2,850 research outputs found
Percolation by cumulative merging and phase transition for the contact process on random graphs
Given a weighted graph, we introduce a partition of its vertex set such that
the distance between any two clusters is bounded from below by a power of the
minimum weight of both clusters. This partition is obtained by recursively
merging smaller clusters and cumulating their weights. For several classical
random weighted graphs, we show that there exists a phase transition regarding
the existence of an infinite cluster.
The motivation for introducing this partition arises from a connection with
the contact process as it roughly describes the geometry of the sets where the
process survives for a long time. We give a sufficient condition on a graph to
ensure that the contact process has a non trivial phase transition in terms of
the existence of an infinite cluster. As an application, we prove that the
contact process admits a sub-critical phase on d-dimensional random geometric
graphs and on random Delaunay triangulations. To the best of our knowledge,
these are the first examples of graphs with unbounded degrees where the
critical parameter is shown to be strictly positive.Comment: 50 pages, many figure
Rate of growth of a transient cookie random walk
We consider a one-dimensional transient cookie random walk. It is known from
a previous paper that a cookie random walk has positive or zero speed
according to some positive parameter or . In this article,
we give the exact rate of growth of in the zero speed regime, namely:
for , converges in law to a
Mittag-Leffler distribution whereas for , converges
in probability to some positive constant
Information homeostasis as a fundamental principle governing the cell division and death
To express genetic information with minimal error is one of the key functions of a cell. Here we propose an information theory based phenomenological model for the expression of genetic information. Based on the model we propose, the concept of "information homeostasis" ensures that genetic information is expressed with minimal error. We suggest that together with energy homeostasis, information homeostasis is a fundamental working principle of a biological cell. This model proposes a novel explanation of why a cell divides and why it stops to divide and thus provides novel insight into oncogenesis and various neuro-degenerative diseases. Moreover, the model suggests a theoretical framework to understand cell division and death, beyond specific biochemical pathways
On the speed of a cookie random walk
We consider the model of the one-dimensional cookie random walk when the
initial cookie distribution is spatially uniform and the number of cookies per
site is finite. We give a criterion to decide whether the limiting speed of the
walk is non-zero. In particular, we show that a positive speed may be obtained
for just 3 cookies per site. We also prove a result on the continuity of the
speed with respect to the initial cookie distribution
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