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“Scaling limits of random walks in the quarter plane”
 

February 13, 2012

SUMMARY

Walks or random walks in the quarter plane present considerable mathematical challenges within the fields of probability theory and analytic combinatorics. Traditional ways to study such walks use complex analysis, functional equations, boundary value problems and conformal mappings. This one-day workshop is about using scaling limits in order to study these walks. Indeed, the use of scaling limits has recently led to several advancements within the field. This typically involves the derivation of a scaling limit and the coupling of the original process with the scaling limit.

ORGANIZERS

Johan van Leeuwaarden (TU Eindhoven - Eurandom)

Kilian Raschel (CNRS and Tours University)

SPEAKERS

Confirmed speakers:

PROGRAMME

REGISTRATION

Speakers and participants of this workshop are kindly invited to extend their stay at Eurandom by a few (or more) days in the context of the Stochastic Activity Month (link to SAM website). Please contact one of the organizers when considering this possibility.

ABSTRACTS

Ivo Adan

Stretching the limits of the compensation approach

This talk reviews some results on the use of compensation arguments to solve the steady-state equations for two-dimensional Markov processes. It also briefly discusses some new results pointing to possible extensions of the applicability of this approach.

Yanting Chen

The Invariant Measure of Homogeneous Markov Processes in The Quarter-Plane: Representation in Geometric Terms

We consider the invariant measure of a continuous-time Markov process in the quarter-plane. The basic solutions of the global balance equation are the geometric distributions. We first show that a finite linear combination of basic geometric distribution cannot be invariant measure unless it consists of a single basic geometric distribution.

Second, we show that a countable linear combination of geometric terms can be an invariant measure only if it consists of pairwise coupled terms. As a consequence, we have obtained a complete characterization of all countable linear combinations of geometric product forms that may yield an invariant measure for a homogeneous continuous-time Markov process in the quarter-plane.

PRESENTATION

Denis Denisov

Probabilistic approach to lattice path enumeration I

We prove a local limit theorem for random walks conditioned to stay in a cone and discuss its applications to the enumeration of lattice paths.

Rodolphe Garbit

Weak convergence to the Brownian meander of a cone

The Brownian meander of a cone is the process obtained from Brownian motion by conditioning it to stay in this cone for a unit of time.
In this talk, I will explain the link between the weak convergence of a conditioned random walk to the Brownian meander of a cone C and the asymptotic behavior of
P(T>n), where T stands for the original random walk exit time from C.

Stella Kapodistria

Erlang arrivals joining the shortest queue

We consider a system in which customers join upon arrival the shortest of two single-server queues. The interarrival times between customers are Erlang distributed and the service times of both servers are exponentially distributed. Under these assumptions, this system gives rise to a Markov chain on a multi-layered quarter plane. For this Markov chain we derive the equilibrium distribution using the compensation approach. The obtained expression for the equilibrium distribution matches and refines heavy-traffic approximations and tail asymptotics obtained earlier in the literature.

PRESENTATION

Kilian Raschel

Some exact asymptotics in the counting of walks in the quarter plane

Enumeration of planar lattice walks is a classical topic in combinatorics, at the cross-roads of several domains (e.g., probability, statistical physics, computer science). Particularly important quantities are the numbers of paths starting at the origin, having a given length, ending at given points or domains (for instance, the origin, or one axis, or else everywhere), and confined to the quarter plane. While expressions for these numbers of paths exist in the literature (via the knowledge of their generating function), there are very few results on their asymptotic behavior, as the length goes to infinity. The aim of this talk is to propose a new approach to obtain some exact asymptotics for such numbers of walks confined to the quarter plane.
This is a joint work with Guy Fayolle.

Vitali Wachtel

Probabilistic approach to lattice path enumeration II

We prove a local limit theorem for random walks conditioned to stay in a cone and discuss its applications to the enumeration of lattice paths

PRESENTATION

PRACTICAL INFORMATION

Conference Location
The workshop location is Eurandom,  Den Dolech 2, 5612 AZ Eindhoven, Laplace Building, 1st floor, LG 1.105.

Eurandom is located on the campus of Eindhoven University of Technology, in the 'Laplacegebouw' building' (LG on the map). The university is located at 10 minutes walking distance from Eindhoven railway station (take the exit north side and walk towards the tall building on the right with the sign TU/e).

For all information on how to come to Eindhoven, please check http://www.eurandom.tue.nl/contact.htm

CONTACT
For more information please contact Mrs. Patty Koorn,
Workshop officer of Eurandom

Last updated 07-04-14,
by PK