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High genus maps : from the KP hierarchy to probabilistic limits

Abstract : This thesis focuses on combinatorial maps, which are defined as embeddings of graphs on surfaces, or equivalently as gluing of polygons. The genus g of the map is defined as the number of handles of the surface on which it is embedded.In addition to being combinatorial objects, the maps can be represented as factorizations of permutations, which also makes them algebraic objects, which one can study in particular thanks to the representation theory of the symmetric group. In particular, these algebraic properties of maps mean that their generating series satisfies the KP hierarchy (and its generalization, the 2-Toda hierarchy). The KP hierarchy is an infinite set of partial differential equations in an infinity of variables. The partial differential equations of the KP hierarchy are then translated into recurrence formulas which make it possible to enumerate maps of any genus.On the other hand, it is interesting to study the geometric properties of maps, and in particular very large random maps. Many works have focused on the geometrical properties of planar maps, ie of genus 0. In this thesis, we study maps of large genus, that is to say whose genus tends towards infinity at the same time as the size of the map. What will particularly interest us is the notion of local limit, which describes the law of the neighborhood of a particular point (the root) of large uniform random maps.The first part of this thesis (Chapters 1 to 3) is an introduction to all the necessary concepts: maps, of course, but also the KP hierarchy and local limits. In a second part (Chapters 4 and 5), we will seek to deepen the relationship between maps and KP hierarchy, either by explaining existing formulas by combinatorial constructions, or by discovering new formulas. The third part (Chapters 6 and 7) focuses on the study of the local limits of large maps, using in particular the results obtained from the KP hier-archy. Finally the manuscript ends with some open problems (Chapter 8).
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Submitted on : Tuesday, March 16, 2021 - 10:05:09 AM
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Baptiste Louf. High genus maps : from the KP hierarchy to probabilistic limits. Computation and Language [cs.CL]. Université Paris Cité, 2020. English. ⟨NNT : 2020UNIP7020⟩. ⟨tel-03170319⟩



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