Periodic Reporting for period 4 - COMBINEPIC (Elliptic Combinatorics: Solving famous models from combinatorics, probability and statistical mechanics, via a transversal approach of special functions)
Reporting period: 2022-08-01 to 2024-01-31
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 communities: 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 communities: 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.
The research group published 57 papers, edited two books and organised 11 international conferences. The PI (Kilian Raschel) was awarded the Marc Yor Prize (2020) by the French Academy of Sciences and the David P. Robbins Prize (2022) by the American Mathematical Society. Since February 2018, the PI, together with Alin Bostan (member of the consortium) and Lucia Di Vizio, has been organising a workshop entitled "Combinatorics and Transcendence". The main objective is to study the transcendence properties of combinatorial models. About forty sessions were held at the Institut Henri Poincaré (Paris) or at Inria Saclay.
The first main area (random walks in cones and, more generally, transcendence of combinatorial domains) benefited from the work of Andrew Elvey Price (postdoc), Viet Hung Hoang (PhD student, in cotutelle with the University of Münster, Germany), Matthieu Dussaule (postdoc) and the PI. They obtained numerous results on confined random walks, reflected diffusions, persistence problems, potential theory. The research group organised several international conferences, such as "Transient Transcendence in Transylvania" (Brasov, 2019). The research group collaborated with many international experts from different fields of mathematics and computer science, with various interests and views on transcendence and random walks.
Regarding the second main area (statistical mechanics), Andrew Elvey Price, Dan Betea (postdoc), Helen Jenne (postdoc) and the PI worked on the famous Ising model (related to the dimer model), on the six-vertex model, on pattern-avoiding permutations. They organised the international conference "State of the art in statistical mechanics" (Paris, 2018) with famous speakers such as the Fields Medalists Wendelin Werner and Hugo Duminil Copin.
Finally, in the field of population biology, Viet Hung Hoang and the PI, in collaboration with several biologists, have studied the extinction problem for some classes of plant populations with inhomogeneous branching mechanisms. The research group also made new contributions to the elephant random walk, a model recently introduced in the physics literature to study the influence of memory in random walk models.
The first main area (random walks in cones and, more generally, transcendence of combinatorial domains) benefited from the work of Andrew Elvey Price (postdoc), Viet Hung Hoang (PhD student, in cotutelle with the University of Münster, Germany), Matthieu Dussaule (postdoc) and the PI. They obtained numerous results on confined random walks, reflected diffusions, persistence problems, potential theory. The research group organised several international conferences, such as "Transient Transcendence in Transylvania" (Brasov, 2019). The research group collaborated with many international experts from different fields of mathematics and computer science, with various interests and views on transcendence and random walks.
Regarding the second main area (statistical mechanics), Andrew Elvey Price, Dan Betea (postdoc), Helen Jenne (postdoc) and the PI worked on the famous Ising model (related to the dimer model), on the six-vertex model, on pattern-avoiding permutations. They organised the international conference "State of the art in statistical mechanics" (Paris, 2018) with famous speakers such as the Fields Medalists Wendelin Werner and Hugo Duminil Copin.
Finally, in the field of population biology, Viet Hung Hoang and the PI, in collaboration with several biologists, have studied the extinction problem for some classes of plant populations with inhomogeneous branching mechanisms. The research group also made new contributions to the elephant random walk, a model recently introduced in the physics literature to study the influence of memory in random walk models.
We have made progress beyond the state of the art in all three main themes of the project, as evidenced by our 59 publications. Some of these papers appeared in top general journals such as Annales Henri Lebesgue and Transactions of the American Mathematical Society, others in top specialised journals such as Journal of Combinatorial Theory. Series A, Journal of Mathematical Biology and Journal of Probability Theory and Related Fields. Let us briefly present three breakthroughs in our research area.
First, since the beginning of the project, we have been working a lot on "Transcendence and Combinatorics". Roughly speaking, we can associate many combinatorial models with a certain (generating) function, and then measure the complexity of this function (and thus of the model) via the concept of transcendence. These ideas are quite new to the combinatorial community. In the context of the ERC project COMBINEPIC, we have focused our efforts in several directions (organisation of a working group, of the international conference "Transient Transcendence in Transylvania"). We now have a central position on these issues.
Secondly, an important part of our work was devoted to harmonic functions and Martin boundary theory. We published several papers on this subject. The most important, in my opinion, is the paper "Martin boundary of random walks in convex cones". In this paper we solve a long-standing open problem and state that (under general assumptions) there is a unique discrete harmonic function for a large class of Laplacian operators in convex cones. This result was obtained by combining several different approaches. Our paper gives a general answer to a 20 year old problem in probability theory.
Third, the paper "On the stationary distribution of reflected Brownian motion in a wedge: differential properties" is a significant breakthrough in the field of reflected diffusions. The probabilistic model was introduced more than 40 years ago, but the simple problem of explicitly computing the stationary distribution remained unsolved in the literature. In this paper we solve this 40 year old open problem. As in the previous point, our transversal approach is really the key; we have used expertise from pure mathematics (modern differential Galois theory, special function theory) and from combinatorics (functional equations).
First, since the beginning of the project, we have been working a lot on "Transcendence and Combinatorics". Roughly speaking, we can associate many combinatorial models with a certain (generating) function, and then measure the complexity of this function (and thus of the model) via the concept of transcendence. These ideas are quite new to the combinatorial community. In the context of the ERC project COMBINEPIC, we have focused our efforts in several directions (organisation of a working group, of the international conference "Transient Transcendence in Transylvania"). We now have a central position on these issues.
Secondly, an important part of our work was devoted to harmonic functions and Martin boundary theory. We published several papers on this subject. The most important, in my opinion, is the paper "Martin boundary of random walks in convex cones". In this paper we solve a long-standing open problem and state that (under general assumptions) there is a unique discrete harmonic function for a large class of Laplacian operators in convex cones. This result was obtained by combining several different approaches. Our paper gives a general answer to a 20 year old problem in probability theory.
Third, the paper "On the stationary distribution of reflected Brownian motion in a wedge: differential properties" is a significant breakthrough in the field of reflected diffusions. The probabilistic model was introduced more than 40 years ago, but the simple problem of explicitly computing the stationary distribution remained unsolved in the literature. In this paper we solve this 40 year old open problem. As in the previous point, our transversal approach is really the key; we have used expertise from pure mathematics (modern differential Galois theory, special function theory) and from combinatorics (functional equations).