# 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

## Members

Kilian Raschel (Principal Investigator)

Andrew Elvey Price (Researcher)

## PhD position to fill! (Sep. 2019/Aug. 2022)

Cotutelle with

Gerold Alsmeyer from the University of Münster, Germany. Details are

here. Please contact me if you would like to apply.

## International conferences funded by the ERC project

- 21-25.9.20 (Kyiv, Ukraine): Regeneration, branching and decomposability (joint organization with Alexander Iksanov and Alexander Marynych)
- 13-17.5.19 (Braşov, Romania): Transient Transcendence in Transylvania (joint with Alin Bostan)
- 1-5.4.19 (Mahdia, Tunisia): Processus stochastiques, géométrie et structures algébriques (joint with Manon Defosseux, Nizar Demni and Rim Essifi)
- 14-15.3.19 (Tours, France): Preharmonic functions and boundaries (joint with Marc Peigné and Wolfgang Woess)
- 27-28.8.18 (Dijon, France): Grands réseaux aléatoires et marches contraintes, in honor of the 75th birthday of Guy Fayolle (joint with Peggy Cenac-Guesdon, Cyril Furtlehner, Arnaud de La Fortelle and Jean-Marc Lasgouttes)

## Other events supported by the ERC project

## They have been visiting the project

Gerold Alsmeyer,

Alin Bostan,

Manfred Buchacher,

Elisabetta Candellero,

Thomas Dreyfus, Andrew Elvey Price,

Sandro Franceschi,

Éric Fusy,

Alexander Marynych,

James Mcredmond,

Marni Mishna, Samuel Simon,

Pierre Tarrago,

Wolfgang Woess,

Yiqiang Q. Zhao
## Publications

- Plane bipolar orientations and quadrant walks, Mireille Bousquet-Mélou, Éric Fusy and Kilian Raschel (arXiv:1905.04256)
- Eulerian orientations and the six-vertex model on planar map, Mireille Bousquet-Mélou, Andrew Elvey Price and Paul Zinn-Justin (arXiv:1902.07369)
- On walks avoiding a quadrant, Kilian Raschel and Amélie Trotignon (arXiv:1807.08610)
- 3D positive lattice walks and spherical triangles, Beniamin Bogosel, Vincent Perrollaz, Kilian Raschel and Amélie Trotignon (arXiv:1804.06245)
- Martin boundary of random walks in convex cones, Kilian Raschel and Pierre Tarrago (arXiv:1803.09253)
- On the least common multiple of several random integers, Alin Bostan, Alexander Marynych and Kilian Raschel.
*Journal of Number Theory* (to appear)
- Differential transcendence & algebraicity criteria for the series counting weighted quadrant walks, Thomas Dreyfus and Kilian Raschel.
*Publications Mathématiques de Besançon* (to appear)
- The extinction problem for a class of distylous plant populations, Gerold Alsmeyer and Kilian Raschel.
*Journal of Mathematical Biology* (2019) **78** 1841-1874

## Related events

## Summary of the ERC project

This 5-year 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.
## Contact

Feel free to contact me if you have questions.