72 research outputs found
A class of random walks in reversible dynamic environment: antisymmetry and applications to the East model
We introduce via perturbation a class of random walks in reversible dynamic
environments having a spectral gap. In this setting one can apply the
mathematical results derived in http://arxiv.org/abs/1602.06322. As first
results, we show that the asymptotic velocity is antisymmetric in the
perturbative parameter and, for a subclass of random walks, we characterize the
velocity and a stationary distribution of the environment seen from the walker
as suitable series in the perturbative parameter. We then consider as a special
case a random walk on the East model that tends to follow dynamical interfaces
between empty and occupied regions. We study the asymptotic velocity and
density profile for the environment seen from the walker. In particular, we
determine the sign of the velocity when the density of the underlying East
process is not 1/2, and we discuss the appearance of a drift in the balanced
setting given by density 1/2
The parabolic Anderson model on the hypercube
We consider the parabolic Anderson model on the -dimensional hypercube
with random i.i.d. potential . We parametrize time by
volume and study at the location of the -th largest potential,
. Our main result is that, for a certain class of potential
distributions, the solution exhibits a phase transition: for short time scales
behaves like a system without diffusion and grows as
, whereas, for long time scales
the growth is dictated by the principle eigenvalue and the corresponding
eigenfunction of the operator , for which we give
precise asymptotics. Moreover, the transition time depends only on the
difference .
One of our main motivations in this article is to investigate the
mutation-selection model of population genetics on a random fitness landscape,
which is given by the ratio of to its total mass, with
corresponding to the fitness landscape. We show that the phase transition of
the solution translates to the mutation-selection model as follows: a
population initially concentrated at moves completely to
on time scales where the transition of growth rates happens. The
class of potentials we consider involves the Random Energy Model (REM) of
statistical physics which is studied as one of the main examples of a random
fitness landscape.Comment: 22 pages, 1 figur
Intertwining wavelets or Multiresolution analysis on graphs through random forests
We propose a new method for performing multiscale analysis of functions
defined on the vertices of a finite connected weighted graph. Our approach
relies on a random spanning forest to downsample the set of vertices, and on
approximate solutions of Markov intertwining relation to provide a subgraph
structure and a filter bank leading to a wavelet basis of the set of functions.
Our construction involves two parameters q and q'. The first one controls the
mean number of kept vertices in the downsampling, while the second one is a
tuning parameter between space localization and frequency localization. We
provide an explicit reconstruction formula, bounds on the reconstruction
operator norm and on the error in the intertwining relation, and a Jackson-like
inequality. These bounds lead to recommend a way to choose the parameters q and
q'. We illustrate the method by numerical experiments.Comment: 39 pages, 12 figure
Symmetric exclusion as a random environment: hydrodynamic limits
We consider a one-dimensional continuous time random walk with transition
rates depending on an underlying autonomous simple symmetric exclusion process
starting out of equilibrium. This model represents an example of a random walk
in a slowly non-uniform mixing dynamic random environment. Under a proper
space-time rescaling in which the exclusion is speeded up compared to the
random walk, we prove a hydrodynamic limit theorem for the exclusion as seen by
this walk and we derive an ODE describing the macroscopic evolution of the
walk. The main difficulty is the proof of a replacement lemma for the exclusion
as seen from the walk without explicit knowledge of its invariant measures. We
further discuss how to obtain similar results for several variants of this
model.Comment: 19 page
On some random forests with determinantal roots
Consider a finite weighted oriented graph. We study a probability measure on the set of spanning rooted oriented forests on the graph. We prove that the set of roots sampled from this measure is a determinantal process, characterized by a possibly non-symmetric kernel with complex eigenvalues. We then derive several results relating this measure to the Markov process associated with the starting graph, to the spectrum of its generator and to hitting times of subsets of the graph. In particular, the mean hitting time of the set of roots turns out to be independent of the starting point, conditioning or not to a given number of roots. Wilson's algorithm provides a way to sample this measure and, in absence of complex eigenvalues of the generator, we explain how to get samples with a number of roots approximating a prescribed integer. We also exploit the properties of this measure to give some probabilistic insight into the proof of an algebraic result due to Micchelli and Willoughby [13]. Further, we present two different related coalescence and fragmentation processes
Laws of Large Numbers for Weighted Sums of Independent Random Variables: A Game of Mass
We consider weighted sums of independent random variables regulated by an increment sequence and provide operative conditions that ensure a strong law of large numbers for such sums in both the centred and non-centred case. The existing criteria for the strong law are either implicit or based on restrictions on the increment sequence. In our setup we allow for an arbitrary sequence of increments, possibly random , provided the random variables regulated by such increments satisfy some mild concentration conditions. In the non-centred case, convergence can be translated into the behaviour of a deterministic sequence and it becomes a game of mass when the expectation of the random variables is a function of the increment sizes. We identify various classes of increments and illustrate them with a variety of concrete examples
Mixing times of random walks on dynamic configuration models
The mixing time of a random walk, with or without backtracking, on a random
graph generated according to the configuration model on vertices, is known
to be of order . In this paper we investigate what happens when the
random graph becomes {\em dynamic}, namely, at each unit of time a fraction
of the edges is randomly rewired. Under mild conditions on the
degree sequence, guaranteeing that the graph is locally tree-like, we show that
for every the -mixing time of random walk
without backtracking grows like
as , provided
that . The latter condition
corresponds to a regime of fast enough graph dynamics. Our proof is based on a
randomised stopping time argument, in combination with coupling techniques and
combinatorial estimates. The stopping time of interest is the first time that
the walk moves along an edge that was rewired before, which turns out to be
close to a strong stationary time.Comment: 23 pages, 6 figures. Previous version contained a mistake in one of
the proofs. In this version we look at nonbacktracking random walk instead of
simple random wal
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