Planar maps and random partitions

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.

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

References in corpus (5)

• The topological structure of scaling limits of large planar maps
• The degree distribution in bipartite planar maps: applications to the Ising model
• The three-point function of planar quadrangulations
• Blocked edges on Eulerian maps and mobiles: Application to spanning trees, hard particles and the Ising model
• Vacancy localization in the square dimer model

Cited by in corpus (2)

• Bijective enumeration of planar bipartite maps with three tight boundaries, or how to slice pairs of pants
• On quasi-polynomials counting planar tight maps