La fonction ` a deux points et ` a trois points des - PowerPoint PPT Presentation

La fonction ` a deux points et ` a trois points des quadrangulations et cartes ´ Eric Fusy (CNRS/LIX) Travaux avec J´ er´ emie Bouttier et Emmanuel Guitter S´ eminaire Calin, LIPN, Mai 2014

Maps Def. Planar map = connected graph embedded on the sphere = Easier to draw in the plane (by choosing a face to be the outer face) ⇒

Maps as random discrete surfaces Natural questions: • Typical distance between (random) vertices in random maps the order of magnitude is n 1 / 4 ( � = n 1 / 2 in random trees) - [Chassaing-Schaeffer’04] probabilistic random { quadrang. - [Bouttier Di Francesco Guitter’03] exact GF expressions • How does a random map (rescaled by n 1 / 4 ) “look like” ? convergence to the “Brownian map” [Le Gall’13, Miermont’13]

Counting (rooted) maps with a marked corner • Very simple counting formulas ([Tutte’60s]), for instance Let q n = # { rooted quadrangulations with n faces } m n = # { rooted maps with n edges } (2 n )! 2 n +2 3 n Then m n = q n = n !( n +1)!

Counting (rooted) maps with a marked corner • Very simple counting formulas ([Tutte’60s]), for instance Let q n = # { rooted quadrangulations with n faces } m n = # { rooted maps with n edges } (2 n )! 2 n +2 3 n Then m n = q n = n !( n +1)! • Proof of m n = q n by easy local bijection: ⇒ ⇒

Counting (rooted) maps with a marked corner • Very simple counting formulas ([Tutte’60s]), for instance Let q n = # { rooted quadrangulations with n faces } m n = # { rooted maps with n edges } (2 n )! 2 n +2 3 n Then m n = q n = n !( n +1)! • Proof of m n = q n by easy local bijection: ⇒ ⇒ But this bijection does not preserve distance-parameters (only bounds)

The k -point function • Let M = ∪ n M [ n ] be a family of maps (quadrangulations, general, ...) where n is a size-parameter ( # faces for quad., # edges for gen. maps) • Let M ( k ) = family of maps from M with k marked vertices v 1 , . . . , v k

The k -point function • Let M = ∪ n M [ n ] be a family of maps (quadrangulations, general, ...) where n is a size-parameter ( # faces for quad., # edges for gen. maps) • Let M ( k ) = family of maps from M with k marked vertices v 1 , . . . , v k Refinement by distances : � k � For D = ( d i,j ) 1 ≤ i<j ≤ k any -tuple of positive integers 2 D := subfamily of M ( k ) where dist( v i , v j ) = d ij for 1 ≤ i < j ≤ k let M ( k ) The counting series G D ≡ G D ( g ) of M ( k ) D with respect to the size is called the k -point function of M v 1 v 1 k = 3 k = 2 d 12 = 2 d 12 = 3 v 2 d 13 = 2 v 2 v 3 d 23 = 3 quadrangulation general map

Exact expressions for the k -point function • For the two-point functions: - quadrangulations [Bouttier Di Francesco Guitter’03] - maps with prescribed (bounded) face-degrees [Bouttier Guitter’08] - general maps [Ambjørn Budd’13] - general hypermaps, general constellations [Bouttier F Guitter’13] • For the three-point functions - quadrangulations [Bouttier Guitter’08] [F Guitter’14] - general maps & bipartite maps

Exact expressions for the k -point function Outline of the talk • For the two-point functions: - quadrangulations [Bouttier Di Francesco Guitter’03] 1 uses Schaeffer’s bijection - maps with prescribed (bounded) face-degrees [Bouttier Guitter’08] based on clever observation - general maps [Ambjørn Budd’13] 3 on Miermont’s bijection - general hypermaps, general constellations [Bouttier F Guitter’13] • For the three-point functions - quadrangulations 2 [Bouttier Guitter’08] uses Miermont’s bijection [F Guitter’14] - general maps & bipartite maps 4 uses AB bijection uses AB bijection uses AB bijection

Computing the two-point function of quadrangulations using the Schaeffer bijection

Well-labelled trees Well-labelled tree = plane tree where - each vertex v has a label ℓ ( v ) ∈ Z - each edge e = { u, v } satisfies | ℓ ( u ) − ℓ ( v ) | ≤ 1 1 0 0 1 1 2 2 - 1 0

Pointed quadrangulations, geodesic labelling Pointed quadrangulation = quadrangulation with a marked vertex v 0 Geodesic labelling with respect to v 0 : ℓ ( v ) = dist( v 0 , v ) Rk: two types of faces 1 2 2 i + 2 i + 1 3 2 1 i + 1 i + 1 i i 2 i i + 1 1 stretched confluent 0 2 v 0 1

The Schaeffer bijection [Schaeffer’99] , also [Cori-Vauquelin’81] Pointed quadrangulation ⇒ well-labelled tree with min-label=1 n edges n faces 1 1 1 2 2 2 2 2 2 3 3 3 2 2 1 2 1 1 2 2 2 1 1 1 0 2 0 2 2 1 1 1 Local rule in each face: 2 i + 1 i + 1 i + 1 i + i i i 1 i +

The 2-point function of quadrangulations (1) Denote by G d ≡ G d ( g ) the two-point function of quadrangulations bijection ⇒ G d ( g ) = GF of well-labelled trees with min-label=1 and with a marked vertex of label d 1 1 2 2 2 2 3 3 d = 3 2 2 1 1 2 2 1 1 0 2 2 1 1 Rk: G d = F d − F d − 1 = ∆ d F d where F d ≡ F d ( g ) = GF of well-labelled trees with positive labels and with a marked vertex of label d

The 2-point function of quadrangulations (2) 1 ⇒ F i = log 1 i - 1 − g ( R i − 1 + R i + R i +1 ) i 1 i - i GF rooted well-labelled i with R i = 1 i - 1 i + trees with positive labels > 0 > 0 and label i at the root

The 2-point function of quadrangulations (2) 1 ⇒ F i = log 1 i - 1 − g ( R i − 1 + R i + R i +1 ) i 1 i - i GF rooted well-labelled i with R i = 1 i - 1 i + trees with positive labels > 0 > 0 and label i at the root 1 R d Equ. for R i : R i = 1 − g ( R i − 1 + R i + R i +1 ) (so F i = log( R i ) , G d = log( R d − 1 ))

The 2-point function of quadrangulations (2) 1 ⇒ F i = log 1 i - 1 − g ( R i − 1 + R i + R i +1 ) i 1 i - i GF rooted well-labelled i with R i = 1 i - 1 i + trees with positive labels > 0 > 0 and label i at the root 1 R d Equ. for R i : R i = 1 − g ( R i − 1 + R i + R i +1 ) (so F i = log( R i ) , G d = log( R d − 1 )) • Exact expression for R i [BDG’03] [ i ] x [ i + 3] x with the notation [ i ] x = 1 − x i R i = R 1 − x [ i + 1] x [ i + 2] x R = 1 + 3 gR 2 { with R ≡ R ( g ) and x ≡ x ( g ) given by x = gR 2 (1 + x + x 2 ) S 1 / 2 √ with S = √ 1 − 12 g √ 1 − (1+6 g ) S − S − 24 g +1 R ( g ) = 1 − S 6 x ( g ) = 2 − 1+ S +6 g 6 g

The 2-point function of quadrangulations (2) 1 ⇒ F i = log 1 i - 1 − g ( R i − 1 + R i + R i +1 ) i 1 i - i GF rooted well-labelled i with R i = 1 i - 1 i + trees with positive labels > 0 > 0 and label i at the root 1 R d Equ. for R i : R i = 1 − g ( R i − 1 + R i + R i +1 ) (so F i = log( R i ) , G d = log( R d − 1 )) • Exact expression for R i [BDG’03] [ i ] x [ i + 3] x with the notation [ i ] x = 1 − x i R i = R 1 − x [ i + 1] x [ i + 2] x R = 1 + 3 gR 2 { with R ≡ R ( g ) and x ≡ x ( g ) given by x = gR 2 (1 + x + x 2 ) S 1 / 2 √ with S = √ 1 − 12 g √ 1 − (1+6 g ) S − S − 24 g +1 R ( g ) = 1 − S 6 x ( g ) = 2 − 1+ S +6 g 6 g � [ d ] 2 � x [ d +3] x Final 2-point function expression: G d = log [ d − 1] x [ d +2] 2 x

Asymptotic considerations • Two-point function of (plane) trees: G d ( g ) = ( gR 2 ) d with R = 1 + gR 2 = 1 −√ 1 − 4 g d = 5 2 g G d is the d th power of a series having a square-root singularity ⇒ d/n 1 / 2 converges in law (Rayleigh law, density α exp( − α 2 ) )

Asymptotic considerations • Two-point function of (plane) trees: G d ( g ) = ( gR 2 ) d with R = 1 + gR 2 = 1 −√ 1 − 4 g d = 5 2 g G d is the d th power of a series having a square-root singularity ⇒ d/n 1 / 2 converges in law (Rayleigh law, density α exp( − α 2 ) ) • Two-point function of quadrangulations: G d ( g ) ∼ d →∞ a 1 x d + a 2 x 2 d + · · · where x = x ( g ) has a quartic singularity ⇒ d/n 1 / 4 converges to an explicit law [BDG’03]

Asymptotic considerations • Two-point function of (plane) trees: G d ( g ) = ( gR 2 ) d with R = 1 + gR 2 = 1 −√ 1 − 4 g d = 5 2 g G d is the d th power of a series having a square-root singularity ⇒ d/n 1 / 2 converges in law (Rayleigh law, density α exp( − α 2 ) ) • Two-point function of quadrangulations: G d ( g ) ∼ d →∞ a 1 x d + a 2 x 2 d + · · · where x = x ( g ) has a quartic singularity ⇒ d/n 1 / 4 converges to an explicit law [BDG’03] Convergence in the two cases “follows” from (proof by Hankel contour) [Banderier, Flajolet, Louchard, Schaeffer’03] : for 0 < s < 1 , � ∞ 1 [ g n ] x αn s ∼ x ( g ) ∼ 1 − (1 − g ) s ⇒ e − t Im(exp( − αt s e iπs ))d t 2 πn g → 1 0

Computing the two-point and three-point function of quadrangulations using Miermont’s bijection

Well-labelled maps Well-labelled map = map where - each vertex v has a label ℓ ( v ) ∈ Z - each edge e = { u, v } satisfies | ℓ ( u ) − ℓ ( v ) | ≤ 1 0 2 1 1 a well-labelled map M with 3 faces 0 0 0 0 Rk: Well-labelled tree = well-labelled map with one face

Very-well-labelled quadrangulations Very-well-labelled quadrangulation = quadrangulation where - each vertex v has a label ℓ ( v ) ∈ Z - each edge e = { u, v } satisfies | ℓ ( u ) − ℓ ( v ) | = 1 Rk: two types of faces 0 2 i + 2 i + 1 1 1 0 0 0 i + 1 i + 1 i i 0 - 1 - 1 i i + 1 stretched confluent 0 a very-well-labelled quadrangulation Q with 3 local min Def: local min= vertex with all neighbours of larger label Rk: Geodesic labelling ⇔ there is just one local min, of label 0