1,139 research outputs found
New infinite family of regular edge-isoperimetric graphs
We introduce a new infinite family of regular graphs admitting nested
solutions in the edge-isoperimetric problem for all their Cartesian powers. The
obtained results include as special cases most of previously known results in
this area
Quenched invariance principles for the random conductance model on a random graph with degenerate ergodic weights
We consider a stationary and ergodic random field
that is parameterized by the edge set of the Euclidean lattice ,
. The random variable , taking values in and
satisfying certain moment bounds, is thought of as the conductance of the edge
. Assuming that the set of edges with positive conductances give rise to a
unique infinite cluster , we prove a quenched
invariance principle for the continuous-time random walk among random
conductances under relatively mild conditions on the structure of the infinite
cluster. An essential ingredient of our proof is a new anchored relative
isoperimetric inequality.Comment: 22 page
Absorption Time of the Moran Process
The Moran process models the spread of mutations in populations on graphs. We
investigate the absorption time of the process, which is the time taken for a
mutation introduced at a randomly chosen vertex to either spread to the whole
population, or to become extinct. It is known that the expected absorption time
for an advantageous mutation is O(n^4) on an n-vertex undirected graph, which
allows the behaviour of the process on undirected graphs to be analysed using
the Markov chain Monte Carlo method. We show that this does not extend to
directed graphs by exhibiting an infinite family of directed graphs for which
the expected absorption time is exponential in the number of vertices. However,
for regular directed graphs, we show that the expected absorption time is
Omega(n log n) and O(n^2). We exhibit families of graphs matching these bounds
and give improved bounds for other families of graphs, based on isoperimetric
number. Our results are obtained via stochastic dominations which we
demonstrate by establishing a coupling in a related continuous-time model. The
coupling also implies several natural domination results regarding the fixation
probability of the original (discrete-time) process, resolving a conjecture of
Shakarian, Roos and Johnson.Comment: minor change
Small spectral radius and percolation constants on non-amenable Cayley graphs
Motivated by the Benjamini-Schramm non-unicity of percolation conjecture we
study the following question. For a given finitely generated non-amenable group
, does there exist a generating set such that the Cayley graph
, without loops and multiple edges, has non-unique percolation,
i.e., ? We show that this is true if
contains an infinite normal subgroup such that is non-amenable.
Moreover for any finitely generated group containing there exists
a generating set of such that . In particular
this applies to free Burnside groups with . We
also explore how various non-amenability numerics, such as the isoperimetric
constant and the spectral radius, behave on various growing generating sets in
the group
Coboundary expanders
We describe a natural topological generalization of edge expansion for graphs
to regular CW complexes and prove that this property holds with high
probability for certain random complexes.Comment: Version 2: significant rewrite. 18 pages, title changed, and main
theorem extended to more general random complexe
Ramanujan Complexes and bounded degree topological expanders
Expander graphs have been a focus of attention in computer science in the
last four decades. In recent years a high dimensional theory of expanders is
emerging. There are several possible generalizations of the theory of expansion
to simplicial complexes, among them stand out coboundary expansion and
topological expanders. It is known that for every d there are unbounded degree
simplicial complexes of dimension d with these properties. However, a major
open problem, formulated by Gromov, is whether bounded degree high dimensional
expanders, according to these definitions, exist for d >= 2. We present an
explicit construction of bounded degree complexes of dimension d = 2 which are
high dimensional expanders. More precisely, our main result says that the
2-skeletons of the 3-dimensional Ramanujan complexes are topological expanders.
Assuming a conjecture of Serre on the congruence subgroup property, infinitely
many of them are also coboundary expanders.Comment: To appear in FOCS 201
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