Abstracts > Raschel Kilian

In combinatorics, and more specifically in the study of lattice walks confined to cones such as the quarter plane, it has become standard to investigate the differential nature of the associated generating functions. When these functions are differentially finite or differentially algebraic, one can derive concrete information about the model, including asymptotics, recurrences, and, in some cases, closed-form expressions. By contrast, the combinatorial significance of differential transcendence remains much less understood.

In this talk, I will present a family of examples for which we provide a combinatorial and probabilistic interpretation of differential transcendence. Focusing on singular walks in the quarter plane, we show that, while the generating function is generically differentially transcendental, it exhibits a stronger form of differential transcendence at the spectral radius. This phenomenon is explained by an underlying probabilistic critical behavior.

This is joint work with Alin Bostan and Lucia Di Vizio (arXiv:2504.13542).