24 research outputs found
Localisation and ageing in the parabolic Anderson model with Weibull potential
The parabolic Anderson model is the Cauchy problem for the heat equation on
the integer lattice with a random potential . We consider the case when
is a collection of independent identically
distributed random variables with Weibull distribution with parameter
, and we assume that the solution is initially localised in the
origin. We prove that, as time goes to infinity, the solution completely
localises at just one point with high probability, and we identify the
asymptotic behaviour of the localisation site. We also show that the intervals
between the times when the solution relocalises from one site to another
increase linearly over time, a phenomenon known as ageing.Comment: Published in at http://dx.doi.org/10.1214/13-AOP882 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
A conditioning principle for Galton-Watson trees
We show that an infinite Galton-Watson tree, conditioned on its martingale
limit being smaller than \eps, converges as \eps\downarrow 0 in law to the
regular -ary tree, where is the essential minimum of the offspring
distribution. This gives an example of entropic repulsion where the limit has
no entropy.Comment: This is now superseded by a new paper, arXiv:1204.3080, written
jointly with Nina Gantert. The new paper contains much stronger results (e.g.
the two-point concentration of the level at which the Galton-Watson tree
ceases to be minimal) based on a significantly more delicate analysis, making
the present paper redundan
Ageing in the parabolic Anderson model
The parabolic Anderson model is the Cauchy problem for the heat equation with
a random potential. We consider this model in a setting which is continuous in
time and discrete in space, and focus on time-constant, independent and
identically distributed potentials with polynomial tails at infinity. We are
concerned with the long-term temporal dynamics of this system. Our main result
is that the periods, in which the profile of the solutions remains nearly
constant, are increasing linearly over time, a phenomenon known as ageing. We
describe this phenomenon in the weak sense, by looking at the asymptotic
probability of a change in a given time window, and in the strong sense, by
identifying the almost sure upper envelope for the process of the time
remaining until the next change of profile. We also prove functional scaling
limit theorems for profile and growth rate of the solution of the parabolic
Anderson model.Comment: 43 pages, 4 figure
Galton-Watson trees with vanishing martingale limit
We show that an infinite Galton-Watson tree, conditioned on its martingale
limit being smaller than \eps, agrees up to generation with a regular
-ary tree, where is the essential minimum of the offspring
distribution and the random variable is strongly concentrated near an
explicit deterministic function growing like a multiple of \log(1/\eps). More
precisely, we show that if then with high probability as \eps
\downarrow 0, takes exactly one or two values. This shows in particular
that the conditioned trees converge to the regular -ary tree, providing an
example of entropic repulsion where the limit has vanishing entropy.Comment: This supersedes an earlier paper, arXiv:1006.2315, written by a
subset of the authors. Compared with the earlier version, the main result
(the two-point concentration of the level at which the Galton-Watson tree
ceases to be minimal) is much stronger and requires significantly more
delicate analysi
A two cities theorem for the parabolic Anderson model
The parabolic Anderson problem is the Cauchy problem for the heat equation
on with random potential . We
consider independent and identically distributed potentials, such that the
distribution function of converges polynomially at infinity. If is
initially localized in the origin, that is, if u(0,{z})={\mathbh1}_0({z}), we
show that, as time goes to infinity, the solution is completely localized in
two points almost surely and in one point with high probability. We also
identify the asymptotic behavior of the concentration sites in terms of a weak
limit theorem.Comment: Published in at http://dx.doi.org/10.1214/08-AOP405 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org