BIRS Workshop Lecture Videos

Banff International Research Station Logo

BIRS Workshop Lecture Videos

Course #4: Probabilistic Tools for Lattice Path Enumeration Raschel, Kilian


In this lecture we will focus on techniques coming from probability theory and analysis to study models of walks confined to multidimensional cones, with arbitrary big steps and possibly with weights. To give an example of the fruitful interaction between the above domains, we will restrict our attention to the computation of critical exponents which appear in the asymptotic behavior of confined walks. For instance, what is the asymptotic behavior of the number of excursions, i.e., of the number of walks starting and ending at given points, remaining in a fixed cone, as the number of steps goes to infinity ? In a first part, using an approximation of random walks by Brownian motion, we will present the seminal work of Denisov and Wachtel providing a solution to the above problem in the case of excursions. We shall also present partial results and conjectures related to the total number of walks confined to a cone. We will show new results concerning the non-D-finiteness of some series counting walks in the quarter plane. In a second part we shall be interested in discrete harmonic functions in cones. The generating function of these harmonic functions satisfies a functional equation, which happens to be closely related to the well-known functional equation that appears in the context of enumeration of confined walks. We shall explain the link between these harmonic functions and a one-parameter family of conformal mappings. These harmonic functions provide a second way to compute the critical exponents. We will present several conjectures.

Item Media

Item Citations and Data


Attribution-NonCommercial-NoDerivatives 4.0 International