ED Mathématiques et Informatique
Study of higher dimensional combinatorial objects
by Thomas MULLER (LaBRI - Laboratoire Bordelais de Recherche en Informatique)
The defense will take place at 14h00 - Amphithéâtre du LABRI Université de Bordeaux, 351 cours de la Libération, Bâtiment A30, 33405 Talence
in front of the jury composed of
- Adrian TANASA - Professeur - Université de Bordeaux - Directeur de these
- Frédérique BASSINO - Professeure - Université Sorbonne Paris Nord - Examinateur
- Valentin BONZOM - Professeur - Université Gustave Eiffel - Examinateur
- Frédéric PATRAS - Directeur de recherche - Université Côte d'Azur - Examinateur
- Jean-François MARCKERT - Directeur de recherche - Université de Bordeaux - Examinateur
- Kasia REJZNER - Professor - University of York - Examinateur
- Razvan GURAU - Professor - Universität Heidelberg - Rapporteur
- Eric FUSY - Directeur de recherche - Université Gustave Eiffel - Rapporteur
This thesis explores higher-dimensional generalizations of classical combinatorial objects such as permutations, combinatorial maps, and mosaic floorplans. Its comprehensive goal is to develop combinatorial tools to better understand these higher dimensional objects. The first part of this thesis focuses on the combinatorial properties of random tensor models, whose Feynman graphs are (D+1)-colored graphs that extend the notion combinatorial maps. This analysis focuses on tensor models with interactions of order six or higher. We begin by studying the 1/N expansion, where N denotes the size of the tensors, of the sextic O(N)^3 model, which generalizes in a non-trivial way the sextic U(N)^3 model. Some of the interactions considered in this model lead to a graph structure at the leading order in the 1/N expansion that differs drastically from previously studied models. We then investigate a second asymptotic expansion known as the double scaling limit. We implement this expansion for the prismatic tensor model, a restricted version of the sextic $O(N)^3$ model in which we consider only the prismatic interaction. Studying this expansion requires analyzing sub-leading order graphs in the $1/N$ expansion, whose structure is generally unknown. To address this, we use the scheme decomposition, originally introduced by Gurau and Schaeffer and which adapts in a non trivial way the one introduced by Chapuy, Marcus and Schaeffer for combinatorial maps. This allows us to characterize the structure of the leading-order graphs in the double scaling limit and to determine the dominant contribution to the two-point function in this expansion. Additionally, we investigate duality properties between different tensor models whose symmetries are governed by the groups O(N) and Sp(N). We prove that the transformation N to -N maps the Feynman graph amplitudes of one model to those of the other. We prove that this duality holds for models with interactions of arbitrary order. The second part of this thesis focuses on d-floorplans and d-permutations, the higher dimensional analogs of mosaic floorplans and permutations. A floorplan is a partition of a rectangle by other rectangles with no empty rooms. Additionally, a floorplan is said to be mosaic if the segments induced by the partitioning of the bounding rectangle are non crossing. Similarly, a d-dimensional floorplan is a partition of a d-dimensional hyperrectangles with n disjoint interior d-dimensional hyperrectangles and no empty rooms. This partitioning induces borders, which are (d-1)-hyperrectangles. A d-floorplan is a d-dimensional floorplan for which there are no borders crossing. A d-permutation is a tuple of (d-1) permutations which can be represented as a d-dimensional point diagram. We first construct a generating tree for d-floorplans. Given a d-floorplan, we can remove its emph{top} box and unequivocally fill the resulting empty space in order to obtain a smaller d-floorplan. The structure of the tree is defined by this operation and generalizes the one of mosaic floorplans. However, the associated labels and the rewriting rule appear to be significantly more involved in higher dimensions. This allow us to find the first numbers of the enumeration sequences of d-floorplans, which not match with any known sequences. Then, we establish a bijection between 2^{d-1}-floorplans and d-permutations characterized by forbidden patterns. This bijection generalizes the one between mosaic floorplans and Baxter permutations. This set of d-permutations is strictly contained within the set of Baxter d-permutations introduced by N. Bonichon and P.-J. Morel, {it J. Integer Sequences} 25 (2022) and contains the separable d-permutations introduced by Asinowski and Mansour.