# ERC starting grant - 2018/2022 - COMBINEPIC - 759702

## Title

Elliptic Combinatorics:
Solving famous models from combinatorics,
probability and statistical mechanics,
via a transversal approach of
special functions

## Principal Investigator

Kilian Raschel
## Summary

This 5-years project is devoted to the use of special functions in combinatorics, probability theory and statistical mechanics. The term "special functions" is understood here in a broad sense, including algebraic, differentially finite, (hyper)elliptic, hypergeometric functions, etc. In this project we focus on two major examples emanating from combinatorics and probability:

- Random walks in cones,
- Integrable models in two-dimensional statistical mechanics.

Though deeply different, these domains have two points in common. First, they are fundamental research domains in combinatorics and probability: random walks in cones appear in the theory of quantum random walks, non-colliding random walks, planar maps, population biology, finance, etc.; integrable models of two-dimensional statistical mechanics (including the dimer model, the Ising model and spanning trees/forests) consist of the few models of the field which are exactly solvable, thus opening the way for remarkable exact formulas. Further, in both domains, the last ten years have seen the development of promising techniques to understand these exactly solvable models: functional equations, special functions and boundary value problems, to cite a few. We also propose applications in finance and population biology.

## Events

## Publications

- Differential transcendence & algebraicity criteria for the series counting weighted quadrant walks, with Thomas Dreyfus (arXiv)

## Past events

## Contact

Feel free to contact me if you have questions.