Bijections for tree-decorated map and applications to random maps. Luis Fredes (Work in progress with Avelio Sepúlveda (Univ. Lyon 1)) LIX 2019 Luis Fredes (Université de Bordeaux) Tree-decorated maps 1 / 30
MAPS Luis Fredes (Université de Bordeaux) Tree-decorated maps 2 / 30
Map A planar map is a proper embedding of Same graph, different embeddings on the sphere a finite connected planar graph in the (sketch by N. Curien) sphere, considered up to direct homeomorphisms of the sphere. Maps seen as different objects (sketch by N. Curien) Luis Fredes (Université de Bordeaux) Tree-decorated maps 3 / 30
Map The faces are the connected components of the complement of the edges. It has a distinguished half-edge: the root edge . The face that is at the left of the root-edge will be called the root-face . Luis Fredes (Université de Bordeaux) Tree-decorated maps 4 / 30
Planar trees A planar tree is a map with one face. The set of trees with a edges. � 2 a � 1 C a = a + 1 a Luis Fredes (Université de Bordeaux) Tree-decorated maps 5 / 30
Quadrangulations The degree of a face is the number of edges adjacent to it. A quadrangulation is a map whose faces have degree 4. Let Q f be the set of all quadrangulations with f faces, then � 2 f � 2 1 |Q f | = 3 f . f + 2 f + 1 f � �� � C f Analytic [Tutte ’60]. This number also counts general maps with a = f edges! Bijective [Tutte ’60, Cori-Vauquelin-Schaeffer ’98]. Luis Fredes (Université de Bordeaux) Tree-decorated maps 6 / 30
Quadrangulations with a boundary A quadrangulation with a boundary is a map where the root-face plays a special role: it has arbitrary degree . The set of quadrangulations with f internal faces and a boundary of size 2 p has cardinality 3 f p � 2 f + p − 1 �� 2 p � . ( f + p + 1 )( f + p ) f p Analytic by [Bender & Canfield ’94; Bouttier & Guitter ’09] and bijective by [ Schaeffer ’97 ; Bettinelli ’15] Luis Fredes (Université de Bordeaux) Tree-decorated maps 7 / 30
Quadrangulations with a simple boundary The set of quadrangulations with f internal faces and a simple boundary of size p (root-face of degre p ) has cardinality � 2 f + p − 1 �� 3 p � 3 f − p 2 p . ( f + 2 p )( f + 2 p − 1 ) f − p + 1 p Analytic [Bouttier & Guitter ’09] Luis Fredes (Université de Bordeaux) Tree-decorated maps 8 / 30
Spanning tree-decorated maps A spanning tree-decorated map ( ST map ) is a pair ( m , t ) where m is a map and t ⊂ M m is a spanning tree of m . The family of ST maps with a edges is counted by C a C a + 1 Analytic by [Mullin ’67] and bijective by [Walsh and Lehman ’72; Cori, Dulucq & Viennot ’86; Bernardi ’06] Luis Fredes (Université de Bordeaux) Tree-decorated maps 9 / 30
Spanning tree-decorated maps A ( f , a ) tree-decorated map is a pair ( m , t ) where m is a map with f faces, and t is a tree with a edges, so that t ⊂ M m containing the root-edge . Luis Fredes (Université de Bordeaux) Tree-decorated maps 10 / 30
Bijection Proposition (F. & Sepúlveda ’19) The set of ( f , a ) tree-decorated maps is in bijection with (the set of maps with a simple boundary of size 2 a and f interior faces) × (the set of trees with a edges). Luis Fredes (Université de Bordeaux) Tree-decorated maps 11 / 30
What do we obtain when the boundary is not simple? Luis Fredes (Université de Bordeaux) Tree-decorated maps 12 / 30
What do we obtain when the boundary is not simple? We introduce BUBBLE-MAPS! Luis Fredes (Université de Bordeaux) Tree-decorated maps 12 / 30
What do we obtain when the boundary is not simple? We introduce BUBBLE-MAPS! f 1 f 3 f 2 + Luis Fredes (Université de Bordeaux) Tree-decorated maps 12 / 30
What do we obtain when the boundary is not simple? We introduce BUBBLE-MAPS! f 1 f 3 f 2 + ↓ f 3 f 2 f 1 Luis Fredes (Université de Bordeaux) Tree-decorated maps 12 / 30
Some remarks and extensions From the map with a boundary the bijection preserves: Internal faces. 1 Internal vertices. 2 Internal edges. 3 It also preserves attributes on them. It works with some subfamilies of trees: Binary tree- decorated Maps. 1 SAW decorated maps (Already done by Curien & Caraceni). 2 Luis Fredes (Université de Bordeaux) Tree-decorated maps 13 / 30
Counting results Corollary (F. & Sepúlveda ’19) The number of ( f , a ) tree-decorated quadrangulations is � 3 a � ( 2 f + a − 1 )! 2 a 3 f − a ( f + 2 a )!( f − a + 1 )! a + 1 a , a , a Luis Fredes (Université de Bordeaux) Tree-decorated maps 14 / 30
Counting results Corollary (F. & Sepúlveda ’19) The number of ( f , a ) tree-decorated quadrangulations is � 3 a � ( 2 f + a − 1 )! 2 a 3 f − a ( f + 2 a )!( f − a + 1 )! a + 1 a , a , a We also count ( f , a ) tree-decorated triangulations. Maps (triangulations and quadrangulations) with a simple boundary decorated in a subtree. Forest-decorated maps. "Tree-decorated general maps". Luis Fredes (Université de Bordeaux) Tree-decorated maps 14 / 30
Re-rooting Q T Q M Q T,M Luis Fredes (Université de Bordeaux) Tree-decorated maps 15 / 30
Re-rooting Q T Q M Q T,M Luis Fredes (Université de Bordeaux) Tree-decorated maps 15 / 30
Re-rooting Q T Q M Q T,M |Q T | × 2 | E | = |Q M | × 2 | T | Luis Fredes (Université de Bordeaux) Tree-decorated maps 15 / 30
CONVERGENCE RESULTS Luis Fredes (Université de Bordeaux) Tree-decorated maps 16 / 30
Local Limits (Benjamini-Schramm Topology ’01) For a map m and r ∈ N , let B r ( m ) denote the ball of radius r from the root-vertex. Consider M a family of finite maps. The local topology on M is the metric space ( M , d loc ) , where d loc ( m 1 , m 2 ) = ( 1 + sup { r ≥ 0 : B r ( m 1 ) = B r ( m 2 ) } ) − 1 k = 0 = d loc ( m , m ′ ) = 2 − 1 m ′ m Luis Fredes (Université de Bordeaux) Tree-decorated maps 17 / 30
Local Limits (Benjamini-Schramm Topology ’01) For a map m and r ∈ N , let B r ( m ) denote the ball of radius r from the root-vertex. Consider M a family of finite maps. The local topology on M is the metric space ( M , d loc ) , where d loc ( m 1 , m 2 ) = ( 1 + sup { r ≥ 0 : B r ( m 1 ) = B r ( m 2 ) } ) − 1 k = 0 = d loc ( m , m ′ ) = 2 − 1 m ′ m Proposition The space ( M , d loc ) is Polish (metric, separable and complete). Luis Fredes (Université de Bordeaux) Tree-decorated maps 17 / 30
Gromov-Hausdorff topology ( E, d E ) Let ( E , d E ) be a metric space and A , B ⊂ E . The Hausdorff distance is � � d H ( A , B ) = inf ε > 0 : A ⊂ B ε , B ⊂ A ε B A Luis Fredes (Université de Bordeaux) Tree-decorated maps 18 / 30
Gromov-Hausdorff topology ( E, d E ) B ε Let ( E , d E ) be a metric space and A , B ⊂ E . The Hausdorff distance is � � d H ( A , B ) = inf ε > 0 : A ⊂ B ε , B ⊂ A ε B A Luis Fredes (Université de Bordeaux) Tree-decorated maps 18 / 30
Gromov-Hausdorff topology Let ( E , d E ) be a metric space and A , B ⊂ E . The Hausdorff distance is � � d H ( A , B ) = inf ε > 0 : A ⊂ B ε , B ⊂ A ε Consider the set S of compact metric spaces up to isometry classes. The Gromov-Hausdorff distance between two metric spaces ( X , d ) and ( X ′ , d ′ ) is defined as d GH (( X , d ) , ( X ′ , d ′ )) = inf d H ( φ ( X ) , φ ′ ( X ′ )) where the infimum is taken over all metric spaces ( E , d E ) and all isometric embeddings φ, φ ′ from X , X ′ respectively into E . Luis Fredes (Université de Bordeaux) Tree-decorated maps 18 / 30
Gromov-Hausdorff topology ( E, d E ) φ ′ φ ( X, d ) ( X ′ , d ′ ) φ ′ ( X ′ ) φ ( X ) Consider the set S of compact metric spaces up to isometry classes. The Gromov-Hausdorff distance between two metric spaces ( X , d ) and ( X ′ , d ′ ) is defined as d GH (( X , d ) , ( X ′ , d ′ )) = inf d H ( φ ( X ) , φ ′ ( X ′ )) where the infimum is taken over all metric spaces ( E , d E ) and all isometric embeddings φ, φ ′ from X , X ′ respectively into E . Luis Fredes (Université de Bordeaux) Tree-decorated maps 18 / 30
Gromov-Hausdorff topology ( E, d E ) ¯ φ ′ ¯ φ ( X, d ) ( X ′ , d ′ ) ¯ φ ( X ) ¯ φ ′ ( X ′ ) Consider the set S of compact metric spaces up to isometry classes. The Gromov-Hausdorff distance between two metric spaces ( X , d ) and ( X ′ , d ′ ) is defined as d GH (( X , d ) , ( X ′ , d ′ )) = inf d H ( φ ( X ) , φ ′ ( X ′ )) where the infimum is taken over all metric spaces ( E , d E ) and all isometric embeddings φ, φ ′ from X , X ′ respectively into E . Luis Fredes (Université de Bordeaux) Tree-decorated maps 18 / 30
Gromov-Hausdorff topology ( E, d E ) ¯ φ ′ ¯ φ ( X, d ) ( X ′ , d ′ ) ¯ φ ( X ) ¯ φ ′ ( X ′ ) Consider the set S of compact metric spaces up to isometry classes. The Gromov-Hausdorff distance between two metric spaces ( X , d ) and ( X ′ , d ′ ) is defined as d GH (( X , d ) , ( X ′ , d ′ )) = inf d H ( φ ( X ) , φ ′ ( X ′ )) where the infimum is taken over all metric spaces ( E , d E ) and all isometric embeddings φ, φ ′ from X , X ′ respectively into E . Proposition The function d GH induces a metric on S . The space ( S , d GH ) is separable and complete. Luis Fredes (Université de Bordeaux) Tree-decorated maps 18 / 30
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