ERC starting grant - 2018/2024 - COMBINEPIC - 759702

Title

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

Members

Andrew Elvey Price (Researcher, Feb 2019/Jan 2024)
Dan Betea (Postdoctoral researcher, Nov 2021/Oct 2022)
Matthieu Dussaule (Postdoctoral researcher, Nov 2020/Oct 2022)
Helen Jenne (Postdoctoral researcher, Sep 2020/Aug 2021)
Viet Hung Hoang (PhD student, Oct 2019/May 2023)
Andreas Nessmann (PhD student, Oct 2020/Nov 2023)
Kilian Raschel (Principal Investigator)
Harriet Walsh (Postdoctoral researcher, Nov 2022/Jan 2024)

Working group

• Groupe de travail Transcendance et Combinatoire, with Alin Bostan and Lucia Di Vizio

International conferences funded by the ERC project

• 13-17.11.23 (Cargèse, France): Combinatoire elliptique et au-delà! (joint organization with Sandro Franceschi and Harriet Walsh)
• 19-23.6.23 (Angers, France): Journées de Probabilités 2023 (joint organization with the probabilists and statisticians of the LAREMA)
• 24-28.4.23 (Roscoff, France): 40 years of reflected Brownian motion and related topics (joint organization with Sandro Franceschi and Hélène Guérin)
• 14-16.3.22 (Saumur, France): Branching and persistence (joint organization with Loïc Chaumont and Rodolphe Garbit)
• 21-25.6.21 (Marseille, France): Lattice Paths, Combinatorics and Interactions (joint organization with Cyril Banderier, Jehanne Dousse, Enrica Duchi, Christian Krattenthaler, Greta Panova and Michael Wallner)
• 20-24.5.19 (Bedlewo, Poland): Probability and Analysis 2019
• 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)
• 10-14.12.18 (IHP, Paris): États de la Recherche en Mécanique Statistique
• 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

• 28.2-5.3.9.21 (Kyiv, Ukraine, cancelled): Regeneration, branching and decomposability (joint with Alexander Iksanov and Alexander Marynych)
• 25-29.6.18 (Tours): Journées de Probabilités 2018

Books edited

• Topics in Statistical Mechanics, Cédric Boutillier, Kilian Raschel and Béatrice de Tilière (editors)
  Panoramas et Synthèses 59 (2023), French Mathematical Society
• Transcendence in Algebra, Combinatorics, Geometry and Number Theory, Alin Bostan and Kilian Raschel (editors)
  Springer Proceedings in Mathematics and Statistics 373 (2021), Springer

Publications

• Invariant measure of gaps in degenerate competing three-particle systems, Sandro Franceschi, Tomoyuki Ichiba, Ioannis Karatzas and Kilian Raschel (arXiv:2401.10734, free pdf version)
• Elephant polynomials, Hélène Guérin, Lucile Laulin and Kilian Raschel (arXiv:2401.04959, pdf)
• Interface fluctuations associated with split Fermi seas, Harriet Walsh (arXiv:2311.02056, pdf)
• Logarithmic terms in discrete heat kernel expansions in the quadrant, Andrew Elvey-Price, Andreas Nessmann and Kilian Raschel (arXiv:2309.15209, pdf)
• A fixed-point equation approach for the superdiffusive elephant random walk, Hélène Guérin, Lucile Laulin and Kilian Raschel (arXiv:2308.14630, pdf)
• Full asymptotic expansion for orbit-summable quadrant walks and discrete polyharmonic functions, Andreas Nessmann (arXiv:2307.11539, pdf)
• Multicritical Schur measures and higher-order analogues of the Tracy-Widom distribution, Dan Betea, Jérémie Bouttier and Harriet Walsh (arXiv:2307.05303, pdf)
• Random Walks in the High-Dimensional Limit II: The Crinkled Subordinator, Zakhar Kabluchko, Alexander Marynych and Kilian Raschel (arXiv:2306.04747, pdf)
• Multivariate multiplicative functions of uniform random vectors in large integer domains, Zakhar Kabluchko, Oleksandr Marynych and Kilian Raschel. Results in Mathematics 78 (2023) Paper No. 201 (arXiv:2301.13586, pdf)
• Polyharmonic Functions in the Quarter Plane, Andreas Nessmann (arXiv:2212.07258, pdf)
• Random walks with drift inside a pyramid: convergence rate for the survival probability, Rodolphe Garbit and Kilian Raschel. ALEA, Latin American Journal of Probability and Mathematical Statistics 20 (2023) 973-988 (arXiv:2211.16050, pdf)
• Limit shapes for skew Howe duality, Dan Betea, Anton Nazarov and Travis Scrimshaw. Séminaire Lotharingien de Combinatoire 89B (2023) Article #35, 12 pp. (Proceedings of the 35th Conference on Formal Power Series and Algebraic Combinatorics) (arXiv:2211.13728, pdf)
• Branching Random Walks on relatively hyperbolic groups, Matthieu Dussaule, Longmin Wang and Wenyuan Yang (arXiv:2211.07213, pdf)
• Marches aléatoires dans un cône et fonctions discrètes harmoniques, Kilian Raschel and Pierre Tarrago. La Gazette de la Société Mathématique de France 175 (2023) 35-48 (arXiv:2211.02925, pdf)
• Harmonic functions for singular quadrant walks, Viet Hung Hoang, Kilian Raschel and Pierre Tarrago. Indagationes Mathematicae 34 (2023) 936-972 (Special issue in honour of J. W. Cohen) (arXiv:2208.00477, pdf)
• A dual skew symmetry for transient reflected Brownian motion in an orthant, Sandro Franceschi and Kilian Raschel. Queueing Systems 102 (2022) 123-141 Special issue in honour of Masakiyo Miyazawa (arXiv:2208.00224, pdf)
• Stationary Brownian motion in a 3/4-plane: Reduction to a Riemann-Hilbert problem via Fourier transforms, Guy Fayolle, Sandro Franceschi and Kilian Raschel. Indagationes Mathematicae 34 (2023) 874-890 (Special issue in honour of J. W. Cohen) (arXiv:2206.12859, pdf)
• A local limit theorem for convergent random walks on relatively hyperbolic groups, Matthieu Dussaule, Marc Peigné and Samuel Tapie (arXiv:2202.11339, pdf)
• Discrete harmonic functions for non-symmetric Laplace operators in the quarter plane, Viet Hung Hoang. Markov Processes Related Fields (2022) no. 4, 605 (arXiv:2201.09985, pdf)
• Asymptotics of arithmetic functions of GCD and LCM of random integers in hyperbolic regions, Alexander Iksanov, Alexander Marynych and Kilian Raschel. Results in Mathematics 77 (2022) Paper No. 165 (arXiv:2112.11892, pdf)
• Lattice walks confined to an octant in dimension 3: (non-)rationality of the second critical exponent, Luc Hillairet, Helen Jenne and Kilian Raschel. Annales de l'Institut Henri Poincaré D (to appear) (arXiv:2112.03567, pdf)
• Persistence for a class of order-one autoregressive processes and Mallows-Riordan polynomials, Gerold Alsmeyer, Alin Bostan, Kilian Raschel and Thomas Simon. Advances in Applied Mathematics 150 (2023) 102555 (arXiv:2112.03016, pdf)
• The generating function of the survival probabilities in a cone is not rational, Rodolphe Garbit and Kilian Raschel. Séminaire Lotharingien de Combinatoire 87B (2022) Special issue for the 9th International Conference on Lattice Path Combinatorics and Applications (arXiv:2111.05027, pdf)
• Double-dimer condensation and the PT-DT correspondence, Helen Jenne, Gautam Webb and Benjamin Young (arXiv:2109.11773, pdf)
• On the stationary distribution of reflected Brownian motion in a non-convex wedge, Guy Fayolle, Sandro Franceschi and Kilian Raschel. Markov Processes and Related Fields 28 (2022) 737-772 (arXiv:2102.11754, pdf)
• On the stationary distribution of reflected Brownian motion in a wedge: differential properties, Mireille Bousquet-Mélou, Andrew Elvey Price, Sandro Franceschi, Charlotte Hardouin and Kilian Raschel (arXiv:2101.01562, pdf)
• Differential transcendence of Bell numbers and relatives: a Galois theoretic approach, Alin Bostan, Lucia Di Vizio and Kilian Raschel (arXiv:2012.15292, pdf)
• The combinatorial PT-DT correspondence, Helen Jenne, Gautam Webb and Benjamin Young. Séminaire Lotharingien de Combinatoire 85B (2021) Article #89, 12 pp. (Proceedings of the 33rd Conference on Formal Power Series and Algebraic Combinatorics) (arXiv:2012.08484, pdf)
• Constructing discrete harmonic functions in wedges, Viet Hung Hoang, Kilian Raschel and Pierre Tarrago. Transactions of the American Mathematical Society 375 (2022) 4741-4782 (arXiv:2012.08947, pdf)
• Reflected random walks and unstable Martin boundary, Irina Ignatiouk-Robert, Irina Kurkova and Kilian Raschel. Annales de l'Institut Henri Poincaré, Probabilités et Statistiques (to appear, 2024) (arXiv:2009.11596, pdf)
• Asymptotics of 3-stack-sortable permutations, Colin Defant, Andrew Elvey Price and Anthony J Guttmann. Electronic Journal of Combinatorics 28 (2021) no.2, Paper No. 2.49, 29 pp. (arXiv:2009.10439, pdf)
• Bijections between walks inside a triangular domain and Motzkin paths of bounded amplitude, Julien Courtiel, Andrew Elvey Price and Irène Marcovici. Electronic Journal of Combinatorics 28 (2021) no. 2, Paper No. 2.6, 44 pp. (arXiv:2007.08868, pdf)
• The six-vertex model on random planar maps revisited, Andrew Elvey Price and Paul Zinn-Justin. Journal of Combinatorial Theory. Series A 196 (2023) Paper No. 105739, 32 pp. (arXiv:2007.07928, pdf)
• On the nature of four models of symmetric walks avoiding a quadrant, Thomas Dreyfus and Amélie Trotignon. Annals of Combinatorics 25 (2021) no.3, 617–644. (arXiv:2006.02299, pdf)
• Martin boundary of random walks in convex cones, Jetlir Duraj, Kilian Raschel, Pierre Tarrago and Vitali Wachtel. Annales Henri Lebesgue 5 (2022) 559-609. (arXiv:2003.03647, pdf)
• Counting lattice walks by winding angle, Andrew Elvey Price. Séminaire Lotharingien de Combinatoire 84B (2020) Article #43, 12 pp. (Proceedings of the 32nd Conference on Formal Power Series and Algebraic Combinatorics) (arXiv:2003.01740, pdf)
• Asymptotics of minimal deterministic finite automata recognizing a finite binary language, Andrew Elvey Price, Wenjie Fang and Michael Wallner. Leibniz International Proceedings in Informatics 159 (2020) 11:1-11:13 (Proceedings of the 31st International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms, AofA 2020) (AofA.2020.11, pdf)
• Polyharmonic functions and random processes in cones, François Chapon, Éric Fusy and Kilian Raschel. Leibniz International Proceedings in Informatics 159 (2020) 9:1-9:19 (Proceedings of the 31st International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms, AofA 2020) (arXiv:2001.07149, pdf)
• Phylogenetic trees, augmented perfect matchings, and a Thron-type continued fraction (T-fraction) for the Ward polynomials, Andrew Elvey Price and Alan D. Sokal. Electronic Journal of Combinatorics 27 (2020) no. 4, Paper No. 4.6, 36 pp. (arXiv:2001.01468, pdf)
• Stieltjes moment sequences for pattern-avoiding permutations, Alin Bostan, Andrew Elvey Price, Anthony John Guttmann and Jean-Marie Maillard. Electronic Journal of Combinatorics 27 (2020), no. 4, Paper No. 4.20, 59 pp. (arXiv:2001.00393, pdf)
• Martin boundary of killed random walks on isoradial graphs, Cédric Boutillier and Kilian Raschel. Potential Analysis 57 (2022) 201-226 (arXiv:1912.10410, pdf)
• Permutations sortable by deques and two stacks in parallel share the same growth rate, Andrew Elvey Price (arXiv:1912.00056, pdf)
• Recurrence of two-dimensional queueing processes, and random walk exit times from the quadrant, Marc Peigné and Wolfgang Woess. The Annals of Applied Probability 31 (2021) no.6 2519-2537 (arXiv:1909.00616, pdf)
• Compacted binary trees admit a stretched exponential, Andrew Elvey Price, Wenjie Fang and Michael Wallner. Journal of Combinatorial Theory, Series A 177 (2021) 105306 (arXiv:1908.11181, pdf)
• The asymptotics of reflectable weighted walks in arbitrary dimension, Marni Mishna and Samuel Simon. Advances in Applied Mathematics 118 (2020) 102043 (arXiv:1907.10184, pdf)
• Plane bipolar orientations and quadrant walks, Mireille Bousquet-Mélou, Éric Fusy and Kilian Raschel. Séminaire Lotharingien de Combinatoire 81 (2020) Art. B81l, 64 pp. (arXiv:1905.04256, pdf)
• Green's functions with oblique Neumann boundary conditions in a wedge, Sandro Franceschi. Journal of Theoretical Probability 34 (2021) no.4, 1775-1810. (arXiv:1905.04049, pdf)
• Boundary behaviour of λ-polyharmonic functions on regular trees, Ecaterina Sava-Huss and Wolfgang Woess. Annali di Matematica Pura ed Applicata. Series IV (4)200 (2021) no.1, 35-50. (arXiv:1904.10290, pdf)
• Eulerian orientations and the six-vertex model on planar maps, Mireille Bousquet-Mélou, Andrew Elvey Price and Paul Zinn-Justin. Séminaire Lotharingien de Combinatoire 82B (2019) Article #70, 12 pp. (Proceedings of the 31st Conference on Formal Power Series and Algebraic Combinatorics, Ljubljana) (arXiv:1902.07369, pdf)
• On the least common multiple of several random integers, Alin Bostan, Alexander Marynych and Kilian Raschel. Journal of Number Theory (2019) 204 113-133 (arXiv:1901.03002, pdf)
• On walks avoiding a quadrant, Kilian Raschel and Amélie Trotignon. Electronic Journal of Combinatorics (2019) 26 Paper 3.31 1-34 (arXiv:1807.08610, pdf)
• The extinction problem for a class of distylous plant populations, Gerold Alsmeyer and Kilian Raschel. Journal of Mathematical Biology (2019) 78 1841-1874 (arXiv:1805.03123, pdf)
• 3D positive lattice walks and spherical triangles, Beniamin Bogosel, Vincent Perrollaz, Kilian Raschel and Amélie Trotignon. Journal of Combinatorial Theory, Series A 172 (2020) 105189 (arXiv:1804.06245, pdf)
• Boundary behavior of random walks in cones, Kilian Raschel and Pierre Tarrago. Markov Processes and Related Fields 26 4 (2020) 711-756 (arXiv:1803.09253, pdf)
• Differential transcendence and algebraicity criteria for the series counting weighted quadrant walks, Thomas Dreyfus and Kilian Raschel. Publications Mathématiques de Besançon (2019) 41-80 (arXiv:1709.06831, pdf)
• Counting quadrant walks via Tutte's invariant method, Olivier Bernardi, Mireille Bousquet-Mélou and Kilian Raschel. Combinatorial Theory 1 (2021) Paper No. 3 (arXiv:1708.08215, pdf)
• The Z-invariant Ising model via dimers, Cédric Boutillier, Béatrice de Tilière and Kilian Raschel. Probability Theory and Related Fields (2019) 174 235-305 (arXiv:1612.09082, pdf)

They have been visiting the project

Gerold Alsmeyer, Beniamin Bogosel, Alin Bostan, Manfred Buchacher, Elisabetta Candellero, François Chapon, Thomas Dreyfus, Matthieu Dussaule, Andrew Elvey Price, Sandro Franceschi, Éric Fusy, Olga Izyumtseva, Helen Jenne, Alexander Marynych, James Mcredmond, Marni Mishna, Stjepan Šebek, Samuel Simon, Pierre Tarrago, Amélie Trotignon, Wolfgang Woess, Yiqiang Q. Zhao

Summary of the ERC project

This 6-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 population biology.

Contact

Feel free to contact me if you have questions.