Scaling limits of random planar maps with a prescribed degree - PowerPoint PPT Presentation

Scaling limits of random planar maps with a prescribed degree sequence Cyril Marzouk CNRS & Université Paris Diderot ERC CombiTop ( � = Guillaume Chapuy) Journées ALÉA 2019, CIRM

Planar maps A (planar) map M is a finite connected (multi-)graph embedded in the 2 D -sphere viewed up to continuous deformations. We assume it to be rooted and bipartite.

Random planar maps A map is also a gluing of polygons:

Random planar maps A map is also a gluing of polygons: For every n , take ( d n ( k )) k � 1 ∈ Z N + such that � k � 1 d n ( k ) = n , then put M d n = { maps with d n ( k ) faces with degree 2 k for all k � 1 } .

Random planar maps A map is also a gluing of polygons: For every n , take ( d n ( k )) k � 1 ∈ Z N + such that � k � 1 d n ( k ) = n , then put M d n = { maps with d n ( k ) faces with degree 2 k for all k � 1 } . Example: Fix some p � 2 and take d n ( k ) = n if p = k and d n ( k ) = 0 otherwise, then M d n is the set of 2 p -angulations with n faces.

Topology Given a map M , we shall consider the (compact) metric measured space M = � Vertices ( M ) , d graph , p unif � , and we look for a limit in law in the sense of Gromov–Hausdorff–Prokhorov when d graph is multiplied by r n → 0 as the number of faces n → ∞ .

A very brief history Fix any p � 2 and sample M n , p a random 2 p -angulation with n faces.

A very brief history Fix any p � 2 and sample M n , p a random 2 p -angulation with n faces. ◮ Le Gall ’07: the sequence ( n − 1 / 4 M n , p ) n admits subsequential limits.

A very brief history Fix any p � 2 and sample M n , p a random 2 p -angulation with n faces. ◮ Le Gall ’07: the sequence ( n − 1 / 4 M n , p ) n admits subsequential limits. ◮ Le Gall ’13 and Miermont ’13 for p = 2 : ( d ) � p ( p − 1 ) n � − 1 / 4 M n , p − − − − → M n →∞ � where M = ( M , D , p ) is ( 2 / 3 times) the Brownian map ,

A very brief history Fix any p � 2 and sample M n , p a random 2 p -angulation with n faces. ◮ Le Gall ’07: the sequence ( n − 1 / 4 M n , p ) n admits subsequential limits. ◮ Le Gall ’13 and Miermont ’13 for p = 2 : ( d ) � p ( p − 1 ) n � − 1 / 4 M n , p − − − − → M n →∞ � where M = ( M , D , p ) is ( 2 / 3 times) the Brownian map , it has ◮ the topology of the sphere (Le Gall & Paulin ’08, Miermont ’08), ◮ Hausdorff dimension 4 (Le Gall ’07).

Back to our model Recall: d n ( k ) � 0 with � k � 1 d n ( k ) = n , then M d n uniform random map with d n ( k ) faces with degree 2 k for all k � 1 . Put � σ 2 k ( k − 1 ) d n ( k ) , n = k � 1 sort of global half-degree variance.

Back to our model Recall: d n ( k ) � 0 with � k � 1 d n ( k ) = n , then M d n uniform random map with d n ( k ) faces with degree 2 k for all k � 1 . Put � σ 2 k ( k − 1 ) d n ( k ) , n = k � 1 sort of global half-degree variance. T heorem . The sequence ( σ − 1 / 2 M d n ) n always admits subsequential limits. n

Back to our model Recall: d n ( k ) � 0 with � k � 1 d n ( k ) = n , then M d n uniform random map with d n ( k ) faces with degree 2 k for all k � 1 . Put � σ 2 k ( k − 1 ) d n ( k ) , n = k � 1 sort of global half-degree variance. T heorem . The sequence ( σ − 1 / 2 M d n ) n always admits subsequential limits. n ( d ) Theorem . Moreover σ − 1 / 2 M d n − − − − → M if and only if n n →∞ n →∞ σ − 1 lim n max { k � 1 : d n ( k ) � 0 } = 0 .

Back to our model Recall: d n ( k ) � 0 with � k � 1 d n ( k ) = n , then M d n uniform random map with d n ( k ) faces with degree 2 k for all k � 1 . Put � σ 2 k ( k − 1 ) d n ( k ) , n = k � 1 sort of global half-degree variance. T heorem . The sequence ( σ − 1 / 2 M d n ) n always admits subsequential limits. n ( d ) Theorem . Moreover σ − 1 / 2 − − − − → M d n M if and only if n n →∞ n →∞ σ − 1 lim n max { k � 1 : d n ( k ) � 0 } = 0 . Corollary. Fix any p � 2 and sample M n , p a random 2 p -angulation with n faces, then ( d ) � p ( p − 1 ) n � − 1 / 4 M n , p − − − − → M . n →∞

Back to our model Recall: d n ( k ) � 0 with � k � 1 d n ( k ) = n , then M d n uniform random map with d n ( k ) faces with degree 2 k for all k � 1 . Put � σ 2 k ( k − 1 ) d n ( k ) , n = k � 1 sort of global half-degree variance. T heorem . The sequence ( σ − 1 / 2 M d n ) n always admits subsequential limits. n ( d ) Theorem . Moreover σ − 1 / 2 M d n − − − − → M if and only if n n →∞ n →∞ σ − 1 lim n max { k � 1 : d n ( k ) � 0 } = 0 . Corollary. Fix any ( p n ) n ∈ { 2 , 3 , . . . } N and sample M n , p n a random 2 p n -angulation with n faces, then ( d ) � p n ( p n − 1 ) n � − 1 / 4 M n , p n − − − − → M . n →∞

Further results ◮ Previous case assuming σ 2 n ∼ σ 2 n plus annoying assumptions; � ’18. ◮ If one but only one face with degree ϱ n ∼ ϱσ n , then Brownian disk with perimeter ϱ of Betinelli & Miermont ’17 instead. ◮ If one face with degree ϱ n ≫ σ n , then scaling ϱ 1 / 2 and Aldous’ Brownian n CRT instead. ◮ Application to size-conditioned critical α -stable Boltzmann maps ( P ( deg � 2 k ) ≈ k − α ), with σ 1 / 2 of order n 1 /( 2 α ) : n ◮ tightness when α ∈ ( 1 , 2 ) (already Le Gall & Miermont ’11 and � ’18 bis), ◮ convergence to the Brownian map when α = 2 (already � ’18 bis), ◮ convergence to the Brownian CRT when α = 1 (new!).

A key tool: bijection with trees Combine the bijections due to Boutier, Di Francesco & Guiter ’04 and to Janson & Stefánsson ’15: − 1 − 2 − 1 0 0 − 2 0 − 1 0 1 − 1 0 − 1 − 2 1 0 0 The tree is chosen uniformly at random amongst those with d n ( k ) vertices with arity k ; given the tree, labels obey the local rule: � − 1 � − 1 � − 1 = 0 � − 1

And trees are coded by paths 4 − 1 − 2 − 1 0 3 2 − 2 − 1 0 0 1 0 4 12 20 24 32 8 16 28 0 1 − 1 0 1 − 1 − 2 1 0 0 − 1 0 − 2

Continuum object: the Brownian tree with Brownian labels

Continuum object: the Brownian map

Convergence of processes ◮ Tightness of maps follows from tightness of the label process (Le Gall ’07).

Convergence of processes ◮ Tightness of maps follows from tightness of the label process (Le Gall ’07). ◮ So does the convergence to a Brownian limit (Le Gall ’13, Betinelli & Miermont ’15).

Convergence of processes ◮ Tightness of maps follows from tightness of the label process (Le Gall ’07). ◮ So does the convergence to a Brownian limit (Le Gall ’13, Betinelli & Miermont ’15). ◮ Tightness of this label process relies only on the Łukasiewicz path of the tree which is a very simple process.

Convergence of processes ◮ Tightness of maps follows from tightness of the label process (Le Gall ’07). ◮ So does the convergence to a Brownian limit (Le Gall ’13, Betinelli & Miermont ’15). ◮ Tightness of this label process relies only on the Łukasiewicz path of the tree which is a very simple process. ◮ Convergence of finite dimensional marginals of this label process needs that of the height process.

Convergence of processes ◮ Tightness of maps follows from tightness of the label process (Le Gall ’07). ◮ So does the convergence to a Brownian limit (Le Gall ’13, Betinelli & Miermont ’15). ◮ Tightness of this label process relies only on the Łukasiewicz path of the tree which is a very simple process. ◮ Convergence of finite dimensional marginals of this label process needs that of the height process. ◮ Important remark: tightness of the height process is not always true! (take size-conditioned sub-critical BGW trees, Kortchemski ’15).

Convergence of processes ◮ Tightness of maps follows from tightness of the label process (Le Gall ’07). ◮ So does the convergence to a Brownian limit (Le Gall ’13, Betinelli & Miermont ’15). ◮ Tightness of this label process relies only on the Łukasiewicz path of the tree which is a very simple process. ◮ Convergence of finite dimensional marginals of this label process needs that of the height process. ◮ Important remark: tightness of the height process is not always true! (take size-conditioned sub-critical BGW trees, Kortchemski ’15). ◮ Large degree regime, ‘Inhomogenous Continuum Random Maps’?