98 research outputs found
An exactly solvable travelling wave equation in the Fisher-KPP class
For a simple one dimensional lattice version of a travelling wave equation,
we obtain an exact relation between the initial condition and the position of
the front at any later time. This exact relation takes the form of an inverse
problem: given the times at which the travelling wave reaches the
positions , one can deduce the initial profile. We show, by means of complex
analysis, that a number of known properties of travelling wave equations in the
Fisher-KPP class can be recovered, in particular Bramson's shifts of the
positions. We also recover and generalize Ebert-van Saarloos' corrections
depending on the initial condition.Comment: For version 2: some typos + clarification of (87
Accessibility percolation with backsteps
Consider a graph in which each site is endowed with a value called
\emph{fitness}. A path in the graph is said to be "open" or "accessible" if the
fitness values along that path is strictly increasing. We say that there is
accessibility percolation between two sites when such a path between them
exists. Motivated by the so called House-of-Cards model from evolutionary
biology, we consider this question on the -hypercube where the
fitness values are independent random variables. We show that, in the large
limit, the probability that an accessible path exists from an arbitrary
starting point to the (random) fittest site is no more than and we conjecture that this probability
does converge to . More precisely, there is a phase transition on
the value of the fitness of the starting site: assuming that the fitnesses
are uniform in , we show that, in the large limit, there is almost
surely no path to the fittest site if and we conjecture that
there are almost surely many paths if .
If one conditions on the fittest site to be on the opposite corner of the
starting site rather than being randomly chosen, the picture remains the same
but with the critical point being now .
Along the way, we obtain a large estimation for the number of self-avoiding
paths joining two opposite corners of the -hypercube
The number of accessible paths in the hypercube
Motivated by an evolutionary biology question, we study the following
problem: we consider the hypercube where each node carries an
independent random variable uniformly distributed on , except
which carries the value and which carries
the value . We study the number of paths from vertex
to the opposite vertex along which the values
on the nodes form an increasing sequence. We show that if the value on
is set to then converges in law as
to times the product of two standard independent
exponential variables. As a first step in the analysis, we study the same
question when the graph is that of a tree where the root has arity , each
node at level 1 has arity , \ldots, and the nodes at level have only
one offspring which are the leaves of the tree (all the leaves are assigned the
value 1, the root the value ).Comment: Published at http://dx.doi.org/10.3150/14-BEJ641 in the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Branching Brownian motion with absorption and the all-time minimum of branching Brownian motion with drift
We study a dyadic branching Brownian motion on the real line with absorption
at 0, drift and started from a single particle at position
When is large enough so that the process has a positive
probability of survival, we consider the number of individuals absorbed
at 0 by time and for the functions We show that if and only of
for some and we study the properties of these functions.
Furthermore, for is
the cumulative distribution function of the all time minimum of the branching
Brownian motion with drift started at 0 without absorption.
We give three descriptions of the family through a
single pair of functions, as the two extremal solutions of the
Kolmogorov-Petrovskii-Piskunov (KPP) traveling wave equation on the half-line,
through a martingale representation and as an explicit series expansion. We
also obtain a precise result concerning the tail behavior of . In
addition, in the regime where almost surely, we show that suitably centered converges to the KPP critical
travelling wave on the whole real line.Comment: Grant information adde
An exactly soluble noisy traveling wave equation appearing in the problem of directed polymers in a random medium
We calculate exactly the velocity and diffusion constant of a microscopic
stochastic model of evolving particles which can be described by a noisy
traveling wave equation with a noise of order . Our model can be
viewed as the infinite range limit of a directed polymer in random medium with
sites in the transverse direction. Despite some peculiarities of the
traveling wave equations in the absence of noise, our exact solution allows us
to test the validity of a simple cutoff approximation and to show that, in the
weak noise limit, the position of the front can be completely described by the
effect of the noise on the first particle.Comment: 5 page
Growth rates of the population in a branching Brownian motion with an inhomogeneous breeding potential
We consider a branching particle system where each particle moves as an
independent Brownian motion and breeds at a rate proportional to its distance
from the origin raised to the power , for . The asymptotic
behaviour of the right-most particle for this system is already known; in this
article we give large deviations probabilities for particles following
"difficult" paths, growth rates along "easy" paths, the total population growth
rate, and we derive the optimal paths which particles must follow to achieve
this growth rate.Comment: 56 pages, 1 figur
How genealogies are affected by the speed of evolution
In a series of recent works it has been shown that a class of simple models
of evolving populations under selection leads to genealogical trees whose
statistics are given by the Bolthausen-Sznitman coalescent rather than by the
well known Kingman coalescent in the case of neutral evolution. Here we show
that when conditioning the genealogies on the speed of evolution, one finds a
one parameter family of tree statistics which interpolates between the
Bolthausen-Sznitman and Kingman's coalescents. This interpolation can be
calculated explicitly for one specific version of the model, the exponential
model. Numerical simulations of another version of the model and a
phenomenological theory indicate that this one-parameter family of tree
statistics could be universal. We compare this tree structure with those
appearing in other contexts, in particular in the mean field theory of spin
glasses
A free boundary problem arising from branching Brownian motion with selection
We study a free boundary problem for a parabolic partial differential
equation in which the solution is coupled to the moving boundary through an
integral constraint. The problem arises as the hydrodynamic limit of an
interacting particle system involving branching Brownian motion with selection,
the so-called Brownian bees model which is studied in a companion paper. In
this paper we prove existence and uniqueness of the solution to the free
boundary problem, and we characterise the behaviour of the solution in the
large time limit.Comment: 53 page
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