Dynamical Windings of Random Walks and Exclusion Models. Part I: Thermodynamic Limit

We consider a system consisting of a planar random walk on a square lattice, submitted to stochastic elementary local deformations. Depending on the deformation transition rates, and specifically on a parameter $\eta$ which breaks the symmetry between the left and right orientation, the winding distribution of the walk is modified, and the system can be in three different phases: folded, stretched and glassy. An explicit mapping is found, leading to consider the system as a coupling of two exclusion processes. For all closed or periodic initial sample paths, a convenient scaling permits to show a convergence in law (or almost surely on a modified probability space) to a continuous curve, the equation of which is given by a system of two non linear stochastic differential equations. The deterministic part of this system is explicitly analyzed via elliptic functions. In a similar way, by using a formal fluid limit approach, the dynamics of the system is shown to be equivalent to a system of two coupled Burgers' equations.Comment: 31 pages, 13 figures. Pages 5,6,8,9,10,12,23 color printed. INRIA Report 460

A Markovian Analysis of IEEE 802.11 Broadcast Transmission Networks with Buffering

The purpose of this paper is to analyze the so-called back-off technique of the IEEE 802.11 protocol in broadcast mode with waiting queues. In contrast to existing models, packets arriving when a station (or node) is in back-off state are not discarded, but are stored in a buffer of infinite capacity. As in previous studies, the key point of our analysis hinges on the assumption that the time on the channel is viewed as a random succession of transmission slots (whose duration corresponds to the length of a packet) and mini-slots during which the back-o? of the station is decremented. These events occur independently, with given probabilities. The state of a node is represented by a two-dimensional Markov chain in discrete-time, formed by the back-off counter and the number of packets at the station. Two models are proposed both of which are shown to cope reasonably well with the physical principles of the protocol. The stabillity (ergodicity) conditions are obtained and interpreted in terms of maximum throughput. Several approximations related to these models are also discussed

Random Walks in the Quarter-Plane: Advances in Explicit Criterions for the Finiteness of the Associated Group in the Genus 1 Case

In the book [FIM], original methods were proposed to determine the invariant measure of random walks in the quarter plane with small jumps, the general solution being obtained via reduction to boundary value problems. Among other things, an important quantity, the so-called group of the walk, allows to deduce theoretical features about the nature of the solutions. In particular, when the \emph{order} of the group is finite, necessary and sufficient conditions have been given in [FIM] for the solution to be rational or algebraic. In this paper, when the underlying algebraic curve is of genus $1$, we propose a concrete criterion ensuring the finiteness of the group. It turns out that this criterion can be expressed as the cancellation of a determinant of a matrix of order 3 or 4, which depends in a polynomial way on the coefficients of the walk.Comment: 2 figure

About a possible analytic approach for walks in the quarter plane with arbitrary big jumps

In this note, we consider random walks in the quarter plane with arbitrary big jumps. We announce the extension to that class of models of the analytic approach of [G. Fayolle, R. Iasnogorodski, and V. Malyshev, Random walks in the quarter plane, Springer-Verlag, Berlin (1999)], initially valid for walks with small steps in the quarter plane. New technical challenges arise, most of them being tackled in the framework of generalized boundary value problems on compact Riemann surfaces.Comment: 7 pages, 3 figures, extended abstrac

Random walks in the quarter plane with zero drift: an explicit criterion for the finiteness of the associated group

In many recent studies on random walks with small jumps in the quarter plane, it has been noticed that the so-called "group" of the walk governs the behavior of a number of quantities, in particular through its "order". In this paper, when the "drift" of the random walk is equal to 0, we provide an effective criterion giving the order of this group. More generally, we also show that in all cases where the "genus" of the algebraic curve defined by the kernel is 0, the group is infinite, except precisely for the zero drift case, where finiteness is quite possible

Random walks in the quarter plane with zero drift: an explicit criterion for the finiteness of the associated group

In many recent studies on random walks with small jumps in the quarter plane, it has been noticed that the so-called "group" of the walk governs the behavior of a number of quantities, in particular through its "order". In this paper, when the "drift" of the random walk is equal to 0, we provide an effective criterion giving the order of this group. More generally, we also show that in all cases where the "genus" of the algebraic curve defined by the kernel is 0, the group is infinite, except precisely for the zero drift case, where finiteness is quite possible

About Hydrodynamic Limit of Some Exclusion Processes via Functional Integration

This article considers some classes of models dealing with the dynamics of discrete curves subjected to stochastic deformations. It turns out that the problems of interest can be set in terms of interacting exclusion processes, the ultimate goal being to derive hydrodynamic limits after proper scalings. A seemingly new method is proposed, which relies on the analysis of specific partial differential operators, involving variational calculus and functional integration: indeed, the variables are the values of some functions at given points, the number of which tends to become infinite, which requires the construction of \emph{generalized measures}. Starting from a detailed analysis of the \textsc{asep} system on the torus Z/N/Z, we claim that the arguments a priori work in higher dimensions (ABC, multi-type exclusion processes, etc), leading to sytems of coupled partial differential equations of Burgers' type.Comment: Proceedings on CD. ISBN 978-5-901158-15-9; Int. Math. Conf. "50 Years of IPPI" (2011