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Workshop
on
“Scaling limits of random walks in the
quarter plane”
February 13, 2012
SUMMARY | REGISTRATION | SPEAKERS |
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.
Johan van Leeuwaarden (TU Eindhoven - Eurandom)
Kilian Raschel (CNRS and Tours University)
Confirmed speakers:
Ivo Adan | TU Eindhoven |
Yanting Chen | University Twente |
Denis Denisov | Cardiff University |
Rodolphe Garbit | Université d'Angers |
Stella Kapodistria | TU Eindhoven |
Kilian Raschel | Universität Bielefeld |
Vitali Wachtel | Universität München |
09.20 - 09.30 | Welcome | |
09.30 - 10.10 | Denis Denisov | Probabilistic approach to lattice path enumeration I |
10.10 - 10.50 | Vitali Wachtel | Probabilistic approach to lattice path enumeration II |
10.50 - 11.20 | Coffee/tea break | |
11.20 - 12.00 | Rodolphe Garbit | Weak convergence to the Brownian meander of a cone |
12.00 - 13.30 | Lunch | |
13.30 - 14.10 | Kilian Raschel | Some exact asymptotics in the counting of walks in the quarter plane |
14.10 - 14.50 | Ivo Adan | Stretching the limits of the compensation approach |
14.50 - 15.20 | Coffee/tea break | |
15.20 - 16.00 | Yanting Chen | The Invariant Measure of Homogeneous Markov Processes in The Quarter-Plane: Representation in Geometric Terms |
16.00 - 16.40 | Stella Kapodistria | Erlang arrivals joining the shortest queue |
18.30 - | Conference dinner |
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.
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.
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
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