Random walks in the quarter plane
Johan van Leeuwaarden
This talk is about two-dimensional random walks in the positive quadrant.
Each step (to a neighboring site) occurs with a certain probability, where
different probabilities may be taken for the interior of the state space and
for the boundaries. The bivariate generating function of the stationary
distribution satisfies a functional equation that is in general hard to solve
(involves conformal mappings and boundary value problems). We present both
exact and approximate results, including asymptotic expressions for rare
events (large deviations). We also discuss connections to lattice path counting
and analytic combinatorics.