vertices
and each edge in is replaced by a matching
representing a bijection so that the edges of are of the form
. Lifts have been studied as a means to efficiently
construct expanders. In this work, we study lifts obtained from groups and
group actions. We derive the spectrum of such lifts via the representation
theory principles of the underlying group. Our main results are:
(1) There is a constant such that for every , there
does not exist an abelian -lift of any -vertex -regular base graph
with being almost Ramanujan (nontrivial eigenvalues of the adjacency matrix
at most in magnitude). This can be viewed as an analogue of the
well-known no-expansion result for abelian Cayley graphs.
(2) A uniform random lift in a cyclic group of order of any -vertex
-regular base graph , with the nontrivial eigenvalues of the adjacency
matrix of bounded by in magnitude, has the new nontrivial
eigenvalues also bounded by in magnitude with probability
. In particular, there is a constant such that for
every , there exists a lift of every Ramanujan graph in
a cyclic group of order with being almost Ramanujan. We use this to
design a quasi-polynomial time algorithm to construct almost Ramanujan
expanders deterministically.
The existence of expanding lifts in cyclic groups of order
can be viewed as a lower bound on the order of the largest abelian group
that produces expanding lifts. Our results show that the lower bound matches
the upper bound for (upto in the exponent)
Fractal Weyl laws in discrete models of chaotic scattering
We analyze simple models of quantum chaotic scattering, namely quantized open
baker's maps. We numerically compute the density of quantum resonances in the
semiclassical r\'{e}gime. This density satisfies a fractal Weyl law, where the
exponent is governed by the (fractal) dimension of the set of trapped
trajectories. This type of behaviour is also expected in the (physically more
relevant) case of Hamiltonian chaotic scattering. Within a simplified model, we
are able to rigorously prove this Weyl law, and compute quantities related to
the "coherent transport" through the system, namely the conductance and "shot
noise". The latter is close to the prediction of random matrix theory.Comment: Invited article in the Special Issue of Journal of Physics A on
"Trends in Quantum Chaotic Scattering
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