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Table of Contents
Intro
Introduction
These Lecture Notes (Do Not) Contain
Contents
Part I (Planar) Maps
1 Discrete Random Surfaces in High Genus
1.1 What Is a Map? Different Points of View
1.1.1 Gluing of Polygons and a First Exploration
Genus
1.1.2 Other Definitions of Maps
Via Permutations
Embedded Graphs
1.1.3 Duality
1.2 Geometry and Topology of Uniform Maps
1.2.1 Enumeration ``à la Tutte''
1.2.2 Uniform Maps Are Almost Uniform Permutations
Geometric and Topological Properties of a Uniform Map
1.3 Exploring Random Maps with Prescribed Faces and a Conjecture
1.3.1 Random Gluing of Prescribed Polygons
1.3.2 Peeling Explorations of MP
1.3.3 Examples of Peeling Explorations
Conclusion: Impose Topological Constraints!
2 Why Are Planar Maps Exceptional?
2.1 Finite and Infinite Planar Maps
2.1.1 Finite Planar Maps
2.1.2 Local Topology and Infinite Maps
2.1.3 Infinite Maps of the Plane and the Half-Plane
2.2 Euler's Formula and Applications
2.2.1 k-Angulations and Bipartite Maps
2.2.2 Platonic Solids
2.2.3 Fàry Theorem
2.2.4 6-5-4 Color Theorem
2.2.5 Moser's circle
2.3 Faithful Representations of Planar Maps
2.3.1 Tutte's Barycentric Embedding
2.3.2 Circle Packing
3 The Miraculous Enumeration of Bipartite Maps
3.1 Maps with a Boundary and a Target
3.1.1 Maps with a Boundary
3.1.2 Maps with a Target
3.2 Counting Planar Maps and Tutte's Equation
3.2.1 The Case of Quadrangulations
3.2.2 Boltzmann Maps and Tutte Slicing Formula
3.3 Formulas for Disk Partition Functions
3.3.1 Boltzmann Measure
3.3.2 Admissibility
3.4 Getting Our Hands on W()
3.4.1 Towards an Expression for W()
3.4.2 Back to the Admissibility Criterion
3.5 Examples
3.5.1 2p-Angulations
3.5.2 Uniform Bipartite Maps
3.5.3 Triangulations
3.5.4 Canonical Stable Maps
Part II Peeling Explorations
4 Peeling of Finite Boltzmann Maps
4.1 Peeling Processes
4.1.1 Gluing Maps with a Boundary
4.1.2 Peeling Process
4.1.3 Peeling Process with a Target and Filled-in Explorations
4.2 Law of the Peeling Under the Boltzmann Measures
4.2.1 q-Boltzmann Maps
4.2.2 q-Boltzmann Maps Without Target
4.2.3 q-Boltzmann Maps with Target
Introduction
These Lecture Notes (Do Not) Contain
Contents
Part I (Planar) Maps
1 Discrete Random Surfaces in High Genus
1.1 What Is a Map? Different Points of View
1.1.1 Gluing of Polygons and a First Exploration
Genus
1.1.2 Other Definitions of Maps
Via Permutations
Embedded Graphs
1.1.3 Duality
1.2 Geometry and Topology of Uniform Maps
1.2.1 Enumeration ``à la Tutte''
1.2.2 Uniform Maps Are Almost Uniform Permutations
Geometric and Topological Properties of a Uniform Map
1.3 Exploring Random Maps with Prescribed Faces and a Conjecture
1.3.1 Random Gluing of Prescribed Polygons
1.3.2 Peeling Explorations of MP
1.3.3 Examples of Peeling Explorations
Conclusion: Impose Topological Constraints!
2 Why Are Planar Maps Exceptional?
2.1 Finite and Infinite Planar Maps
2.1.1 Finite Planar Maps
2.1.2 Local Topology and Infinite Maps
2.1.3 Infinite Maps of the Plane and the Half-Plane
2.2 Euler's Formula and Applications
2.2.1 k-Angulations and Bipartite Maps
2.2.2 Platonic Solids
2.2.3 Fàry Theorem
2.2.4 6-5-4 Color Theorem
2.2.5 Moser's circle
2.3 Faithful Representations of Planar Maps
2.3.1 Tutte's Barycentric Embedding
2.3.2 Circle Packing
3 The Miraculous Enumeration of Bipartite Maps
3.1 Maps with a Boundary and a Target
3.1.1 Maps with a Boundary
3.1.2 Maps with a Target
3.2 Counting Planar Maps and Tutte's Equation
3.2.1 The Case of Quadrangulations
3.2.2 Boltzmann Maps and Tutte Slicing Formula
3.3 Formulas for Disk Partition Functions
3.3.1 Boltzmann Measure
3.3.2 Admissibility
3.4 Getting Our Hands on W()
3.4.1 Towards an Expression for W()
3.4.2 Back to the Admissibility Criterion
3.5 Examples
3.5.1 2p-Angulations
3.5.2 Uniform Bipartite Maps
3.5.3 Triangulations
3.5.4 Canonical Stable Maps
Part II Peeling Explorations
4 Peeling of Finite Boltzmann Maps
4.1 Peeling Processes
4.1.1 Gluing Maps with a Boundary
4.1.2 Peeling Process
4.1.3 Peeling Process with a Target and Filled-in Explorations
4.2 Law of the Peeling Under the Boltzmann Measures
4.2.1 q-Boltzmann Maps
4.2.2 q-Boltzmann Maps Without Target
4.2.3 q-Boltzmann Maps with Target