[Paper Review] Planar maps and random partitions

Jérémie Bouttier|arXiv (Cornell University)|Dec 14, 2019

Random Matrices and Applications174 references4 citations

TL;DR

This habilitation thesis synthesizes research from 2005–2019 on random planar maps and random partitions, introducing a unified bijective approach via slice decomposition, computing the three-point function of quadrangulations, and analyzing the O(n) loop model on maps. It further establishes connections between Schur processes, domino tilings, and fermionic systems, revealing universal edge and bulk asymptotics via determinantal and pfaffian point processes.

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.

Motivation & Objective

• To unify and extend bijective methods for studying random planar maps, particularly through the slice decomposition framework.
• To compute the three-point function of random quadrangulations and connect it to continued fractions and Boltzmann map models.
• To analyze the O(n) loop model on random planar maps, deriving its phase diagram and loop nesting statistics.
• To explore the link between steep domino tilings, Schur processes, and fermionic systems, including bulk and edge asymptotics.
• To investigate universal scaling limits in Schur processes, especially in the presence of periodic and free boundaries, and their connections to KPZ universality.

Proposed method

• Employing a general map-mobile bijection to translate metric properties of planar maps into combinatorial structures amenable to analysis.
• Applying the slice decomposition—a bijective, recursive method—to irreducible and general planar maps, enabling enumeration and statistical analysis.
• Using recursive combinatorial decompositions to solve the O(n) loop model on random maps, yielding exact phase diagrams and nesting statistics.
• Formulating Schur processes via growth diagrams and domino tilings, with a focus on periodic and free boundary conditions.
• Applying saddle-point methods and determinantal/pfaffian point process techniques to derive bulk and edge asymptotics in Schur processes.
• Leveraging connections to integrable probability, including the Robinson-Schensted-Knuth correspondence and the domino shuffling algorithm, to enable efficient sampling and asymptotic analysis.

Experimental results

Research questions

• RQ1 How can the three-point function of random quadrangulations be computed using a general map-mobile bijection, and what is its relation to continued fractions?
• RQ2 What is the structure and universality of the phase diagram of the O(n) loop model on random planar maps, and how do loop nesting statistics behave?
• RQ3 How do bulk and edge asymptotics of Schur processes with periodic and free boundaries relate to known distributions such as Tracy–Widom and Airy kernels?
• RQ4 What is the connection between the Schur process and last passage percolation, and how does it relate to the KPZ equation and finite-temperature kernels?
• RQ5 Can the methods developed for Schur processes be extended to nondeterminantal models such as Macdonald processes, particularly with two free boundaries?

Key findings

• The three-point function of random quadrangulations is computed via a general map-mobile bijection, revealing deep connections to continued fractions and Boltzmann map models.
• The slice decomposition provides a unified bijective framework for analyzing irreducible and general planar maps, enabling recursive enumeration and metric analysis.
• The O(n) loop model on random planar maps exhibits a rich phase diagram, with exact solutions derived through recursive combinatorial decomposition.
• Loop nesting statistics in the O(n) model are analytically characterized, showing universal scaling behavior across different regimes.
• Bulk and edge asymptotics of Schur processes with one and two free boundaries are derived, with edge limits expressed via Fredholm pfaffians and interpolating Tracy–Widom distributions.
• A connection is established between the Schur process and the KPZ equation through the appearance of the finite-temperature Airy kernel in edge asymptotics, suggesting a deeper universality link.