Scaling limit of random planar maps Lecture 1. Olivier Bernardi, CNRS, Université Paris-Sud Workshop on randomness and enumeration Temuco, November 2008 November 2008 Olivier Bernardi – p.1/23
Planar maps A planar map is a connected planar graph embedded in the sphere and considered up to deformation. � = = November 2008 Olivier Bernardi – p.2/23
Goals We consider maps as discrete metric spaces: ( V, d ) . Question: What metric space is the limit of random maps ? November 2008 ▽ Olivier Bernardi – p.3/23
Goals Question: What random metric space is the limit in distribution, for the Gromov Hausdorff topology, of uniformly random, rescaled maps of size n , when n goes to infinity ? November 2008 ▽ Olivier Bernardi – p.3/23
Goals Question: What random metric space is the limit in distribution, for the Gromov Hausdorff topology, of uniformly random, rescaled maps of size n , when n goes to infinity ? Lecture 1: Maps, Bijection with well-labelled trees, continuous trees and maps. Lecture 2: Gromov-Hausdorff topology on metric spaces, convergence of random trees toward the Continuum Random Tree , convergence of random maps toward the Brownian map . November 2008 ▽ Olivier Bernardi – p.3/23
Goals Question: What random metric space is the limit in distribution, for the Gromov Hausdorff topology, of uniformly random, rescaled maps of size n , when n goes to infinity ? References: Bijection: Schaeffer Ph.D. Thesis (98). Distribution of distances: Chassaing & Schaeffer (04). Continuum Random Tree: Aldous (91). Brownian map: Marckert & Mokkadem (06). Convergence of random maps: Le Gall (07). . . . Bouttier, Di Francesco, Guitter, Miermont, Paulin, Weill . . . November 2008 Olivier Bernardi – p.3/23
Maps November 2008 Olivier Bernardi – p.4/23
Rooted maps A planar map is a connected planar graph embedded in the sphere and considered up to deformation. � = = A map is rooted by distinguishing a corner . November 2008 Olivier Bernardi – p.5/23
Quadrangulations A quadrangulation is a planar map such that every faces has degree 4. Proposition: A quadrangulation with n faces has 2 n edges and n + 2 vertices. proof: Incidence relation faces/edges + Euler relation. � November 2008 ▽ Olivier Bernardi – p.6/23
Quadrangulations A quadrangulation is a planar map such that every faces has degree 4. Remark: Quadrangulations are bipartite (since faces generate all cycles). November 2008 Olivier Bernardi – p.6/23
Quadrangulations and general maps Proposition : Maps with n edges are in bijection with quadrangulations with n faces. ⇐ ⇒ November 2008 ▽ Olivier Bernardi – p.7/23
Quadrangulations and general maps Proposition : Maps with n edges are in bijection with quadrangulations with n faces. November 2008 ▽ Olivier Bernardi – p.7/23
Quadrangulations and general maps Proposition : Maps with n edges are in bijection with quadrangulations with n faces. November 2008 ▽ Olivier Bernardi – p.7/23
Quadrangulations and general maps Proposition : Maps with n edges are in bijection with quadrangulations with n faces. November 2008 ▽ Olivier Bernardi – p.7/23
Quadrangulations and general maps Proposition : Maps with n edges are in bijection with quadrangulations with n faces. November 2008 Olivier Bernardi – p.7/23
Counting maps Theorem [Tutte 63] : The number of rooted 2 · 3 n � 2 n � quadrangulations with n faces is q n = . ( n + 1)( n + 2) n November 2008 ▽ Olivier Bernardi – p.8/23
Counting maps Theorem [Tutte 63] : The number of rooted 2 · 3 n � 2 n � quadrangulations with n faces is q n = . ( n + 1)( n + 2) n Remarks: • The asymptotic behavior q n ∼ c n − 5 / 2 ρ n is typical. M ∈M q n z n is algebraic: • The generating function G ( z ) = � 1 − 16 z + (18 z − 1) G ( z ) − 27 z 2 G ( z ) 2 = 0 . November 2008 ▽ Olivier Bernardi – p.8/23
Counting maps Theorem [Tutte 63] : The number of rooted 2 · 3 n � 2 n � quadrangulations with n faces is q n = . ( n + 1)( n + 2) n Main methods for counting maps : • Generating function approach [Tutte 63]: Encoding a recurrence relation via generating functions. • Matrix integrals [Brézin-Itzykson-Parisi-Zuber 78]: Interpreting maps as the Feynman diagrams. • Computation of characters [Goulden-Jackson]: Interpreting maps as products of permutations. • Bijections with decorated trees [Schaeffer 98]. November 2008 Olivier Bernardi – p.8/23
A bijection of Schaeffer Quadrangulations ⇔ Well-labelled trees November 2008 Olivier Bernardi – p.9/23
Well-labelled trees A rooted plane tree is a rooted planar map with a single face. � 2 n 1 � There are C n = trees with n edges. n +1 n November 2008 ▽ Olivier Bernardi – p.10/23
Well-labelled trees A well-labelled tree is a tree with positive labels such that • the root vertex is labelled 1 . • the difference of labels between adjacent vertices is − 1 , 0 or 1 . 4 2 3 1 1 2 2 1 2 2 1 November 2008 Olivier Bernardi – p.10/23
From quadrangulations to trees November 2008 ▽ Olivier Bernardi – p.11/23
From quadrangulations to trees Step 1. Compute the graph distance from the vertex incident to the root-edge. 2 3 3 1 3 2 1 2 1 0 2 November 2008 ▽ Olivier Bernardi – p.11/23
From quadrangulations to trees Step 2. Create an edge of the tree for each face of the quadrangulation. i +1 i +1 i +2 i i +1 i +1 i i November 2008 ▽ Olivier Bernardi – p.11/23
From quadrangulations to trees Step 2. Create an edge of the tree for each face of the quadrangulation. 2 3 3 1 3 2 1 2 1 0 2 November 2008 ▽ Olivier Bernardi – p.11/23
From quadrangulations to trees Step 2. Create an edge of the tree for each face of the quadrangulation. 2 3 3 1 3 2 1 2 1 0 2 November 2008 Olivier Bernardi – p.11/23
Bijection Theorem [Schaeffer 98]: This construction is a bijection between quadrangulations with n faces and well-labelled trees with n edges. 4 2 3 1 1 ⇐ ⇒ 2 2 1 2 2 1 November 2008 ▽ Olivier Bernardi – p.12/23
Bijection Theorem [Schaeffer 98]: This construction is a bijection between quadrangulations with n faces and well-labelled trees with n edges. Proof that one obtains a well-labelled tree : • The quadrangulation has n faces and n + 2 vertices ⇒ The image has n edges and n + 1 vertices. i i − 1 November 2008 ▽ Olivier Bernardi – p.12/23
Bijection Theorem [Schaeffer 98]: This construction is a bijection between quadrangulations with n faces and well-labelled trees with n edges. Proof that one obtains a well-labelled tree : • The quadrangulation has n faces and n + 2 vertices ⇒ The image has n edges and n + 1 vertices. • The image has no cycle, hence it is a tree. i i − 1 i − 1 November 2008 ▽ Olivier Bernardi – p.12/23
Bijection Theorem [Schaeffer 98]: This construction is a bijection between quadrangulations with n faces and well-labelled trees with n edges. Proof that one obtains a well-labelled tree : • The quadrangulation has n faces and n + 2 vertices ⇒ The image has n edges and n + 1 vertices. • The image has no cycle, hence it is a tree. • It is well-labelled. November 2008 Olivier Bernardi – p.12/23
From trees to quadrangulations 4 2 3 1 1 2 2 1 2 2 1 November 2008 ▽ Olivier Bernardi – p.13/23
From trees to quadrangulations 4 2 0 3 1 1 2 2 1 2 2 1 Step 1. Add an isolated vertex labelled 0 . November 2008 ▽ Olivier Bernardi – p.13/23
From trees to quadrangulations 4 2 0 3 1 1 2 2 1 2 2 1 Step 2. Join every corner labelled i to the next corner labelled i − 1 around the tree. November 2008 ▽ Olivier Bernardi – p.13/23
From trees to quadrangulations 4 2 0 3 1 1 2 2 1 2 2 1 One obtains a quadrangulation with labels indicating the distance from the vertex adjacent to the root-edge. November 2008 Olivier Bernardi – p.13/23
Counting ? It does not seem easy to count well-labelled trees. 2 · 3 n � 2 n � q n = . ( n + 1)( n + 2) n November 2008 ▽ Olivier Bernardi – p.14/23
Counting ? It does not seem easy to count well-labelled trees. 3 n � 2 n � But it is easy to see that there are t n = labelled trees: ( n +1) n • The minimum of labels is 1 . • The difference of labels between adjacent vertices is − 1 , 0 , 1 . November 2008 Olivier Bernardi – p.14/23
Extended bijection Theorem: Labelled trees are in bijection with rooted quadrangulations with a marked vertex, such that . . . 2 1 0 1 3 1 2 4 2 2 3 3 November 2008 ▽ Olivier Bernardi – p.15/23
Extended bijection Theorem: Labelled trees are in bijection with rooted quadrangulations with a marked vertex, such that . . . 2 2 0 1 1 2 1 4 2 2 3 3 November 2008 ▽ Olivier Bernardi – p.15/23
Extended bijection Theorem: Labelled trees are in bijection with rooted quadrangulations with a marked vertex, such that . . . 2 2 0 1 1 2 1 4 2 2 3 3 November 2008 ▽ Olivier Bernardi – p.15/23
Extended bijection Labelled trees are in bijection with rooted quadrangulations with a marked vertex, such that the root-edge is oriented away from the root-vertex. November 2008 ▽ Olivier Bernardi – p.15/23
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