419 research outputs found
Sublogarithmic fluctuations for internal DLA
We consider internal diffusion limited aggregation in dimension larger than
or equal to two. This is a random cluster growth model, where random walks
start at the origin of the d-dimensional lattice, one at a time, and stop
moving when reaching a site that is not occupied by previous walks. It is known
that the asymptotic shape of the cluster is a sphere. When the dimension is two
or more, we have shown in a previous paper that the inner (resp., outer)
fluctuations of its radius is at most of order [resp.,
]. Using the same approach, we improve the upper bound
on the inner fluctuation to when d is larger
than or equal to three. The inner fluctuation is then used to obtain a similar
upper bound on the outer fluctuation.Comment: Published in at http://dx.doi.org/10.1214/11-AOP735 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
On outer fluctuations for internal DLA
We had established inner and outer fluctuation for the internal DLA cluster
when all walks are launched from the origin. In obtaining the outer
fluctuation, we had used a deep lemma of Jerison, Levine and Sheffield, which
estimate roughly the possibility of fingering, and had provided a simple proof
using an interesting estimate for crossing probability for a simple random
walk. The application of the crossing probability to the fingering for the
internal DLA cluster contains a flaw discovered recently, that we correct in
this note. We take the opportunity to make a self-contained exposition.Comment: 10 page
Metastable states, quasi-stationary distributions and soft measures
We establish metastability in the sense of Lebowitz and Penrose under
practical and simple hypothesis for (families of) Markov chains on finite
configuration space in some asymptotic regime, including the case of
configuration space size going to infinity. By comparing restricted ensemble
and quasi-stationary measures, we study point-wise convergence velocity of
Yaglom limits and prove asymptotic exponential exit law. We introduce soft
measures as interpolation between restricted ensemble and quasi-stationary
measure to prove an asymptotic exponential transition law on a generally
different time scale. By using potential theoretic tools, we prove a new
general Poincar\'e inequality and give sharp estimates via two-sided
variational principles on relaxation time as well as mean exit time and
transition time. We also establish local thermalization on a shorter time scale
and give mixing time asymptotics up to a constant factor through a two-sided
variational principal. All our asymptotics are given with explicit quantitative
bounds on the corrective terms.Comment: 41 page
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
Estimating the inverse trace using random forests on graphs
Some data analysis problems require the computation of (regularised) inverse
traces, i.e. quantities of the form \Tr (q \bI + \bL)^{-1}. For large
matrices, direct methods are unfeasible and one must resort to approximations,
for example using a conjugate gradient solver combined with Girard's trace
estimator (also known as Hutchinson's trace estimator). Here we describe an
unbiased estimator of the regularized inverse trace, based on Wilson's
algorithm, an algorithm that was initially designed to draw uniform spanning
trees in graphs. Our method is fast, easy to implement, and scales to very
large matrices. Its main drawback is that it is limited to diagonally dominant
matrices \bL.Comment: Submitted to GRETSI conferenc
Active Phase for Activated Random Walks on the Lattice in all Dimensions
We show that the critical density of the Activated Random Walk model on
is strictly less than one when the sleep rate is small
enough, and tends to when , in any dimension .
As far as we know, the result is new for .
We prove this by showing that, for high enough density and small enough sleep
rate, the stabilization time of the model on the -dimensional torus is
exponentially large. To do so, we fix the the set of sites where the particles
eventually fall asleep, which reduces the problem to a simpler model with
density one. Taking advantage of the Abelian property of the model, we show
that the stabilization time stochastically dominates the escape time of a
one-dimensional random walk with a negative drift. We then check that this slow
phase for the finite volume dynamics implies the existence of an active phase
on the infinite lattice.Comment: 27 pages, new version with minor correction
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
Entre biologistes, militaires et industriels : l’introduction de la pénicilline en France à la Libération
IntroductionEn France, les premières années après la Seconde Guerre mondiale sont des années d’intense débat sur la « modernisation » du pays, ses dimensions politiques, économiques ou sociales. Les sciences n’échappent pas à ce mouvement. Dans ce domaine comme dans les autres, l’heure est au bilan critique de l’entre-deux-guerres, à la mise en accusation du retard accumulé, à la rédaction de toutes sortes de projets et programmes d’action. Pour les sciences de la vie et de la santé, la secon..
The Critical Density for Activated Random Walks is always less than 1
Activated Random Walks, on for any , is an
interacting particle system, where particles can be in either of two states:
active or frozen. Each active particle performs a continuous-time simple random
walk during an exponential time of parameter , after which it stays
still in the frozen state, until another active particle shares its location,
and turns it instantaneously back into activity. This model is known to have a
phase transition, and we show that the critical density, controlling the phase
transition, is less than one in any dimension and for any value of the sleep
rate . We provide upper bounds for the critical density in both the
small and large regimes.Comment: 48 pages, 1 figur
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