9,315 research outputs found
Number variance for hierarchical random walks and related fluctuations
We study an infinite system of independent symmetric random walks on a
hierarchical group, in particular, the c-random walks. Such walks are used,
e.g., in population genetics. The number variance problem consists in
investigating if the variance of the number of "particles" N_n(L) lying in the
ball of radius L at a given time n remains bounded, or even better, converges
to a finite limit, as . We give a necessary and sufficient
condition and discuss its relationship to transience/recurrence property of the
walk. Next we consider normalized fluctuations of N_n(L) around the mean as
and L is increased in an appropriate way. We prove convergence of
finite dimensional distributions to a Gaussian process whose properties are
discussed. As the c-random walks mimic symmetric stable processes on R, we
compare our results to those obtained by Hambly and Jones (2007,2009), where
the number variance problem for an infinite system of symmetric stable
processes on R was studied. Since the hierarchical group is an ultrametric
space, corresponding results for symmetric stable processes and hierarchical
random walks may be analogous or quite different, as has been observed in other
contexts. An example of a difference in the present context is that for the
stable processes a fluctuation limit process is a centered Gaussian process
which is not Markovian and has long range dependent stationary increments, but
the counterpart for hierarchical random walks is Markovian, and in a special
case it has independent increments
Short note on the emergence of fractional kinetics
In the present Short Note an idea is proposed to explain the emergence and
the observation of processes in complex media that are driven by fractional
non-Markovian master equations. Particle trajectories are assumed to be solely
Markovian and described by the Continuous Time Random Walk model. But, as a
consequence of the complexity of the medium, each trajectory is supposed to
scale in time according to a particular random timescale. The link from this
framework to microscopic dynamics is discussed and the distribution of
timescales is computed. In particular, when a stationary distribution is
considered, the timescale distribution is uniquely determined as a function
related to the fundamental solution of the space-time fractional diffusion
equation. In contrast, when the non-stationary case is considered, the
timescale distribution is no longer unique. Two distributions are here
computed: one related to the M-Wright/Mainardi function, which is Green's
function of the time-fractional diffusion equation, and another related to the
Mittag-Leffler function, which is the solution of the fractional-relaxation
equation
The evolution of the cover time
The cover time of a graph is a celebrated example of a parameter that is easy
to approximate using a randomized algorithm, but for which no constant factor
deterministic polynomial time approximation is known. A breakthrough due to
Kahn, Kim, Lovasz and Vu yielded a (log log n)^2 polynomial time approximation.
We refine this upper bound, and show that the resulting bound is sharp and
explicitly computable in random graphs. Cooper and Frieze showed that the cover
time of the largest component of the Erdos-Renyi random graph G(n,c/n) in the
supercritical regime with c>1 fixed, is asymptotic to f(c) n \log^2 n, where
f(c) tends to 1 as c tends to 1. However, our new bound implies that the cover
time for the critical Erdos-Renyi random graph G(n,1/n) has order n, and shows
how the cover time evolves from the critical window to the supercritical phase.
Our general estimate also yields the order of the cover time for a variety of
other concrete graphs, including critical percolation clusters on the Hamming
hypercube {0,1}^n, on high-girth expanders, and on tori Z_n^d for fixed large
d. For the graphs we consider, our results show that the blanket time,
introduced by Winkler and Zuckerman, is within a constant factor of the cover
time. Finally, we prove that for any connected graph, adding an edge can
increase the cover time by at most a factor of 4.Comment: 14 pages, to appear in CP
On the spatial Markov property of soups of unoriented and oriented loops
We describe simple properties of some soups of unoriented Markov loops and of
some soups of oriented Markov loops that can be interpreted as a spatial Markov
property of these loop-soups. This property of the latter soup is related to
well-known features of the uniform spanning trees (such as Wilson's algorithm)
while the Markov property of the former soup is related to the Gaussian Free
Field and to identities used in the foundational papers of Symanzik, Nelson,
and of Brydges, Fr\"ohlich and Spencer or Dynkin, or more recently by Le Jan
The lineage process in Galton--Watson trees and globally centered discrete snakes
We consider branching random walks built on Galton--Watson trees with
offspring distribution having a bounded support, conditioned to have nodes,
and their rescaled convergences to the Brownian snake. We exhibit a notion of
``globally centered discrete snake'' that extends the usual settings in which
the displacements are supposed centered. We show that under some additional
moment conditions, when goes to , ``globally centered discrete
snakes'' converge to the Brownian snake. The proof relies on a precise study of
the lineage of the nodes in a Galton--Watson tree conditioned by the size, and
their links with a multinomial process [the lineage of a node is the vector
indexed by giving the number of ancestors of having children
and for which is a descendant of the th one]. Some consequences
concerning Galton--Watson trees conditioned by the size are also derived.Comment: Published in at http://dx.doi.org/10.1214/07-AAP450 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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