88 research outputs found
Entropy of eigenfunctions on quantum graphs
We consider families of finite quantum graphs of increasing size and we are
interested in how eigenfunctions are distributed over the graph. As a measure
for the distribution of an eigenfunction on a graph we introduce the entropy,
it has the property that a large value of the entropy of an eigenfunction
implies that it cannot be localised on a small set on the graph. We then derive
lower bounds for the entropy of eigenfunctions which depend on the topology of
the graph and the boundary conditions at the vertices. The optimal bounds are
obtained for expanders with large girth, the bounds are similar to the ones
obtained by Anantharaman et.al. for eigenfunctions on manifolds of negative
curvature, and are based on the entropic uncertainty principle. For comparison
we compute as well the average behaviour of entropies on Neumann star graphs,
where the entropies are much smaller. Finally we compare our lower bounds with
numerical results for regular graphs and star graphs with different boundary
conditions.Comment: 28 pages, 3 figure
The necessity of conditions for graph quantum ergodicity and Cartesian products with an infinite graph
Anantharaman and Le Masson proved that any family of eigenbases of the
adjacency operators of a family of graphs is quantum ergodic (a form of
delocalization) assuming the graphs satisfy conditions of expansion and high
girth. In this paper, we show that neither of these two conditions is
sufficient by itself to necessitate quantum ergodicity. We also show that
having conditions of expansion and a specific relaxation of the high girth
constraint present in later papers on quantum ergodicity is not sufficient. We
do so by proving new properties of the Cartesian product of two graphs where
one is infinite.Comment: 12 pages, 3 figure
Many nodal domains in random regular graphs
Let be a random -regular graph. We prove that for every constant
, with high probability every eigenvector of the adjacency matrix
of with eigenvalue less than has
polylog nodal domains.Comment: 18 pages. Minor changes to the introductio
The limit points of the top and bottom eigenvalues of regular graphs
We prove that for each the set of all limit points of the second
largest eigenvalue of growing sequences of -regular graphs is
. A similar argument shows that the set of all limit points of
the smallest eigenvalue of growing sequences of -regular graphs with growing
(odd) girth is . The more general question of identifying
all vectors which are limit points of the vectors of the top eigenvalues of
sequences of -regular graphs is considered as well. As a by product, in the
study of discrete counterpart of the "scarring" phenomenon observed in the
investigation of quantum ergodicity on manifolds, our technique provides a
method to construct -regular almost Ramanujan graphs with large girth and
localized eigenvectors corresponding to eigenvalues larger than ,
strengthening a result of Alon, Ganguly, and Srivastava
High-Girth Near-Ramanujan Graphs with Lossy Vertex Expansion
Kahale proved that linear sized sets in -regular Ramanujan graphs have
vertex expansion and complemented this with construction of
near-Ramanujan graphs with vertex expansion no better than .
However, the construction of Kahale encounters highly local obstructions to
better vertex expansion. In particular, the poorly expanding sets are
associated with short cycles in the graph. Thus, it is natural to ask whether
high-girth Ramanujan graphs have improved vertex expansion. Our results are
two-fold:
1. For every for prime and infinitely many , we exhibit an
-vertex -regular graph with girth and vertex
expansion of sublinear sized sets bounded by whose nontrivial
eigenvalues are bounded in magnitude by .
2. In any Ramanujan graph with girth , all sets of size bounded by
have vertex expansion .
The tools in analyzing our construction include the nonbacktracking operator
of an infinite graph, the Ihara--Bass formula, a trace moment method inspired
by Bordenave's proof of Friedman's theorem, and a method of Kahale to study
dispersion of eigenvalues of perturbed graphs.Comment: 15 pages, 1 figur
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