639 research outputs found
Entropy-driven cutoff phenomena
In this paper we present, in the context of Diaconis' paradigm, a general
method to detect the cutoff phenomenon. We use this method to prove cutoff in a
variety of models, some already known and others not yet appeared in
literature, including a chain which is non-reversible w.r.t. its stationary
measure. All the given examples clearly indicate that a drift towards the
opportune quantiles of the stationary measure could be held responsible for
this phenomenon. In the case of birth- and-death chains this mechanism is
fairly well understood; our work is an effort to generalize this picture to
more general systems, such as systems having stationary measure spread over the
whole state space or systems in which the study of the cutoff may not be
reduced to a one-dimensional problem. In those situations the drift may be
looked for by means of a suitable partitioning of the state space into classes;
using a statistical mechanics language it is then possible to set up a kind of
energy-entropy competition between the weight and the size of the classes.
Under the lens of this partitioning one can focus the mentioned drift and prove
cutoff with relative ease.Comment: 40 pages, 1 figur
Universality of trap models in the ergodic time scale
Consider a sequence of possibly random graphs , ,
whose vertices's have i.i.d. weights with a distribution
belonging to the basin of attraction of an -stable law, .
Let , , be a continuous time simple random walk on which
waits a \emph{mean} exponential time at each vertex . Under
considerably general hypotheses, we prove that in the ergodic time scale this
trap model converges in an appropriate topology to a -process. We apply this
result to a class of graphs which includes the hypercube, the -dimensional
torus, , random -regular graphs and the largest component of
super-critical Erd\"os-R\'enyi random graphs
Random subcube intersection graphs I: cliques and covering
We study random subcube intersection graphs, that is, graphs obtained by
selecting a random collection of subcubes of a fixed hypercube to serve
as the vertices of the graph, and setting an edge between a pair of subcubes if
their intersection is non-empty. Our motivation for considering such graphs is
to model `random compatibility' between vertices in a large network. For both
of the models considered in this paper, we determine the thresholds for
covering the underlying hypercube and for the appearance of s-cliques. In
addition we pose some open problems.Comment: 38 pages, 1 figur
Covering Partial Cubes with Zones
A partial cube is a graph having an isometric embedding in a hypercube.
Partial cubes are characterized by a natural equivalence relation on the edges,
whose classes are called zones. The number of zones determines the minimal
dimension of a hypercube in which the graph can be embedded. We consider the
problem of covering the vertices of a partial cube with the minimum number of
zones. The problem admits several special cases, among which are the problem of
covering the cells of a line arrangement with a minimum number of lines, and
the problem of finding a minimum-size fibre in a bipartite poset. For several
such special cases, we give upper and lower bounds on the minimum size of a
covering by zones. We also consider the computational complexity of those
problems, and establish some hardness results
Almost uniform sampling via quantum walks
Many classical randomized algorithms (e.g., approximation algorithms for
#P-complete problems) utilize the following random walk algorithm for {\em
almost uniform sampling} from a state space of cardinality : run a
symmetric ergodic Markov chain on for long enough to obtain a random
state from within total variation distance of the uniform
distribution over . The running time of this algorithm, the so-called {\em
mixing time} of , is , where
is the spectral gap of .
We present a natural quantum version of this algorithm based on repeated
measurements of the {\em quantum walk} . We show that it
samples almost uniformly from with logarithmic dependence on
just as the classical walk does; previously, no such
quantum walk algorithm was known. We then outline a framework for analyzing its
running time and formulate two plausible conjectures which together would imply
that it runs in time when is
the standard transition matrix of a constant-degree graph. We prove each
conjecture for a subclass of Cayley graphs.Comment: 13 pages; v2 added NSF grant info; v3 incorporated feedbac
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