Bijections for tree-decorated map and applications to random maps. Luis Fredes (Work in progress with Avelio Sepúlveda (Univ. Lyon 1)) ALEA 2019 Luis Fredes (Université de Bordeaux) Tree-decorated maps 1 / 24
MAPS Luis Fredes (Université de Bordeaux) Tree-decorated maps 2 / 24
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 / 24
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 / 24
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 / 24
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] and Bijective [Cori-Vauquelin-Schaeffer ’98]. Luis Fredes (Université de Bordeaux) Tree-decorated maps 6 / 24
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 / 24
Quadrangulations with a simple boundary The set of quadrangulations with f internal faces and a simple boundary of size 2 p (root-face of degre 2 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 / 24
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 / 24
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 / 24
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 / 24
← → Luis Fredes (Université de Bordeaux) Tree-decorated maps 12 / 24
What do we obtain when the boundary is not simple? Luis Fredes (Université de Bordeaux) Tree-decorated maps 13 / 24
What do we obtain when the boundary is not simple? We introduce BUBBLE-MAPS! Luis Fredes (Université de Bordeaux) Tree-decorated maps 13 / 24
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 13 / 24
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 13 / 24
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 14 / 24
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 15 / 24
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 15 / 24
Re-rooting Q T Q M Q T,M Luis Fredes (Université de Bordeaux) Tree-decorated maps 16 / 24
Re-rooting Q T Q M Q T,M Luis Fredes (Université de Bordeaux) Tree-decorated maps 16 / 24
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 16 / 24
Counting results In the case of spanning tree decorated quadrangulations rooted in the tree we obtain � 3 f � 2 C 2 , f = ( f + 1 )( f + 2 ) f , f , f Luis Fredes (Université de Bordeaux) Tree-decorated maps 17 / 24
Counting results In the case of spanning tree decorated quadrangulations rooted in the tree we obtain � 3 f � 2 C 2 , f = ( f + 1 )( f + 2 ) f , f , f and remember that the Catalan numbers are � 2 f � 1 C 1 , f = ( f + 1 ) f Luis Fredes (Université de Bordeaux) Tree-decorated maps 17 / 24
Counting results In the case of spanning tree decorated quadrangulations rooted in the tree we obtain � 3 f � 2 C 2 , f = ( f + 1 )( f + 2 ) f , f , f and remember that the Catalan numbers are � 2 f � 1 C 1 , f = ( f + 1 ) f A possible generalization of Catalan numbers: � m � � ( m + 1 ) n � − 1 � ( m + 1 ) n � � m + n � 1 � C m , n = m ! = ( n + i ) n , n , . . . , n n n , n , . . . , n i = 1 � �� � � �� � m + 1 times m + 1 times Luis Fredes (Université de Bordeaux) Tree-decorated maps 17 / 24
CONVERGENCE RESULTS Luis Fredes (Université de Bordeaux) Tree-decorated maps 18 / 24
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 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 Meaning that a sequence of maps ( m i ) i ∈ N converges if for all r ∈ N , B r ( m i ) is constant from certain point on. Luis Fredes (Université de Bordeaux) Tree-decorated maps 19 / 24
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 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 Meaning that a sequence of maps ( m i ) i ∈ N converges if for all r ∈ N , B r ( m i ) is constant from certain point on. Proposition The space ( M , d loc ) is Polish (metric, separable and complete). Luis Fredes (Université de Bordeaux) Tree-decorated maps 19 / 24
EX 1: Uniform trees t a = Unif. tree with a edges. Theorem (Kesten ’86) ( d ) − − − − − → t a t ∞ local Luis Fredes (Université de Bordeaux) Tree-decorated maps 20 / 24
EX 1: Uniform trees t a = Unif. tree with a edges. Theorem (Kesten ’86) ( d ) − − − − − → t a t ∞ local Properties t ∞ is an infinite tree. It has one infinite branch (the spine) which divides the tree in independent critical geometric Galton-Watson trees. t ∞ construction. Luis Fredes (Université de Bordeaux) Tree-decorated maps 20 / 24
EX 2: Uniform quadrangulation with a boundary q f , p = Unif. quadrangulations with a boundary of size 2 p and f faces. Theorem (Curien & Miermont ’12) ( d ) ( d ) − − − − − − − → − local ( p →∞ ) UIHPQ − − − − − − → q f , p local ( f →∞ ) q ∞ , p Luis Fredes (Université de Bordeaux) Tree-decorated maps 21 / 24
EX 2: Uniform quadrangulation with a boundary q f , p = Unif. quadrangulations with a boundary of size 2 p and f faces. Theorem (Curien & Miermont ’12) ( d ) ( d ) − − − − − − − → − local ( p →∞ ) UIHPQ − − − − − − → q f , p local ( f →∞ ) q ∞ , p Properties (Curien & Miermont ’12) q p ∞ = Uniform Infinite Planar Quadrangulation with perimeter 2 p . They also obtain the convergences for the simple boundary case. UIHPQ (sketch by N. Curien & A. Caraceni). Luis Fredes (Université de Bordeaux) Tree-decorated maps 21 / 24
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