Merging costs for the additive Marcus-Lushnikov process, and Union-Find algorithms

Starting with a monodisperse configuration with $n$ size-1 particles, an additive Marcus-Lushnikov process evolves until it reaches its final state (a unique particle with mass $n$). At each of the $n-1$ steps of its evolution, a merging cost is incurred, that depends on the sizes of the two particles involved, and on an independent random factor. This paper deals with the asymptotic behaviour of the cumulated costs up to the $k$th clustering, under various regimes for $(n,k)$, with applications to the study of Union--Find algorithms.Comment: 28 pages, 1 figur

The center of mass of the ISE and the Wiener index of trees

We derive the distribution of the center of mass $S$ of the integrated superBrownian excursion (ISE) {from} the asymptotic distribution of the Wiener index for simple trees. Equivalently, this is the distribution of the integral of a Brownian snake. A recursion formula for the moments and asymptotics for moments and tail probabilities are derived.Comment: 11 page

Local limit of labeled trees and expected volume growth in a random quadrangulation

Exploiting a bijective correspondence between planar quadrangulations and well-labeled trees, we define an ensemble of infinite surfaces as a limit of uniformly distributed ensembles of quadrangulations of fixed finite volume. The limit random surface can be described in terms of a birth and death process and a sequence of multitype Galton--Watson trees. As a consequence, we find that the expected volume of the ball of radius $r$ around a marked point in the limit random surface is $\Theta(r^4)$.Comment: Published at http://dx.doi.org/10.1214/009117905000000774 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

The impatient collector

In the coupon collector problem with $n$ items, the collector needs a random number of tries $T_{n}\simeq n\ln n$ to complete the collection. Also, after $nt$ tries, the collector has secured approximately a fraction $\zeta_{\infty}(t)=1-e^{-t}$ of the complete collection, so we call $\zeta_{\infty}$ the (asymptotic) \emph{completion curve}. In this paper, for $\nu>0$, we address the asymptotic shape $\zeta (\nu,.)$ of the completion curve under the condition $T_{n}\leq \left( 1+\nu \right) n$, i.e. assuming that the collection is \emph{completed unlikely fast}. As an application to the asymptotic study of complete accessible automata, we provide a new derivation of a formula due to Kor\v{s}unov

On the Convergence of Population Protocols When Population Goes to Infinity

Population protocols have been introduced as a model of sensor networks consisting of very limited mobile agents with no control over their own movement. A population protocol corresponds to a collection of anonymous agents, modeled by finite automata, that interact with one another to carry out computations, by updating their states, using some rules. Their computational power has been investigated under several hypotheses but always when restricted to finite size populations. In particular, predicates stably computable in the original model have been characterized as those definable in Presburger arithmetic. We study mathematically the convergence of population protocols when the size of the population goes to infinity. We do so by giving general results, that we illustrate through the example of a particular population protocol for which we even obtain an asymptotic development. This example shows in particular that these protocols seem to have a rather different computational power when a huge population hypothesis is considered.Comment: Submitted to Applied Mathematics and Computation. 200

Segmentation and tracking of video objects for a content-based video indexing context

This paper examines the problem of segmentation and tracking of video objects for content-based information retrieval. Segmentation and tracking of video objects plays an important role in index creation and user request definition steps. The object is initially selected using a semi-automatic approach. For this purpose, a user-based selection is required to define roughly the object to be tracked. In this paper, we propose two different methods to allow an accurate contour definition from the user selection. The first one is based on an active contour model which progressively refines the selection by fitting the natural edges of the object while the second used a binary partition tree with aPeer ReviewedPostprint (published version

A non-ergodic probabilistic cellular automaton with a unique invariant measure

To appear in Stochastic Processes and their Applications.International audienceWe exhibit a Probabilistic Cellular Automaton (PCA) on the integers with an alphabet and a neighborhood of size 2 which is non-ergodic although it has a unique invariant measure. This answers by the negative an old open question on whether uniqueness of the invariant measure implies ergodicity for a PCA

Asynchronous Cellular Automata and Brownian Motion

International audienceThis paper deals with some very simple interacting particle systems, \emphelementary cellular automata, in the fully asynchronous dynamics: at each time step, a cell is randomly picked, and updated. When the initial configuration is simple, we describe the asymptotic behavior of the random walks performed by the borders of the black/white regions. Following a classification introduced by Fatès \emphet al., we show that four kinds of asymptotic behavior arise, two of them being related to Brownian motion