Apparaît dans la collection : The Pentagram Map and Ramifications

The subject of this talk is the discrete Schwarzian Kadomtsev–Petviashvili (dSKP) recurrence. We will first derive an explicit expression for its solution in terms of the initial data. More precisely, we will show that the solution is the ratio of two partition functions associated with a model of oriented dimers. Each of these partition functions involves cancellations, and we will derive an alternative expression without cancellations, involving trees and forests. Beyond its combinatorial interest, this expression is used to establish results on the singularities of the recurrence. We will then explain how the dSKP equation arises in a number of discrete geometric systems, including discrete holomorphic functions, polygon recutting, and the pentagram map. We will discuss the implications of our results for these systems. This is joint work with Niklas Affolter (TU Wien) and Paul Melotti (Université Paris-Saclay).