First-passage percolation in random triangulations Jean-Franois Le - - PowerPoint PPT Presentation

First-passage percolation in random triangulations

Jean-François Le Gall (joint with Nicolas Curien)

Université Paris-Sud Orsay and Institut universitaire de France

Random Geometry and Physics, Paris 2016

Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP , Paris 2016 1 / 30

To define a canonical random geometry in two dimensions (motivations from physics: 2D quantum gravity) Replace the sphere S2 by a discretization, namely a graph drawn on the sphere (= planar map). Choose such a planar map uniformly at random in a suitable class and equip its vertex set with the graph distance.

Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP , Paris 2016 2 / 30

To define a canonical random geometry in two dimensions (motivations from physics: 2D quantum gravity) Replace the sphere S2 by a discretization, namely a graph drawn on the sphere (= planar map). Choose such a planar map uniformly at random in a suitable class and equip its vertex set with the graph distance. Let the size of the graph tend to infinity and pass to the limit after rescaling to get a random metric space: the Brownian map. This convergence is robust: it still holds if we make local modifications of the graph

• distance. (Assign random lengths to the

edges: first-passage percolation distance.) Goal of the lecture: Explain this robustness property.

Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP , Paris 2016 2 / 30

• 1. The geometry of large random planar maps

Definition

A planar map is a proper embedding of a finite connected graph into the two-dimensional sphere (considered up to orientation-preserving homeomorphisms of the sphere). Loops and multiple edges allowed.

Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP , Paris 2016 3 / 30

• 1. The geometry of large random planar maps

Definition

A planar map is a proper embedding of a finite connected graph into the two-dimensional sphere (considered up to orientation-preserving homeomorphisms of the sphere). Loops and multiple edges allowed.

root edge root vertex

A rooted triangulation with 20 faces Faces = connected components of the complement of edges p-angulation: each face is incident to p edges p = 3: triangulation p = 4: quadrangulation Rooted map: distinguished

• riented edge

Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP , Paris 2016 3 / 30

A large triangulation of the sphere Can we get a continuous model out of this ?

Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP , Paris 2016 4 / 30

Planar maps as metric spaces

M planar map V(M) = set of vertices of M dgr graph distance on V(M) (V(M), dgr) is a (finite) metric space

1 1 2 1 2 2 3 4

In blue : distances from root

Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP , Paris 2016 5 / 30

Planar maps as metric spaces

M planar map V(M) = set of vertices of M dgr graph distance on V(M) (V(M), dgr) is a (finite) metric space

1 1 2 1 2 2 3 4

In blue : distances from root Mp

n = {rooted p − angulations with n faces}

Mp

n is a finite set (finite number of possible “shapes”)

Choose Mn uniformly at random in Mp

n.

Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP , Paris 2016 5 / 30

Planar maps as metric spaces

M planar map V(M) = set of vertices of M dgr graph distance on V(M) (V(M), dgr) is a (finite) metric space

1 1 2 1 2 2 3 4

In blue : distances from root Mp

n = {rooted p − angulations with n faces}

Mp

n is a finite set (finite number of possible “shapes”)

Choose Mn uniformly at random in Mp

n.

View (V(Mn), dgr) as a random variable with values in K = {compact metric spaces, modulo isometries} which is equipped with the Gromov-Hausdorff distance. (A sequence (En) of compact metric spaces converges if one can embed all En’s isometrically in the same big space E so that they converge for the Hausdorff metric on compact subsets of E.)

Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP , Paris 2016 5 / 30

The Brownian map

Mp

n = {rooted p − angulations with n faces}

Mn uniform over Mp

n, V(Mn) vertex set of Mn, dgr graph distance

Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP , Paris 2016 6 / 30

The Brownian map

Mp

n = {rooted p − angulations with n faces}

Mn uniform over Mp

n, V(Mn) vertex set of Mn, dgr graph distance

Theorem (LG 2013, Miermont 2013)

Suppose that either p = 3 (triangulations) or p ≥ 4 is even. Set c3 = 61/4 , cp =

• 9

p(p − 2) 1/4 if p is even. Then, (V(Mn), cp n−1/4 dgr)

(d)

− →

n→∞ (m∞, D∗)

in the Gromov-Hausdorff sense. The limit (m∞, D∗) is a random compact metric space that does not depend on p (universality) and is called the Brownian map (after Marckert-Mokkadem).

Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP , Paris 2016 6 / 30

The Brownian map

Mp

n = {rooted p − angulations with n faces}

Mn uniform over Mp

n, V(Mn) vertex set of Mn, dgr graph distance

Theorem (LG 2013, Miermont 2013)

Suppose that either p = 3 (triangulations) or p ≥ 4 is even. Set c3 = 61/4 , cp =

• 9

p(p − 2) 1/4 if p is even. Then, (V(Mn), cp n−1/4 dgr)

(d)

− →

n→∞ (m∞, D∗)

in the Gromov-Hausdorff sense. The limit (m∞, D∗) is a random compact metric space that does not depend on p (universality) and is called the Brownian map (after Marckert-Mokkadem). Remarks

• The case p = 3 (triangulations) solves a question of Schramm (2006)
• Extensions to other classes of random planar maps: Abraham,

Addario-Berry-Albenque, Beltran-LG, Bettinelli-Jacob-Miermont, etc.

Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP , Paris 2016 6 / 30

Two properties of the Brownian map

Theorem (Hausdorff dimension)

dim(m∞, D∗) = 4 a.s.

Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP , Paris 2016 7 / 30

Two properties of the Brownian map

Theorem (Hausdorff dimension)

dim(m∞, D∗) = 4 a.s.

Theorem (topological type)

Almost surely, (m∞, D∗) is homeomorphic to the 2-sphere S2.

Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP , Paris 2016 7 / 30

Construction of the Brownian map

The Brownian map (m∞, D∗) is constructed as a quotient space of Aldous’ Brownian continuum random tree (the CRT), for an equivalence relation defined in terms of Brownian labels assigned to the vertices of the CRT.

A simulation of the CRT

Two points a and b of the CRT are glued if they have the same label and if one can go from a to b around the tree (clockwise or counterclockwise) meeting

• nly points with greater

label.

Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP , Paris 2016 8 / 30

Recent progress

Miller, Sheffield (2015-2016) have developed a program aiming to relate the Brownian map with Liouville quantum gravity: new construction of the Brownian map via the random growth process called Quantum Loewner Evolution (an analog of the celebrated SLE processes studied by Lawler, Schramm and Werner) this construction makes it possible to equip the Brownian map with a conformal structure, and in fact to show that this conformal structure is determined by the Brownian map. Still open: To show that this conformal structure also arises in the limit

• f conformal embeddings of discrete graphs (e.g. triangulations) that

can be produced via circle packings or the uniformization theorem of the theory of Riemann surfaces.

Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP , Paris 2016 9 / 30

• 2. Modifications of distances on random planar maps

(joint work with Nicolas Curien, arXiv:1511.04264) Assign i.i.d. random weights (lengths) we to the edges of a (random) planar map M. Define the weight w(γ) of a path γ as the sum of the weights of the edges it contains. The first passage percolation distance dFPP is defined on the vertex set V(M) by dFPP(v, v′) = inf{w(γ) : γ path from v to v′}.

Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP , Paris 2016 10 / 30

• 2. Modifications of distances on random planar maps

(joint work with Nicolas Curien, arXiv:1511.04264) Assign i.i.d. random weights (lengths) we to the edges of a (random) planar map M. Define the weight w(γ) of a path γ as the sum of the weights of the edges it contains. The first passage percolation distance dFPP is defined on the vertex set V(M) by dFPP(v, v′) = inf{w(γ) : γ path from v to v′}. Goal: In large scales, dFPP behaves like the graph distance dgr (asymptotically, balls for dFPP are close to balls for dgr). This is not expected to be true in deterministic lattices such as Zd, but random planar maps are in a sense more isotropic.

Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP , Paris 2016 10 / 30

• 2. Modifications of distances on random planar maps

(joint work with Nicolas Curien, arXiv:1511.04264) Assign i.i.d. random weights (lengths) we to the edges of a (random) planar map M. Define the weight w(γ) of a path γ as the sum of the weights of the edges it contains. The first passage percolation distance dFPP is defined on the vertex set V(M) by dFPP(v, v′) = inf{w(γ) : γ path from v to v′}. Goal: In large scales, dFPP behaves like the graph distance dgr (asymptotically, balls for dFPP are close to balls for dgr). This is not expected to be true in deterministic lattices such as Zd, but random planar maps are in a sense more isotropic. Consequence: The scaling limit of the metric space associated with dFPP will again be the Brownian map! (Universality of the limit!)

Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP , Paris 2016 10 / 30

The Uniform Infinite Planar Triangulation (UIPT)

Let ∆n be uniformly distributed over {triangulations with n faces}. For every r ≥ 1, let Br(∆n) be the ball of radius r in ∆n, defined as the union of all faces incident to a vertex at graph distance strictly less than r from the root vertex ρ. One can prove (Angel-Schramm 2003, Stephenson 2014) that ∆n

(d)

− →

n→∞ ∆∞

in the local limit sense, where ∆∞ is a (rooted) infinite random triangulation called the UIPT for Uniform Infinite Planar Triangulation.

Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP , Paris 2016 11 / 30

The Uniform Infinite Planar Triangulation (UIPT)

Let ∆n be uniformly distributed over {triangulations with n faces}. For every r ≥ 1, let Br(∆n) be the ball of radius r in ∆n, defined as the union of all faces incident to a vertex at graph distance strictly less than r from the root vertex ρ. One can prove (Angel-Schramm 2003, Stephenson 2014) that ∆n

(d)

− →

n→∞ ∆∞

in the local limit sense, where ∆∞ is a (rooted) infinite random triangulation called the UIPT for Uniform Infinite Planar Triangulation. The convergence holds in the sense of local limits: for every r and for every fixed planar map M, P(Br(∆n) = M) − →

n→∞ P(Br(∆∞) = M).

This is very different from the Gromov-Hausdorff convergence: Here we do no rescaling and thus the limit is a non-compact (infinite) random lattice.

Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP , Paris 2016 11 / 30

An artistic representation of the UIPT (artist: N. Curien)

Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP , Paris 2016 12 / 30

First-passage percolation in the UIPT

Assign i.i.d. weights we with common distribution ν to the edges of the UIPT ∆∞ and consider the associated first-passage percolation distance dFPP. Assume ν is supported on [c, C], where 0 < c ≤ C < ∞. For every real r ≥ 0, let BFPP

r

(∆∞) be the ball of radius r for dFPP.

Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP , Paris 2016 13 / 30

First-passage percolation in the UIPT

Assign i.i.d. weights we with common distribution ν to the edges of the UIPT ∆∞ and consider the associated first-passage percolation distance dFPP. Assume ν is supported on [c, C], where 0 < c ≤ C < ∞. For every real r ≥ 0, let BFPP

r

(∆∞) be the ball of radius r for dFPP.

Theorem

There exists a constant c0 with c ≤ c0 ≤ C, such that, for every ε > 0, we have lim

r→∞ P

• sup

x,y∈Br(∆∞)

• dFPP(x, y) − c0 dgr(x, y)
• > εr
• = 0.

In particular, B(1−ε)r/c0(∆∞) ⊂ BFPP

r

(∆∞) ⊂ B(1+ε)r/c0(∆∞) with probability tending to 1 as r → ∞.

Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP , Paris 2016 13 / 30

First-passage percolation in the UIPT

Assign i.i.d. weights we with common distribution ν to the edges of the UIPT ∆∞ and consider the associated first-passage percolation distance dFPP. Assume ν is supported on [c, C], where 0 < c ≤ C < ∞. For every real r ≥ 0, let BFPP

r

(∆∞) be the ball of radius r for dFPP.

Theorem

There exists a constant c0 with c ≤ c0 ≤ C, such that, for every ε > 0, we have lim

r→∞ P

• sup

x,y∈Br(∆∞)

• dFPP(x, y) − c0 dgr(x, y)
• > εr
• = 0.

In particular, B(1−ε)r/c0(∆∞) ⊂ BFPP

r

(∆∞) ⊂ B(1+ε)r/c0(∆∞) with probability tending to 1 as r → ∞. The ball of radius r for the FPP distance is asymptotically close to the ball of radius r/c0 for the graph distance.

• Remark. In general one cannot compute the constant c0, except in

special cases.

Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP , Paris 2016 13 / 30

The Eden model

The Eden growth model corresponds to first-passage percolation with exponential edge weights on the dual map of the UIPT. At rate 1 for each edge of the boundary, one “reveals” the triangle incident to this edge (the chosen edge is thus uniform on the boundary).

Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP , Paris 2016 14 / 30

The Eden model

The Eden growth model corresponds to first-passage percolation with exponential edge weights on the dual map of the UIPT. At rate 1 for each edge of the boundary, one “reveals” the triangle incident to this edge (the chosen edge is thus uniform on the boundary).

Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP , Paris 2016 15 / 30

The Eden model

The Eden growth model corresponds to first-passage percolation with exponential edge weights on the dual map of the UIPT. At rate 1 for each edge of the boundary, one “reveals” the triangle incident to this edge (the chosen edge is thus uniform on the boundary). This is an instance of the so-called peeling process (studied by Angel and others) and one can use known results for the volume of the discovered region at step n (Curien-LG) to compute c0 = 2 √ 3 (the set discovered at time r in the Eden model is close to the ball of radius r/(2 √ 3) for the graph distance on the UIPT)

Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP , Paris 2016 15 / 30

First-passage percolation in finite triangulations

∆n is uniformly distributed over {triangulations with n faces} dFPP first-passage percolation distance on V(∆n) defined using weights i.i.d. according to ν (same assumption on ν).

Theorem

(V(∆n), 61/4 n−1/4 dFPP)

(d)

− →

n→∞ (m∞, c0 D∗)

in the Gromov-Hausdorff sense. Here c0 is the same constant as in the UIPT case, and (m∞, D∗) is the Brownian map. Idea of the proof: Use absolute continuity arguments to relate large (finite) triangulations to the UIPT, and then apply the theorem about the UIPT.

Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP , Paris 2016 16 / 30

Other (potential) applications of the method

(Without edge weights) can compare the graph distances on a large triangulation (or the UIPT) and on its dual map. In large scales, they are proportional, with a scaling constant c0 = 2 √ 3 + 1 FPP on quadrangulations (similar techniques should apply). Tutte’s bijection relates quadrangulations to general planar maps: compare the graph distances on a quadrangulation and on the associated general planar map. Here one expects c0 = 1 Tutte’s bijection

A general planar map The associated quadrangulation (red edges)

Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP , Paris 2016 17 / 30

• 3. Geometric ingredients: Balls and hulls in the UIPT

In view of studying the first-passage percolation distance on the UIPT,

• ne needs more information about its geometry.

The hull of radius r, denoted by B•

r (∆∞), is obtained by filling in the

“holes” in the ball Br(∆∞).

ρ

The shaded part is the ball B2(∆∞) (all triangles that contain a vertex at distance ≤ 1 from ρ)

Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP , Paris 2016 18 / 30

• 3. Geometric ingredients: Balls and hulls in the UIPT

In view of studying the first-passage percolation distance on the UIPT,

• ne needs more information about its geometry.

The hull of radius r, denoted by B•

r (∆∞), is obtained by filling in the

“holes” in the ball Br(∆∞).

ρ

The hull B•

2(∆∞) is

the union of B2(∆∞) and the two holes.

Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP , Paris 2016 19 / 30

Layers in the UIPT – related to Krikun (2005)

∂B•

ℓ

B•

k

For k < ℓ, the successive layers between B•

k

and B•

ℓ are the sets

B•

j \B• j−1

for k < j ≤ ℓ. (Here 3 layers)

Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP , Paris 2016 20 / 30

Downward triangles in the layers

∂B•

ℓ

B•

k

k < ℓ fixed For each edge of ∂B•

ℓ

consider the unique triangle (called a downward triangle) containing this edge whose third vertex is on ∂B•

ℓ−1

(We do not get all triangles in the layer B•

ℓ \B• ℓ−1, only

those that have an edge in the exterior boundary of the layer) Do the same for the layer B•

ℓ−1\B• ℓ−2

And so on.

Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP , Paris 2016 21 / 30

Downward triangles in the layers

∂B•

ℓ

B•

k

Remove the edges not

• n the downward

(colored) triangles. This creates “white” slots.

Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP , Paris 2016 22 / 30

The forest coding downward triangles

∂B•

ℓ

B•

k

e e′

Can represent the configuration of downward triangles by a forest of trees whose vertices are the edges of ∂B•

j for all k ≤ j ≤ ℓ.

An edge e of ∂B•

j is the

parent of an edge e′ of ∂B•

j−1 if the white slot

whose boundary contains e′ is bounded

• n its right by the

downward triangle associated with e.

Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP , Paris 2016 23 / 30

The forest coding downward triangles

B•

k

∂B•

ℓ

The forest representing the structure of layers between B•

k and B• ℓ .

The roots of trees in the forest are all edges of ∂B•

ℓ .

To reconstruct B•

ℓ \B• k

• ne only needs

the forest coding the configuration of downward triangles, the triangulations (with boundaries) filling in the white slots.

Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP , Paris 2016 24 / 30

The Galton-Watson structure

Let T1, T2, . . . , TPℓ be the forest coding the configuration of downward triangles between ∂B•

k and ∂B• ℓ . Here k < ℓ, Pℓ, resp. Pk, is the size of

∂B•

ℓ , resp. ∂B• k.

τ1, . . . , τp deterministic forest with height ℓ − k and q vertices at height ℓ − k. Write V∗(τi) for all vertices of τi except those at height ℓ − k.

Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP , Paris 2016 25 / 30

The Galton-Watson structure

Let T1, T2, . . . , TPℓ be the forest coding the configuration of downward triangles between ∂B•

k and ∂B• ℓ . Here k < ℓ, Pℓ, resp. Pk, is the size of

∂B•

ℓ , resp. ∂B• k.

τ1, . . . , τp deterministic forest with height ℓ − k and q vertices at height ℓ − k. Write V∗(τi) for all vertices of τi except those at height ℓ − k.

Proposition (related to Krikun (2005))

P

• (T1, T2, . . . , TPℓ) = (τ1, . . . , τp)
• Pk = q
• = h(p)

h(q)

• v∈V∗(τ1)∪···∪V∗(τp)

θ(nv) where nv is the number of offspring of v; θ is the critical offspring distribution with generating function

• θ(n) xn = 1 −
• 1 +

1 √ 1 − x −2 ; h(p) = 4−p (2p)!

(p!)2 , for p ≥ 1 is the stationary distribution for the

Galton-Watson process with offspring distribution θ.

Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP , Paris 2016 25 / 30

• 4. The half-plane model

Construct a triangulation L of the lower half-plane as follows. Each horizontal edge on the line Z × {−k} belongs to a downward triangle whose third vertex is on the line Z × {−k − 1}.

−1 −2 −3 −4

(0, 0) (1, 0) (−1, 0)

The trees characterizing the configuration of downward triangles are independent Galton-Watson trees with offspring distribution θ. Slots are filled in with “free triangulations” with a boundary (probab. of a given triangul. with n inner vertices is K (12 √ 3)−n).

Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP , Paris 2016 26 / 30

First-passage percolation in the half-plane model

Assign i.i.d. weights we to the edges of L, with common distribution ν. Consider the associated first-passage percolation distance dFPP.

Proposition

Let ρ = (0, 0) be the root and for every k ≥ 0, let Lk be the horizontal line at vertical coordinate −k. Then 1 k dFPP(ρ, Lk)

a.s.

− →

k→∞ c0 ∈ [c, C].

Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP , Paris 2016 27 / 30

First-passage percolation in the half-plane model

Assign i.i.d. weights we to the edges of L, with common distribution ν. Consider the associated first-passage percolation distance dFPP.

Proposition

Let ρ = (0, 0) be the root and for every k ≥ 0, let Lk be the horizontal line at vertical coordinate −k. Then 1 k dFPP(ρ, Lk)

a.s.

− →

k→∞ c0 ∈ [c, C].

Proof: Kingman’s subadditive ergodic theorem.

−k − ℓ −k

geodesic from to Lk vk geodesic from vk to Lk+ℓ (in region below Lk )

ρ

ρ

dFPP(ρ, Lk+ℓ) ≤ dFPP(ρ, Lk)+Zk,ℓ where Zk,ℓ

(d)

= dFPP(ρ, Lℓ) and Zk,ℓ is independent of dFPP(ρ, Lk).

Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP , Paris 2016 27 / 30

• 5. A few ideas of the proof for FPP on the UIPT

Locally (say below the boundary of the hull of radius r), the UIPT looks like the half-plane model.

r r − 1 r − 2 r − 3

∞

distance from

r ρ

ρ

Can use the result in the half-plane model to estimate the FPP distance between a typical point of ∂B•

r (∆∞) and ∂B• (1−ε)r(∆∞).

Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP , Paris 2016 28 / 30

Ideas of the proof II

distance from

∞

ρ ρ

We consider slices of the UIPT corresponding to sets of the form B•

r (∆∞)\B• (1−ε)r(∆∞)

For each slice, one can control the distance between an arbitrary point of the upper boundary and the lower boundary of the slice.

Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP , Paris 2016 29 / 30

Ideas of the proof II

distance from

∞

ρ ρ

We consider slices of the UIPT corresponding to sets of the form B•

r (∆∞)\B• (1−ε)r(∆∞)

For each slice, one can control the distance between an arbitrary point of the upper boundary and the lower boundary of the slice. Main technical issue: In order to compare with the half-plane model, one should exclude the possibility that FPP geodesics move too fast in the “horizontal direction” (turn around the slice)

Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP , Paris 2016 29 / 30

A few references

BENJAMINI: Random planar metrics. Proc. ICM 2010. BOUTTIER, DI FRANCESCO, GUITTER: Planar maps as labeled

• mobiles. Electr. J. Combinatorics (2004)

BOUTTIER, GUITTER: The three-point function of planar

• quadrangulations. J. Stat. Mech. Theory Exp. (2008)

CURIEN, LE GALL: First-passage percolation and local modifications of distances in random triangulations (2015) LE GALL: Geodesics in large planar maps ... Acta Math. (2010) LE GALL: Uniqueness and universality of the Brownian map. Ann.

• Probab. (2013)

MILLER, SHEFFIELD: An axiomatic characterization of the Brownian map (2015) MILLER, SHEFFIELD: Liouville quantum gravity and the Brownian map I, II, III (2015-2016) MIERMONT: The Brownian map is the scaling limit of uniform random plane quadrangulations. Acta Math. (2013) SCHRAMM: Conformally invariant scaling limits. Proc. ICM 2006.

Jean-François Le Gall (Université Paris-Sud) First-passage percolation in triangulations RGP , Paris 2016 30 / 30