Counting walks in a quadrant: a unified approach via boundary value problems

@article{Raschel2012,
abstract = {The aim of this article is to introduce a unified method to obtain explicit integral representations of the trivariate generating function counting the walks with small steps which are confined to a quarter plane. For many models, this yields for the first time an explicit expression of the counting generating function. Moreover, the nature of the integrand of the integral formulations is shown to be directly dependent on the finiteness of a naturally attached group of birational transformations as well as on the sign of the covariance of the walk.},
author = {Raschel, Kilian},
journal = {Journal of the European Mathematical Society},
keywords = {lattice walk; counting generating function; boundary value problem; conformal mapping; Weierstrass elliptic function; Riemann surface; uniformization; lattice walk; counting generating function; boundary value problem; conformal mapping; Weierstrass elliptic function; Riemann surface; uniformization},
language = {eng},
number = {3},
pages = {749-777},
publisher = {European Mathematical Society Publishing House},
title = {Counting walks in a quadrant: a unified approach via boundary value problems},
url = {http://eudml.org/doc/277202},
volume = {014},
year = {2012},
}

TY - JOUR
AU - Raschel, Kilian
TI - Counting walks in a quadrant: a unified approach via boundary value problems
JO - Journal of the European Mathematical Society
PY - 2012
PB - European Mathematical Society Publishing House
VL - 014
IS - 3
SP - 749
EP - 777
AB - The aim of this article is to introduce a unified method to obtain explicit integral representations of the trivariate generating function counting the walks with small steps which are confined to a quarter plane. For many models, this yields for the first time an explicit expression of the counting generating function. Moreover, the nature of the integrand of the integral formulations is shown to be directly dependent on the finiteness of a naturally attached group of birational transformations as well as on the sign of the covariance of the walk.
LA - eng
KW - lattice walk; counting generating function; boundary value problem; conformal mapping; Weierstrass elliptic function; Riemann surface; uniformization; lattice walk; counting generating function; boundary value problem; conformal mapping; Weierstrass elliptic function; Riemann surface; uniformization
UR - http://eudml.org/doc/277202
ER -