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1. Multicritical Schur measures and higher-order analogues of the Tracy-Widom distribution

Author

Betea, Dan, Bouttier, Jérémie, Walsh, Harriet, Institut de Physique Théorique - UMR CNRS 3681 (IPHT), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Physique de l'ENS Lyon (Phys-ENS), École normale supérieure de Lyon (ENS de Lyon)-Université de Lyon-Centre National de la Recherche Scientifique (CNRS), Laboratoire Angevin de Recherche en Mathématiques (LAREMA), Université d'Angers (UA)-Centre National de la Recherche Scientifique (CNRS), ANR-18-CE40-0033,DIMERS,Dimères : de la combinatoire à la mécanique quantique(2018), ANR-19-CE48-0011,COMBINE,Combinatoire enumerative en interaction avec l'algebre, la theorie des nombres et la physique(2019), European Project: ERC-2016-STG 716083,CombiTop, and European Project: 759702,COMBINEPIC

Subjects
[MATH.MATH-PR]Mathematics [math]/Probability [math.PR], [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], Probability (math.PR), [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO], FOS: Mathematics, Mathematics - Combinatorics, FOS: Physical sciences, Combinatorics (math.CO), Mathematical Physics (math-ph), Mathematical Physics, Mathematics - Probability
Abstract

We introduce multicritical Schur measures, which are probability laws on integer partitions which give rise to non-generic fluctuations at their edge. They are in the same universality classes as one-dimensional momentum-space models of free fermions in flat confining potentials, studied by Le Doussal, Majumdar and Schehr. These universality classes involve critical exponents of the form 1/(2m+1), with m a positive integer, and asymptotic distributions given by Fredholm determinants constructed from higher order Airy kernels, extending the generic Tracy-Widom GUE distribution recovered for m=1. We also compute limit shapes for the multicritical Schur measures, discuss the finite temperature setting, and exhibit an exact mapping to the multicritical unitary matrix models previously encountered by Periwal and Shevitz., Comment: 50 pages, 8 figures, long version of arXiv:2012.01995 [math.CO]

Published
2023

2. From point processes to quantum optics and back

Author

Bardenet, Rémi, Feller, Alexandre, Bouttier, Jérémie, Degiovanni, Pascal, Hardy, Adrien, Rançon, Adam, Roussel, Benjamin, Schehr, Grégory, Westbrook, Christoph I., Centre de Recherche en Informatique, Signal et Automatique de Lille - UMR 9189 (CRIStAL), Centrale Lille-Université de Lille-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS), Institut de Physique Théorique - UMR CNRS 3681 (IPHT), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Physique de l'ENS Lyon (Phys-ENS), École normale supérieure de Lyon (ENS de Lyon)-Université de Lyon-Centre National de la Recherche Scientifique (CNRS), Qube Research and Technologies, Laboratoire de Physique des Lasers, Atomes et Molécules - UMR 8523 (PhLAM), Université de Lille-Centre National de la Recherche Scientifique (CNRS), Aalto University, Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS), Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), Laboratoire Charles Fabry / Gaz Quantiques, Laboratoire Charles Fabry (LCF), Institut d'Optique Graduate School (IOGS)-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS)-Institut d'Optique Graduate School (IOGS)-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), ANR-20-CHIA-0002,Baccarat,Apprentissage bayésien pour les modèles coûteux, avec applications à la biologie cellulaire(2020), ANR-14-CE32-0017,1 shot reloaded,Optique quantique électronique : effets à N corps à l'échelle de quelques particules(2014), ANR-10-LABX-0039,PALM,Physics: Atoms, Light, Matter(2010), ANR-10-LABX-0070,MILYON,Community of mathematics and fundamental computer science in Lyon(2010), ANR-21-CE47-0012,QuSig4QuSense,Le radar électronique(2021), ANR-16-IDEX-0004,ULNE,ULNE(2016), ANR-20-CE47-0001,COSQUA,Cosmologie et Simulation Quantique(2020), ANR-22-QUA2-0008,MENTA,Quantification d'intrication multipartite dans des systèmes atomiques(2022), ANR-17-CE30-0027,RaMaTraF,Matrices aléatoires et fermions piégés(2017), ANR-18-CE40-0033,DIMERS,Dimères : de la combinatoire à la mécanique quantique(2018), ANR-19-CE48-0011,COMBINE,Combinatoire enumerative en interaction avec l'algebre, la theorie des nombres et la physique(2019), European Project: ERC-2019-STG-851866,Blackjack, European Project: 307392,EC:FP7:ERC,ERC-2012-StG_20111012,SEQUOIA(2013), Laboratoire de Physique Théorique et Hautes Energies (LPTHE), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), and Institut d'Optique Graduate School (IOGS)-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS)

Subjects
Signal Processing (eess.SP), [PHYS]Physics [physics], Quantum Physics, Probability (math.PR), [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], FOS: Physical sciences, Mathematical Physics (math-ph), [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], [PHYS.QPHY]Physics [physics]/Quantum Physics [quant-ph], [MATH.MATH-ST]Mathematics [math]/Statistics [math.ST], FOS: Electrical engineering, electronic engineering, information engineering, FOS: Mathematics, Electrical Engineering and Systems Science - Signal Processing, Quantum Physics (quant-ph), Mathematical Physics, Mathematics - Probability
Abstract

Some fifty years ago, in her seminal PhD thesis, Odile Macchi introduced permanental and determinantal point processes. Her initial motivation was to provide models for the set of detection times in fundamental bosonic or fermionic optical experiments, respectively. After two rather quiet decades, these point processes have quickly become standard examples of point processes with nontrivial, yet tractable, correlation structures. In particular, determinantal point processes have been since the 1990s a technical workhorse in random matrix theory and combinatorics, and a standard model for repulsive point patterns in machine learning and spatial statistics since the 2010s. Meanwhile, our ability to experimentally probe the correlations between detection events in bosonic and fermionic optics has progressed tremendously. In Part I of this survey, we provide a modern introduction to the concepts in Macchi's thesis and their physical motivation, under the combined eye of mathematicians, physicists, and signal processers. Our objective is to provide a shared basis of knowledge for later cross-disciplinary work on point processes in quantum optics, and reconnect with the physical roots of permanental and determinantal point processes., Disclaimer: this is Part I of a cross-disciplinary survey that is work in progress, all comments are welcome

Published
2022

3. Bijective enumeration of planar bipartite maps with three tight boundaries, or how to slice pairs of pants

Author

Bouttier, Jérémie, Guitter, Emmanuel, Miermont, Grégory, Institut de Physique Théorique - UMR CNRS 3681 (IPHT), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Physique de l'ENS Lyon (Phys-ENS), École normale supérieure - Lyon (ENS Lyon)-Centre National de la Recherche Scientifique (CNRS)-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université de Lyon, Unité de Mathématiques Pures et Appliquées (UMPA-ENSL), Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Lyon (ENS Lyon), ANR-10-LABX-0070,MILYON,Community of mathematics and fundamental computer science in Lyon(2010), ANR-18-CE40-0033,DIMERS,Dimères : de la combinatoire à la mécanique quantique(2018), ANR-19-CE48-0011,COMBINE,Combinatoire enumerative en interaction avec l'algebre, la theorie des nombres et la physique(2019), École normale supérieure de Lyon (ENS de Lyon)-Université de Lyon-Centre National de la Recherche Scientifique (CNRS), École normale supérieure de Lyon (ENS de Lyon)-Centre National de la Recherche Scientifique (CNRS), École normale supérieure - Lyon (ENS Lyon)-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université de Lyon-Centre National de la Recherche Scientifique (CNRS), and École normale supérieure - Lyon (ENS Lyon)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Probability (math.PR), [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], FOS: Physical sciences, Ocean Engineering, Mathematical Physics (math-ph), Mathematics::Geometric Topology, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO], FOS: Mathematics, Mathematics - Combinatorics, Combinatorics (math.CO), Mathematical Physics, Mathematics - Probability, 2020 MSC: 05A15, 05A19, 60C05, 60F05
Abstract

We consider planar maps with three boundaries, colloquially called pairs of pants. In the case of bipartite maps with controlled face degrees, a simple expression for their generating function was found by Eynard and proved bijectively by Collet and Fusy. In this paper, we obtain an even simpler formula for \emph{tight} pairs of pants, namely for maps whose boundaries have minimal length in their homotopy class. We follow a bijective approach based on the slice decomposition, which we extend by introducing new fundamental building blocks called bigeodesic triangles and diangles, and by working on the universal cover of the triply punctured sphere. We also discuss the statistics of the lengths of minimal separating loops in (non necessarily tight) pairs of pants and annuli, and their asymptotics in the large volume limit., 76 pages, 29 figures, final version

Published
2021
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4. Planar maps and random partitions

Author

Bouttier, Jérémie, Bouttier, Jérémie, Jeunes Chercheuses et Jeunes Chercheurs - Combinatoire des cartes et applications - - CARTAPLUS2012 - ANR-12-JS02-0001 - JC - VALID, Appel à projets générique - GRaphes et Arbres ALéatoires - - GRAAL2014 - ANR-14-CE25-0014 - Appel à projets générique - VALID, Dimères : de la combinatoire à la mécanique quantique - - DIMERS2018 - ANR-18-CE40-0033 - AAPG2018 - VALID, Institut de Physique Théorique - UMR CNRS 3681 (IPHT), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Physique de l'ENS Lyon (Phys-ENS), École normale supérieure - Lyon (ENS Lyon)-Centre National de la Recherche Scientifique (CNRS)-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université de Lyon, Ville de Paris (programme Émergences Combinatoire à Paris), CNRS INSMI (PEPS CARMA), Université Paris XI, Philippe Chassaing, ANR-12-JS02-0001,CARTAPLUS,Combinatoire des cartes et applications(2012), ANR-14-CE25-0014,GRAAL,GRaphes et Arbres ALéatoires(2014), and ANR-18-CE40-0033,DIMERS,Dimères : de la combinatoire à la mécanique quantique(2018)

Subjects
Schur processes, [MATH.MATH-PR] Mathematics [math]/Probability [math.PR], modèles de dimères, bijections, cartes planaires, random partitions, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], [MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO], [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO], [MATH.MATH-MP] Mathematics [math]/Mathematical Physics [math-ph], dimer models, planar maps, partitions aléatoires, processus de Schur
Abstract

This habilitation thesis summarizes the research that I have carried out from 2005 to 2019. It is organized in four chapters. The first three deal with random planar maps. Chapter 1 is about their metric properties: from a general map-mobile bijection, we compute the three-point function of quadrangulations, before discussing the connection with continued fractions. Chapter 2 presents the slice decomposition, a unified bijective approach that applies notably to irreducible maps. Chapter 3 concerns the O(n) loop model on planar maps: by a combinatorial decomposition, we obtain the phase diagram before studying loop nesting statistics. Chapter 4 deals with random partitions and Schur processes, from steep domino tilings to fermionic systems., Ce mémoire d'habilitation est une synthèse des travaux de recherche que j'ai effectués entre 2005 et 2019. Il s'organise en quatre chapitres. Les trois premiers portent sur les cartes planaires aléatoires. Le chapitre 1 s'intéresse à leurs propriétés métriques: à partir d'une bijection générale entre cartes et mobiles, on calcule la fonction à trois points des quadrangulations, avant d'évoquer le lien avec les fractions continues. Le chapitre 2 présente la décomposition en tranches, une approche bijective unifiée qui s'applique notamment aux cartes irréductibles. Le chapitre 3 a pour sujet le modèle de boucles O(n) sur les cartes planaires: par une décomposition combinatoire récursive on obtient le diagramme de phase, puis on étudie les statistiques d'emboîtements entre boucles. Le chapitre 4 porte quant à lui sur les partitions aléatoires et processus de Schur, en allant des pavages de dominos pentus aux systèmes fermioniques.

Published
2019

5. Cartes planaires et partitions aléatoires

Author

Bouttier, Jérémie, Institut de Physique Théorique - UMR CNRS 3681 (IPHT), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Physique de l'ENS Lyon (Phys-ENS), École normale supérieure de Lyon (ENS de Lyon)-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université de Lyon-Centre National de la Recherche Scientifique (CNRS), Ville de Paris (programme Émergences Combinatoire à Paris), CNRS INSMI (PEPS CARMA), Université Paris XI, Philippe Chassaing, ANR-12-JS02-0001,CARTAPLUS,Combinatoire des cartes et applications(2012), ANR-14-CE25-0014,GRAAL,GRaphes et Arbres ALéatoires(2014), ANR-18-CE40-0033,DIMERS,Dimères : de la combinatoire à la mécanique quantique(2018), École normale supérieure - Lyon (ENS Lyon)-Centre National de la Recherche Scientifique (CNRS)-Université Claude Bernard Lyon 1 (UCBL), and Université de Lyon-Université de Lyon

Subjects
Schur processes, modèles de dimères, bijections, cartes planaires, random partitions, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO], Mathematics - Combinatorics, dimer models, Mathematical Physics, Mathematics - Probability, planar maps, partitions aléatoires, processus de Schur
Abstract

This habilitation thesis summarizes the research that I have carried out from 2005 to 2019. It is organized in four chapters. The first three deal with random planar maps. Chapter 1 is about their metric properties: from a general map-mobile bijection, we compute the three-point function of quadrangulations, before discussing the connection with continued fractions. Chapter 2 presents the slice decomposition, a unified bijective approach that applies notably to irreducible maps. Chapter 3 concerns the $O(n)$ loop model on planar maps: by a combinatorial decomposition, we obtain the phase diagram before studying loop nesting statistics. Chapter 4 deals with random partitions and Schur processes, from steep domino tilings to fermionic systems., Comment: Habilitation thesis, written in English except an introduction in French, 107 pages, many figures. Pages numbers differ from the printed copies given at the defence on 2 December 2019

Published
2019

6. Perfect sampling algorithm for Schur processes

Author

Betea, Dan, Boutillier, Cédric, Bouttier, Jérémie, Chapuy, Guillaume, Corteel, Sylvie, Vuletić, Mirjana, Laboratoire de Probabilités et Modèles Aléatoires (LPMA), Centre National de la Recherche Scientifique (CNRS)-Université Paris Diderot - Paris 7 (UPD7)-Université Pierre et Marie Curie - Paris 6 (UPMC), Laboratoire de Probabilités, Statistiques et Modélisations (LPSM (UMR_8001)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Institut de Physique Théorique - UMR CNRS 3681 (IPHT), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), Département de Mathématiques et Applications - ENS Paris (DMA), École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), Laboratoire d'informatique Algorithmique : Fondements et Applications (LIAFA), Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Department of Mathematics (-), University of Massachusetts [Boston] (UMass Boston), University of Massachusetts System (UMASS)-University of Massachusetts System (UMASS), Projet Émergences Combinatoire à Paris (Ville de Paris)., ANR-08-JCJC-0011,Icomb(2008), ANR-10-BLAN-0123,MAC2,Modèles aléatoires critiques bi-dimensionnels(2010), ANR-12-JS02-0001,CARTAPLUS,Combinatoire des cartes et applications(2012), ANR-14-CE25-0014,GRAAL,GRaphes et Arbres ALéatoires(2014), Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Probabilités, Statistique et Modélisation (LPSM (UMR_8001)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), and École normale supérieure - Paris (ENS-PSL)

Subjects
FOS: Computer and information sciences, Schur processes, Mathematics::Combinatorics, Statistical Mechanics (cond-mat.stat-mech), Discrete Mathematics (cs.DM), vertex operators, Probability (math.PR), 05A17, 05E10, 60C05, 60J10, 68U20, 82B20, FOS: Physical sciences, [INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM], perfect sampling, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], interlaced partitions, [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO], FOS: Mathematics, MSC: 05A17, 05E10, 60C05, 60J10, 68U20, 82B20, Mathematics - Combinatorics, Combinatorics (math.CO), [PHYS.COND.CM-SM]Physics [physics]/Condensed Matter [cond-mat]/Statistical Mechanics [cond-mat.stat-mech], Mathematics - Probability, Condensed Matter - Statistical Mechanics, Markov dynamics, Computer Science - Discrete Mathematics
Abstract

We describe random generation algorithms for a large class of random combinatorial objects called Schur processes, which are sequences of random (integer) partitions subject to certain interlacing conditions. This class contains several fundamental combinatorial objects as special cases, such as plane partitions, tilings of Aztec diamonds, pyramid partitions and more generally steep domino tilings of the plane. Our algorithm, which is of polynomial complexity, is both exact (i.e. the output follows exactly the target probability law, which is either Boltzmann or uniform in our case), and entropy optimal (i.e. it reads a minimal number of random bits as an input). The algorithm encompasses previous growth procedures for special Schur processes related to the primal and dual RSK algorithm, as well as the famous domino shuffling algorithm for domino tilings of the Aztec diamond. It can be easily adapted to deal with symmetric Schur processes and general Schur processes involving infinitely many parameters. It is more concrete and easier to implement than Borodin's algorithm, and it is entropy optimal. At a technical level, it relies on unified bijective proofs of the different types of Cauchy and Littlewood identities for Schur functions, and on an adaptation of Fomin's growth diagram description of the RSK algorithm to that setting. Simulations performed with this algorithm suggest interesting limit shape phenomena for the corresponding tiling models, some of which are new., Comment: 26 pages, 19 figures (v3: final version, corrected a few misprints present in v2)

Published
2018

7. The free boundary Schur process and applications (extended abstract)

Author

Betea, Dan, Bouttier, Jérémie, Nejjar, Peter, Vuletić, Mirjana, Département de Mathématiques et Applications - ENS Paris (DMA), Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), Institut de Recherche en Informatique Fondamentale (IRIF (UMR_8243)), Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Institut de Physique Théorique - UMR CNRS 3681 (IPHT), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Physique de l'ENS Lyon (Phys-ENS), École normale supérieure - Lyon (ENS Lyon)-Centre National de la Recherche Scientifique (CNRS)-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université de Lyon, Institute of Science and Technology [Austria] (IST Austria), University of Massachusetts [Boston] (UMass Boston), University of Massachusetts System (UMASS), Projet Combinatoire à Paris (Ville de Paris), ANR-12-JS02-0001,CARTAPLUS,Combinatoire des cartes et applications(2012), ANR-14-CE25-0014,GRAAL,GRaphes et Arbres ALéatoires(2014), European Project: 338804,EC:FP7:ERC,ERC-2013-ADG,RANMAT(2014), École normale supérieure - Paris (ENS-PSL), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), École normale supérieure de Lyon (ENS de Lyon)-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université de Lyon-Centre National de la Recherche Scientifique (CNRS), Institute of Science and Technology [Klosterneuburg, Austria] (IST Austria), École normale supérieure - Paris (ENS Paris), École normale supérieure - Paris (ENS Paris)-Centre National de la Recherche Scientifique (CNRS), and École normale supérieure - Lyon (ENS Lyon)-Université Claude Bernard Lyon 1 (UCBL)

Subjects
[MATH.MATH-PR]Mathematics [math]/Probability [math.PR], [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO], Mathematics::Representation Theory
Abstract

International audience; We study the Schur process with two free boundaries, a generalization of the original Schur process of Okounkov and Reshetikhin. We compute its correlation functions for arbitrary specializations and apply the result to the asymptotics of symmetric last passage percolation models, symmetric plane partitions and plane overpartitions. This is (with minor modifications) an extended abstract accepted for the proceedings of FPSAC 2017.

Published
2017

8. Nesting statistics in the $O(n)$ loop model on random planar maps

Author

Borot, Gaëtan, Bouttier, Jérémie, Duplantier, Bertrand, Max Planck Institute for Mathematics (MPIM), Max-Planck-Gesellschaft, Département de Mathématiques et Applications - ENS Paris (DMA), École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), Institut de Physique Théorique - UMR CNRS 3681 (IPHT), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), CNRS Projet international de coopération scientifique (PICS) 'Conformal Liouville Quantum Gravity' N° PICS06769, T16/170, ANR-12-JS02-0001,CARTAPLUS,Combinatoire des cartes et applications(2012), ANR-14-CE25-0014,GRAAL,GRaphes et Arbres ALéatoires(2014), École normale supérieure - Paris (ENS-PSL), Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Paris (ENS Paris), and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)

Subjects
High Energy Physics - Theory, [PHYS]Physics [physics], High Energy Physics - Theory (hep-th), Probability (math.PR), FOS: Mathematics, Mathematics - Combinatorics, FOS: Physical sciences, Mathematical Physics (math-ph), Combinatorics (math.CO), Mathematical Physics, Mathematics - Probability, 05Axx, 60K35, 81T40
Abstract

In the O(n) loop model on random planar maps, we study the depth - in terms of the number of levels of nesting - of the loop configuration, by means of analytic combinatorics. We focus on the 'refined' generating series of pointed disks or cylinders, which keep track of the number of loops separating the marked point from the boundary (for disks), or the two boundaries (for cylinders). For the general O(n) loop model, we show that these generating series satisfy functional relations obtained by a modification of those satisfied by the unrefined generating series. In a more specific O(n) model where loops cross only triangles and have a bending energy, we explicitly compute the refined generating series. We analyse their non generic critical behavior in the dense and dilute phases, and obtain the large deviations function of the nesting distribution, which is expected to be universal. Using the framework of Liouville quantum gravity (LQG), we show that a rigorous functional KPZ relation can be applied to the multifractal spectrum of extreme nesting in the conformal loop ensemble (CLE) in the Euclidean unit disk, as obtained by Miller, Watson and Wilson, or to its natural generalisation to the Riemann sphere. It allows us to recover the large deviations results obtained for the critical O(n) random planar map models. This offers, at the refined level of large deviations theory, a rigorous check of the fundamental fact that the universal scaling limits of random planar map models as weighted by partition functions of critical statistical models are given by LQG random surfaces decorated by independent CLEs., Comment: 71 pages, 11 figures. v2: minor text and abstract edits, references added. v2: 105 pages, revised and expanded, including justification of delta analyticity

Published
2016

9. Physique statistique des surfaces aléatoires et combinatoire bijective des cartes planaires

Author

Bouttier, Jérémie, Service de Physique Théorique (SPhT), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Centre National de la Recherche Scientifique (CNRS), Université Pierre et Marie Curie - Paris VI, Di Francesco Philippe, and Bouttier, Jérémie

Subjects
gravité quantique, random geometry, graph theory, géométrie aléatoire, [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], [PHYS.MPHY] Physics [physics]/Mathematical Physics [math-ph], enumerative combinatorics, systèmes intégrables, integrable systems, quantum gravity, combinatoire énumérative, théorie des graphes, modèles de matrices, matrix models
Abstract

Maps are combinatorial objects arising in physics as the natural discretization of random surfaces used in two-dimensional quantum gravity or string theory, as well as in matrix models. After recalling these relations, we establish correspondences between various classes of maps and trees, that are other combinatorial objects with a simple structure. A first mathematical outcome of these constructions are bijective, elementary and rigorous proofs of several results in map enumeration. Moreover, we access to some fine information on the intrinsic geometry of maps, leading to analytical exact results thanks to an unexpected integrability property. Finally we address the question of the existence of a universal continuum limit., Les cartes sont des objets combinatoires apparaissant en physique comme discrétisation naturelle des surfaces aléatoires employées pour la gravité quantique bidimensionnelle ou la théorie des cordes, ainsi que dans les modèles de matrices. Après rappel de ces relations, nous établissons des correspondances entre diverses classes de cartes et d'arbres, autres objets combinatoires de structure simple. Un premier intérêt mathématique de ces constructions est de donner des preuves bijectives, élémentaires et rigoureuses, de plusieurs résultats d'énumération de cartes. Par ailleurs, nous accédons ainsi à une information fine sur la géométrie intrinsèque des cartes, conduisant à des résultats analytiques exacts grâce à une propriété inattendue d'intégrabilité. Nous abordons enfin la question de l'existence d'une limite continue universelle.

Published
2005

10. Convergence de cartes et tas de sable

Author

Selig, Thomas, Laboratoire Bordelais de Recherche en Informatique (LaBRI), Université de Bordeaux (UB)-Centre National de la Recherche Scientifique (CNRS)-École Nationale Supérieure d'Électronique, Informatique et Radiocommunications de Bordeaux (ENSEIRB), Université de Bordeaux, Jean-François Marckert, Marckert, Jean-François, Duchon, Philippe, Bonichon, Nicolas, Bouttier, Jérémie, Duquesne, Thomas, Curien, Nicolas, and Gittenberger, Bernhard

Subjects
Markov chain, Probabilites, [SCCO.COMP]Cognitive science/Computer science, Tas de sable, Limit theorems, Cartes, Sendpile, Combinatories, Graphes, Théorèmes limites, Maps, Chaine de Markov, Combinatoire, Graphs, Probability
Abstract

This Thesis studies various problems located at the boundary between Combinatorics and Probability Theory. It is formed of two independent parts. In the first part, we study the asymptotic properties of some families of \maps" (from a non traditional viewpoint). In thesecond part, we introduce and study a natural stochastic extension of the so-called Sandpile Model, which is a dynamic process on a graph. While these parts are independent, they exploit the same thrust, which is the many interactions between Combinatorics and Discrete Probability, with these two areas being of mutual benefit to each other. Chapter 1 is a general introduction to such interactions, and states the main results of this Thesis. Chapter 2 is an introduction to the convergence of random maps. The main contributions of this Thesis can be found in Chapters 3, 4 (for the convergence of maps) and 5 (for the Stochastic Sandpile model).; Cette thèse est dédiée à l'étude de divers problèmes se situant à la frontière entre combinatoire et théorie des probabilités. Elle se compose de deux parties indépendantes : la première concerne l'étude asymptotique de certaines familles de \cartes" (en un sens non traditionnel), la seconde concerne l'étude d'une extension stochastique naturelle d'un processus dynamique classique sur un graphe appelé modèle du tas de sable. Même si ces deux parties sont a priori indépendantes, elles exploitent la même idée directrice, à savoir les interactions entre les probabilités et la combinatoire, et comment ces domaines sont amenés à se rendreservice mutuellement. Le Chapitre introductif 1 donne un bref aperçu des interactions possibles entre combinatoire et théorie des probabilités, et annonce les principaux résultats de la thèse. Le Chapitres 2 donne une introduction au domaine de la convergence des cartes. Les contributions principales de cette thèse se situent dans les Chapitres 3, 4 (pour les convergences de cartes) et 5 (pour le modèle stochastique du tas de sable).

Published
2014

11. A note on irreducible maps with several boundaries

Author

Emmanuel Guitter, Jérémie Bouttier, Institut de Physique Théorique - UMR CNRS 3681 (IPHT), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), Département de Mathématiques et Applications - ENS Paris (DMA), École normale supérieure - Paris (ENS-PSL), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), ANR-12-JS02-0001,CARTAPLUS,Combinatoire des cartes et applications(2012), ANR-08-JCJC-0011,Icomb(2008), Bouttier, Jérémie, Jeunes Chercheuses et Jeunes Chercheurs - Combinatoire des cartes et applications - - CARTAPLUS2012 - ANR-12-JS02-0001 - JC - VALID, Jeunes chercheuses et jeunes chercheurs - - Icomb2008 - ANR-08-JCJC-0011 - JCJC - VALID, Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), and École normale supérieure - Paris (ENS Paris)

Subjects
Quasi-open map, Degree (graph theory), Applied Mathematics, Generating function, 010103 numerical & computational mathematics, 0102 computer and information sciences, Girth (graph theory), 01 natural sciences, Theoretical Computer Science, Interpretation (model theory), Combinatorics, [MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO], Computational Theory and Mathematics, 010201 computation theory & mathematics, [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO], Bipartite graph, FOS: Mathematics, Mathematics - Combinatorics, Discrete Mathematics and Combinatorics, Geometry and Topology, Tree (set theory), Multiple edges, Combinatorics (math.CO), 0101 mathematics, Mathematics
Abstract

We derive a formula for the generating function of d-irreducible bipartite planar maps with several boundaries, i.e. having several marked faces of controlled degrees. It extends a formula due to Collet and Fusy for the case of arbitrary (non necessarily irreducible) bipartite planar maps, which we recover by taking d=0. As an application, we obtain an expression for the number of d-irreducible bipartite planar maps with a prescribed number of faces of each allowed degree. Very explicit expressions are given in the case of maps without multiple edges (d=2), 4-irreducible maps and maps of girth at least 6 (d=4). Our derivation is based on a tree interpretation of the various encountered generating functions., Comment: 18 pages, 8 figures

Published
2014

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