Discrete polyharmonic functions and lattice paths in the quarter plane
A talk in the Oberseminar Wahrscheinlichkeitstheorie series by
Andreas Nessmann
| Abstract: | Counting the number of lattice paths of a fixed length between two points in a certain domain has by now become a standard problem in combinatorics. The quarter plane case in particular has garnered a lot of interest lately, where instead of an exact enumeration we are interested in asymptotic estimates. While due to Denisov and Wachtel we know that the dependency on start- and endpoint of the first order term is given by harmonic functions, it turns out that in many cases a similar phenomenon occurs for higher order estimates as well, where this dependency is then given by so-called discrete polyharmonic functions.
In this talk, after a brief introduction to the setting and methods used, I will then demonstrate this relation for so-called orbit-summable models, and for one class of more complex ones. If time allows, I will then outline the more general question of how discrete polyharmonic functions can be constructed.
Within the CRC this talk is associated to the project(s): B10 |