## Periodic Reporting for period 3 - COMBINEPIC (Elliptic Combinatorics: Solving famous models from combinatorics, probability and statistical mechanics, via a transversal approach of special functions)

Reporting period: 2021-02-01 to 2022-07-31

What is the problem/issue being addressed?

I am willing to solve several well-known models from combinatorics, probability theory and statistical mechanics: the Ising model on isoradial graphs, dimer models, spanning forests, random walks in cones, occupation time problems, etc. Although completely unrelated a priori, these models have the common feature of being presumed "exactly solvable" models, for which surprising and spectacular formulas should exist for quantities of interest. This is captured by the title "Elliptic Combinatorics", the wording elliptic referring to the use of special functions, in a broad sense.

Besides the exciting nature of the models which we aim at solving, one main strength of our project lies in the variety of modern methods and fields that we cover: combinatorics, probability, algebra (representation theory), computer algebra, algebraic geometry, with a spectrum going from applied to pure mathematics. We propose in addition one major application, in population biology (evolution of multitype populations). We plan to work in close collaborations with researchers from this field, to eventually apply our results (study of extinction probabilities for selfincompatible flower populations, for instance).

Several of our works are about the transcendental nature of the generating functions counting walks in cones (in particular in the quarter plane). This goal is actually part of a bigger project, consisting in studying transcendence in combinatorics and probability theory. The key idea is that many combinatorial or probabilistic models may be studied via generating functions, whose transcendence appears as a measurement of the complexity of the associated model.

Why is it important for society?

I would like to briefly describe the numerous impacts of our project. To summarize, they are triple:

.The first group impacted by our results will be, indisputably, the (French and international) mathematical community: indeed, we aim at obtaining various results from quite different fields.

.The second community which will benefit from this project is the group of PhD students and postdocs who I have hired: they are receiving a strong formation at the forefront of mathematics, with the opportunity to stay in the academic world or to work in companies.

.Finally, considering the applied part of our results, we plan to work in close collaborations with researchers from these fields. Researchers in biology will be the last group impacted by our results.

What are the overall objectives?

Our many scientific objectives may be classified within three thematic sections: random walks in cones, statistical mechanics, applications in population biology. In each of these domains, we have many concrete mathematical questions, which together look in the same direction of a better understanding of these exactly solvable models.

I am willing to solve several well-known models from combinatorics, probability theory and statistical mechanics: the Ising model on isoradial graphs, dimer models, spanning forests, random walks in cones, occupation time problems, etc. Although completely unrelated a priori, these models have the common feature of being presumed "exactly solvable" models, for which surprising and spectacular formulas should exist for quantities of interest. This is captured by the title "Elliptic Combinatorics", the wording elliptic referring to the use of special functions, in a broad sense.

Besides the exciting nature of the models which we aim at solving, one main strength of our project lies in the variety of modern methods and fields that we cover: combinatorics, probability, algebra (representation theory), computer algebra, algebraic geometry, with a spectrum going from applied to pure mathematics. We propose in addition one major application, in population biology (evolution of multitype populations). We plan to work in close collaborations with researchers from this field, to eventually apply our results (study of extinction probabilities for selfincompatible flower populations, for instance).

Several of our works are about the transcendental nature of the generating functions counting walks in cones (in particular in the quarter plane). This goal is actually part of a bigger project, consisting in studying transcendence in combinatorics and probability theory. The key idea is that many combinatorial or probabilistic models may be studied via generating functions, whose transcendence appears as a measurement of the complexity of the associated model.

Why is it important for society?

I would like to briefly describe the numerous impacts of our project. To summarize, they are triple:

.The first group impacted by our results will be, indisputably, the (French and international) mathematical community: indeed, we aim at obtaining various results from quite different fields.

.The second community which will benefit from this project is the group of PhD students and postdocs who I have hired: they are receiving a strong formation at the forefront of mathematics, with the opportunity to stay in the academic world or to work in companies.

.Finally, considering the applied part of our results, we plan to work in close collaborations with researchers from these fields. Researchers in biology will be the last group impacted by our results.

What are the overall objectives?

Our many scientific objectives may be classified within three thematic sections: random walks in cones, statistical mechanics, applications in population biology. In each of these domains, we have many concrete mathematical questions, which together look in the same direction of a better understanding of these exactly solvable models.

Let me now describe a selection of mathematical results obtained by my group.

The first main domain (random walks in cones) has benefited from the (ERC funded) PhD thesis of Viet Hung Hoang (started in October 2019 and in cotutelle with the University of Münster, Germany). Viet Hung Hoang made several key steps in the understanding of random walks with big jumps. We also invented a new approach to understand asymptotically three dimensional positive walks, using various mathematical tools, from spherical geometry to spectral geometry. Another aspect on which we have worked in priority is discrete harmonic functions. Such functions allow in particular to construct random processes conditionned on staying in cones. In particular, we have solved a long-standing open problem, consisting in proving the uniqueness of the positive harmonic function for a general class of random walks in cones.

Regarding our second main domain (statistical mechanics), we have worked both on the famous Ising model (in relation with the dimer model) and on the six-vertex model on planar maps.

Finally, in the domain of population biology, together with Gerold Alsmeyer we have studied the extinction problem for a class of distylous plant populations and we have collaborated with several biologists.

The first main domain (random walks in cones) has benefited from the (ERC funded) PhD thesis of Viet Hung Hoang (started in October 2019 and in cotutelle with the University of Münster, Germany). Viet Hung Hoang made several key steps in the understanding of random walks with big jumps. We also invented a new approach to understand asymptotically three dimensional positive walks, using various mathematical tools, from spherical geometry to spectral geometry. Another aspect on which we have worked in priority is discrete harmonic functions. Such functions allow in particular to construct random processes conditionned on staying in cones. In particular, we have solved a long-standing open problem, consisting in proving the uniqueness of the positive harmonic function for a general class of random walks in cones.

Regarding our second main domain (statistical mechanics), we have worked both on the famous Ising model (in relation with the dimer model) and on the six-vertex model on planar maps.

Finally, in the domain of population biology, together with Gerold Alsmeyer we have studied the extinction problem for a class of distylous plant populations and we have collaborated with several biologists.

During the second half of the project, the ERC project will benefit from a larger research group: two new postdoctoral researchers (Matthieu Dussaule and Helen Jenne) will be joining the group in the Fall 2020; moreover, Andrew Elvey Price, who was just hired by the CNRS, will actually stay in our group as a permanent researcher! Finally, another new PhD student, Andreass Nessmann (not directly funded by the ERC project but who will work on very close topics) will join in November 2020, in cotutelle with Vienna University of Technology, Austria. Let me also recall that my research group is working within the very active team SPACE of the Institute Denis Poisson (https://www.idpoisson.fr/).

We have numerous research projects for the upcoming period, in relation with the three main domains: random walks in cones, statistical mechanics, applications in population biology. We also plan to strengthen interactions with other areas, such as algorithmics, physics, spectral geometry, etc.

We have numerous research projects for the upcoming period, in relation with the three main domains: random walks in cones, statistical mechanics, applications in population biology. We also plan to strengthen interactions with other areas, such as algorithmics, physics, spectral geometry, etc.