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 2D-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 (dn(k))k1 ∈ ZN

+ such that k1 dn(k) = n, then put

Mdn = {maps with dn(k) faces with degree 2k for all k 1}.

Random planar maps

A map is also a gluing of polygons: For every n, take (dn(k))k1 ∈ ZN

+ such that k1 dn(k) = n, then put

Mdn = {maps with dn(k) faces with degree 2k for all k 1}. Example: Fix some p 2 and take dn(k) = n if p = k and dn(k) = 0

• therwise, then Mdn is the set of 2p-angulations with n faces.

Topology

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

A very brief history

Fix any p 2 and sample Mn,p a random 2p-angulation with n faces.

A very brief history

Fix any p 2 and sample Mn,p a random 2p-angulation with n faces.

◮ Le Gall ’07: the sequence (n−1/4Mn,p)n admits subsequential limits.

A very brief history

Fix any p 2 and sample Mn,p a random 2p-angulation with n faces.

◮ Le Gall ’07: the sequence (n−1/4Mn,p)n admits subsequential limits. ◮ Le Gall ’13 and Miermont ’13 for p = 2:

p(p − 1)n−1/4Mn,p

(d)

− − − − →

n→∞

M where M = (M, D,p) is (

• 2/3 times) the Brownian map,

A very brief history

Fix any p 2 and sample Mn,p a random 2p-angulation with n faces.

◮ Le Gall ’07: the sequence (n−1/4Mn,p)n admits subsequential limits. ◮ Le Gall ’13 and Miermont ’13 for p = 2:

p(p − 1)n−1/4Mn,p

(d)

− − − − →

n→∞

M 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: dn(k) 0 with

k1 dn(k) = n, then Mdn uniform random map with

dn(k) faces with degree 2k for all k 1. Put σ 2

n =

• k1

k(k − 1)dn(k), sort of global half-degree variance.

Back to our model

Recall: dn(k) 0 with

k1 dn(k) = n, then Mdn uniform random map with

dn(k) faces with degree 2k for all k 1. Put σ 2

n =

• k1

k(k − 1)dn(k), sort of global half-degree variance.

• Theorem. The sequence (σ −1/2

n

Mdn)n always admits subsequential limits.

Back to our model

Recall: dn(k) 0 with

k1 dn(k) = n, then Mdn uniform random map with

dn(k) faces with degree 2k for all k 1. Put σ 2

n =

• k1

k(k − 1)dn(k), sort of global half-degree variance.

• Theorem. The sequence (σ −1/2

n

Mdn)n always admits subsequential limits.

• Theorem. Moreover σ −1/2

n

Mdn

(d)

− − − − →

n→∞

M if and only if lim

n→∞σ −1 n max{k 1 : dn(k) 0} = 0.

Back to our model

Recall: dn(k) 0 with

k1 dn(k) = n, then Mdn uniform random map with

dn(k) faces with degree 2k for all k 1. Put σ 2

n =

• k1

k(k − 1)dn(k), sort of global half-degree variance.

• Theorem. The sequence (σ −1/2

n

Mdn)n always admits subsequential limits.

• Theorem. Moreover σ −1/2

n

Mdn

(d)

− − − − →

n→∞

M if and only if lim

n→∞σ −1 n max{k 1 : dn(k) 0} = 0.

• Corollary. Fix any p 2 and sample Mn,p a random 2p-angulation with n

faces, then p(p − 1)n−1/4Mn,p

(d)

− − − − →

n→∞

M.

Back to our model

Recall: dn(k) 0 with

k1 dn(k) = n, then Mdn uniform random map with

dn(k) faces with degree 2k for all k 1. Put σ 2

n =

• k1

k(k − 1)dn(k), sort of global half-degree variance.

• Theorem. The sequence (σ −1/2

n

Mdn)n always admits subsequential limits.

• Theorem. Moreover σ −1/2

n

Mdn

(d)

− − − − →

n→∞

M if and only if lim

n→∞σ −1 n max{k 1 : dn(k) 0} = 0.

• Corollary. Fix any (pn)n ∈ {2, 3, . . . }N and sample Mn,pn a random

2pn-angulation with n faces, then pn(pn − 1)n−1/4Mn,pn

(d)

− − − − →

n→∞

M.

Further results

◮ Previous case assuming σ 2 n ∼ σ 2n plus annoying assumptions; ’18. ◮ If one but only one face with degree ϱn ∼ ϱσn, then Brownian disk with

perimeter ϱ of Betinelli & Miermont ’17 instead.

◮ If one face with degree ϱn ≫ σn, then scaling ϱ1/2 n

and Aldous’ Brownian CRT instead.

◮ Application to size-conditioned critical α-stable Boltzmann maps

(P(deg 2k) ≈ k−α), with σ 1/2

n

• f order n1/(2α):

◮ 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 −2 −1 1 −2 −1 1 −1

The tree is chosen uniformly at random amongst those with dn(k) vertices with arity k; given the tree, labels obey the local rule:

−1 −1 = 0 −1 −1

And trees are coded by paths

−1 −2 −1 −2 −1 1 −2 −1 1 −1 1 2 3 4 4 8 12 16 20 24 28 32 −2 −1 1

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’?