Discrete polyharmonic functions and their applications in combinatorics

Abstract

The starting point for this thesis was the article by Chapon, Fusy and Raschel, where the authors noticed that in some a symptotic expansions of lattice path models in the quarter plane the dependency on the endpoint is given by so-called discrete polyharmonic functions, and went to show that the continuous analogue holds (i.e. the heat kernel allows for an expansion using continuous polyharmonic functions). It was already shown by Denisov and Wachtel that, under some slight technical assumptions, the asymptotics of the number of lattice paths in a cone are in a first-order approximation directly tied to discrete harmonic functions. This would in a sense be an analogue to the continuous case, where an asymptotic heat kernel expansion allows for a very similar representation.Furthermore, unlike in the continuous case, the computation of discrete polyharmonic functions in cones has, to the author’s knowledge, not been studied before. These two topics will form the main part of this thesis. After introducing some basic notions and going through some technical prerequisites in Chapters 1 and 2, it will be shown in Chapter 3 how one can construct a basis of the space of all discrete polyharmonic functions. This will be done in two ways; one of them arguably more straightforward and purely algebraicin nature, which works in any case and leads to a basis consisting of functions with algebraic generating function. The downside to this basis, however, is that the functions therein does not allow for a scaling limit, and that they do not allow an easy representation of the polyharmonic functions appearing in the asymptotics of lattice paths in the quarter plane. The second method uses decoupling functions and allows us to construct discrete polyharmonicfunctions for all finite group models. If the correlation coefficient is an integer fraction of p, then the resulting functions will even have rational generating functions of a rather nice shape.In Chapters 4 and 5, the question about the form of asymptotics of the number of lattice paths in the quarter plane will be addressed. For so-called orbit-summable models, which exhibit a remarkable algebraic property tied to their reflection group, we will see in Chapter 4 using a saddle point method that one can indeed find an asymptotic expansion.In Chapter 5, a class of infinite group models is treated, using a parametrization of the kernel curve via Jacobi ?-functions. Perhaps surprisingly, it turns outthat in this case the asymptotic expansion is somewhat more complicated, including logarithmic terms. Nonetheless, the dependency on the endpoint is still given by discrete polyharmonic functions.