Isoperimetric inequalities in random geometry Jean-Franois Le Gall, - - PowerPoint PPT Presentation

Isoperimetric inequalities in random geometry

Jean-François Le Gall, Thomas Lehéricy

Université Paris-Sud Orsay

Les Diablerets 2018

J.F. Le Gall, T. Lehéricy (Univ. Paris-Sud) Isoperimetric inequalities in random geometry Les Diablerets 2018 1 / 24

Outline

The Uniform Infinite Planar Triangulation (UIPT, Angel-Schramm 2003) and the Uniform Infinite Planar Quadrangulation (UIPQ, Krikun 2005) are standard models of discrete random geometry — there are other models which are expected to share the same qualitative properties.

J.F. Le Gall, T. Lehéricy (Univ. Paris-Sud) Isoperimetric inequalities in random geometry Les Diablerets 2018 2 / 24

Outline

The Uniform Infinite Planar Triangulation (UIPT, Angel-Schramm 2003) and the Uniform Infinite Planar Quadrangulation (UIPQ, Krikun 2005) are standard models of discrete random geometry — there are other models which are expected to share the same qualitative properties. We are interested in isoperimetric bounds in these models: Given a set K which is a (simply connected) finite union of faces of the random lattice, how small can the size |∂K| of the boundary be if the volume |K| is fixed (and large) ? We get |∂K| ≥ |K|1/4 up to logarithmic corrections.

J.F. Le Gall, T. Lehéricy (Univ. Paris-Sud) Isoperimetric inequalities in random geometry Les Diablerets 2018 2 / 24

Outline

The Uniform Infinite Planar Triangulation (UIPT, Angel-Schramm 2003) and the Uniform Infinite Planar Quadrangulation (UIPQ, Krikun 2005) are standard models of discrete random geometry — there are other models which are expected to share the same qualitative properties. We are interested in isoperimetric bounds in these models: Given a set K which is a (simply connected) finite union of faces of the random lattice, how small can the size |∂K| of the boundary be if the volume |K| is fixed (and large) ? We get |∂K| ≥ |K|1/4 up to logarithmic corrections. Similar (sharper) results for the Brownian plane, which is a continuous model expected to be the Gromov-Hausdorff scaling limit of the discrete models of random geometry (work in progress of A. Riera).

J.F. Le Gall, T. Lehéricy (Univ. Paris-Sud) Isoperimetric inequalities in random geometry Les Diablerets 2018 2 / 24

• 1. The Uniform Infinite Planar Quadrangulation

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).

Root vertex Root edge

A rooted quadrangulation with 7 faces Faces = connected components of the complement of edges p-angulation: each face is bounded by p edges p = 3: triangulation p = 4: quadrangulation Rooted map: distinguished

• riented edge

J.F. Le Gall, T. Lehéricy (Univ. Paris-Sud) Isoperimetric inequalities in random geometry Les Diablerets 2018 3 / 24

Key observation. Two planar maps M1 and M2 are identified if there is an (orientation-preserving) homeomorphism of the sphere that maps M1 to M2. − → One is interested in the “shape” of the planar map, not in the details of the embedding (though there are important questions concerning canonical embeddings!). The same planar map Because of this, there are (for instance) finitely many quadrangulations with a fixed number n of faces: It makes sense to choose one of them uniformly at random.

J.F. Le Gall, T. Lehéricy (Univ. Paris-Sud) Isoperimetric inequalities in random geometry Les Diablerets 2018 4 / 24

The Uniform Infinite Planar Quadrangulation (UIPQ)

Let Qn be uniformly distributed over {quadrangulations with n faces}. For every integer r ≥ 1, let Br(Qn) be the ball of radius r in Qn, defined as the union of all faces incident to a vertex at graph distance strictly less than r from the root vertex ρ.

J.F. Le Gall, T. Lehéricy (Univ. Paris-Sud) Isoperimetric inequalities in random geometry Les Diablerets 2018 5 / 24

The Uniform Infinite Planar Quadrangulation (UIPQ)

Let Qn be uniformly distributed over {quadrangulations with n faces}. For every integer r ≥ 1, let Br(Qn) be the ball of radius r in Qn, 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 for the UIPT, Krikun 2005) that Qn

(d)

− →

n→∞ Q∞

in the local limit sense, where Q∞ is a (random) infinite quadrangulation called the UIPQ for Uniform Infinite Planar Quadrangulation.

J.F. Le Gall, T. Lehéricy (Univ. Paris-Sud) Isoperimetric inequalities in random geometry Les Diablerets 2018 5 / 24

The Uniform Infinite Planar Quadrangulation (UIPQ)

Let Qn be uniformly distributed over {quadrangulations with n faces}. For every integer r ≥ 1, let Br(Qn) be the ball of radius r in Qn, 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 for the UIPT, Krikun 2005) that Qn

(d)

− →

n→∞ Q∞

in the local limit sense, where Q∞ is a (random) infinite quadrangulation called the UIPQ for Uniform Infinite Planar Quadrangulation. The convergence holds in the sense of local limits: for every r and for every fixed (finite) planar map M, P(Br(Qn) = M) − →

n→∞ P(Br(Q∞) = M).

The distribution of what one sees in a fixed neighborhood of the root vertex of Qn “stabilizes” when n → ∞

J.F. Le Gall, T. Lehéricy (Univ. Paris-Sud) Isoperimetric inequalities in random geometry Les Diablerets 2018 5 / 24

An artistic view of the UIPQ

J.F. Le Gall, T. Lehéricy (Univ. Paris-Sud) Isoperimetric inequalities in random geometry Les Diablerets 2018 6 / 24

Balls and hulls in the UIPQ

The ball Br(Q∞) is the union of all faces incident to a vertex at graph distance strictly less than r from the root vertex ρ. The hull B•

r (Q∞) is obtained by “filling in” the bounded holes in the ball

Br(Q∞).

ρ

The ball B2(Q∞)

J.F. Le Gall, T. Lehéricy (Univ. Paris-Sud) Isoperimetric inequalities in random geometry Les Diablerets 2018 7 / 24

Balls and hulls in the UIPQ

The ball Br(Q∞) is the union of all faces incident to a vertex at graph distance strictly less than r from the root vertex ρ. The hull B•

r (Q∞) is obtained by “filling in” the bounded holes in the ball

Br(Q∞).

ρ

The hull B•

2(Q∞)

J.F. Le Gall, T. Lehéricy (Univ. Paris-Sud) Isoperimetric inequalities in random geometry Les Diablerets 2018 8 / 24

Volume and boundary sizes of hulls

To get a set with a “large” volume but a “small” boundary size, one may first think of a hull B•

r of large radius r.

The volume |B•

r | is the number of faces in B• r .

The perimeter |∂B•

r | is the number of edges in ∂B• r .

distance from

∞

ρ ρ

radius

r

r

hull of

B•

r

∂B•

r

It turns out that, for r large, |B•

r | ≈ r 4

|∂B•

r | ≈ r 2

Hence |∂B•

r | ≈ |B• r |1/2

(Remark. If one replaces hulls by balls one has: |Br| ≈ r 4 |∂Br| ≈ r 3)

J.F. Le Gall, T. Lehéricy (Univ. Paris-Sud) Isoperimetric inequalities in random geometry Les Diablerets 2018 9 / 24

More precise asymptotics (Curien-LG 2015)

For every λ > 0, lim

r→∞ E[e−λc1 r−4|B•

r |] = 33/2 cosh(λ1/4)

• cosh2(λ1/4) + 2

−3/2 lim

r→∞ E[e−λc2r −2|∂B•

r |] = (1 + λ)−3/2

(where c1 and c2 are known constants — the same results hold for the UIPT with different constants c1, c2)

J.F. Le Gall, T. Lehéricy (Univ. Paris-Sud) Isoperimetric inequalities in random geometry Les Diablerets 2018 10 / 24

Smaller separating cycles

distance from

∞

ρ ρ

radius separating cycle

r

r

hull of

∂B•

r

(of size ∼ r2)

B•

r

Krikun (2005) constructed cycles that separate the hull B•

r from infinity, whose

length is roughly linear in r. He conjectured that one cannot do better than linear.

Theorem (LG-Lehéricy)

Let Lr be the minimal length of a cycle separating B•

r from ∞.

P(Lr ≤ εr) ≤ Cδ εδ, for every δ < 2 P(Lr ≥ u r) ≤ C exp(−c u).

J.F. Le Gall, T. Lehéricy (Univ. Paris-Sud) Isoperimetric inequalities in random geometry Les Diablerets 2018 11 / 24

• 2. Isoperimetric bounds

If Γr is a cycle separating the hull B•

r from ∞ with minimal length , and

if Ar is the (finite) region bounded by Γr, one has |Ar| ≥ |B•

r | ≈ r 4

|∂Ar| = |Γr| ≈ r so that in that case |∂Ar| ≤ |Ar|1/4. This is essentially the worse possible situation if one considers regions containing the root ρ.

Theorem (LG-Lehéricy)

Let K be the class of all simply connected regions that are finite unions

• f faces of the UIPQ and contain the root ρ. For every δ > 0,

inf

A∈K

|∂A| |A|1/4(1 + log |A|)3/4+δ > 0 , almost surely.

J.F. Le Gall, T. Lehéricy (Univ. Paris-Sud) Isoperimetric inequalities in random geometry Les Diablerets 2018 12 / 24

Sketch of the proof of the isoperimetric bounds

We explain how to get the following weaker result. Recall that K is the class of all simply connected regions that are finite unions of faces of the UIPQ and contain the root ρ.

Proposition

Let ε > 0. There exists a constant cε > 0 such that, for every n ≥ 1, P

• |∂A| ≥ cε n1/4, ∀A ∈ K such that |A| ≥ n
• ≥ 1 − ε.

(In view of proving the theorem, one in fact needs somewhat more precise estimates, in order to use Borel-Cantelli type arguments.)

J.F. Le Gall, T. Lehéricy (Univ. Paris-Sud) Isoperimetric inequalities in random geometry Les Diablerets 2018 13 / 24

Sketch of the proof of the isoperimetric bounds 2

Choose first α > 0 small enough so that |B•

2αn1/4| < n with high probability,

then 0 < c < α such that the probability that the hull B•

αn1/4 is

separated from ∞ by a cycle of length ≤ c n1/4 is very small. Let A ∈ K such that |A| ≥ n. Then there are two cases:

J.F. Le Gall, T. Lehéricy (Univ. Paris-Sud) Isoperimetric inequalities in random geometry Les Diablerets 2018 14 / 24

Sketch of the proof of the isoperimetric bounds 2

Choose first α > 0 small enough so that |B•

2αn1/4| < n with high probability,

then 0 < c < α such that the probability that the hull B•

αn1/4 is

separated from ∞ by a cycle of length ≤ c n1/4 is very small. Let A ∈ K such that |A| ≥ n. Then there are two cases:

distance from

∞

ρ ρ

radius hull of

∂A αn1/4

αn1/4

If ∂A is at distance greater than αn1/4 from the root vertex ρ : then ∂A separates the hull B•

αn1/4 from ∞, and

we know that we must have |∂A| > c n1/4 except on a set of very small probability.

J.F. Le Gall, T. Lehéricy (Univ. Paris-Sud) Isoperimetric inequalities in random geometry Les Diablerets 2018 14 / 24

Sketch of the proof of the isoperimetric bounds 3

distance from

∞

ρ ρ

radius hull of

∂A αn1/4

αn1/4

If ∂A is at distance smaller than αn1/4 from the root vertex ρ : then we must have |∂A| > c n1/4, because if not the fact that every point of ∂A is at distance ≤ (α + c)n1/4 from ρ implies A ⊂ B•

2αn1/4,

which contradicts |A| ≥ n (except on the event where |B•

2αn1/4| < n, which has small

probability)

J.F. Le Gall, T. Lehéricy (Univ. Paris-Sud) Isoperimetric inequalities in random geometry Les Diablerets 2018 15 / 24

• 3. Estimates for separating cycles

Basic idea. Although the UIPQ is a random lattice, it exhibits certain regularity properties, and in particular a decomposition into layers (already studied by Krikun (2005)) − → This decomposition gives insight into the structure of annuli, where an annulus consists of the part of the UIPQ between the boundaries of the hulls of radius r and r ′, with r < r ′. − → It is more convenient to replace hulls by the (closely related) truncated hulls.

J.F. Le Gall, T. Lehéricy (Univ. Paris-Sud) Isoperimetric inequalities in random geometry Les Diablerets 2018 16 / 24

Truncated hulls

1 1 2 3 2 2 3 1 2 3 4 4 5 3 3 2 1 2 3 2 3 2 3 2 2 3 4 2

ρ

C2

In grey, the hull of radius r = 2. Label each vertex by its distance from ρ. In each face with corners labeled r, r − 1, r, r + 1, draw a diagonal between the two corners labeled r (in blue for r = 2 in the picture). These diagonals form a collection

• f disjoint cycles, and there is a

maximal one denoted by Cr. The truncated hull of radius r is the part of the UIPQ inside the cycle

• Cr. (Remark: Cr has length ≈ r 2.)

For r < r ′, the annulus A(r, r ′) is the part of the UIPQ between the cycles Cr and Cr ′.

J.F. Le Gall, T. Lehéricy (Univ. Paris-Sud) Isoperimetric inequalities in random geometry Les Diablerets 2018 17 / 24

Decomposition in layers

Cr′ Cr

The annulus A(r, r ′) bounded by the cycles Cr and Cr ′ (here r ′ = r + 3). The red dotted curves are the cycles Cs for r < s < r ′. A layer corresponds to the part of the UIPQ between Cs and Cs+1.

J.F. Le Gall, T. Lehéricy (Univ. Paris-Sud) Isoperimetric inequalities in random geometry Les Diablerets 2018 18 / 24

Decomposition in layers

Cr′ Cr

The annulus A(r, r ′) bounded by the cycles Cr and Cr ′ (here r ′ = r + 3). The red dotted curves are the cycles Cs for r < s < r ′. A layer corresponds to the part of the UIPQ between Cs and Cs+1. A triangle having an edge belonging to Cs (for some r < s ≤ r ′) and the third vertex on Cs−1 is called a downward triangle.

J.F. Le Gall, T. Lehéricy (Univ. Paris-Sud) Isoperimetric inequalities in random geometry Les Diablerets 2018 18 / 24

Decomposition in layers

Cr′ Cr

The annulus A(r, r ′) bounded by the cycles Cr and Cr ′ (here r ′ = r + 3). The red dotted curves are the cycles Cs for r < s < r ′. A layer corresponds to the part of the UIPQ between Cs and Cs+1. A triangle having an edge belonging to Cs (for some r < s ≤ r ′) and the third vertex on Cs−1 is called a downward triangle.

J.F. Le Gall, T. Lehéricy (Univ. Paris-Sud) Isoperimetric inequalities in random geometry Les Diablerets 2018 19 / 24

Decomposition in layers 2

Cr′ Cr

Downward triangles are represented as white triangles. The remaining part of the annulus (in the grey slots) has been erased.

J.F. Le Gall, T. Lehéricy (Univ. Paris-Sud) Isoperimetric inequalities in random geometry Les Diablerets 2018 20 / 24

Decomposition in layers 3

Cr′ Cr

Downward triangles are represented as white triangles. The remaining part of the annulus (in the grey slots) has been erased. The configuration of white triangles can be described by a forest of trees denoted by Fr,r′ (Vertices of the trees are the edges of the cycles Cs for r ≤ s ≤ r ′, the roots of the trees are the edges of Cr ′)

J.F. Le Gall, T. Lehéricy (Univ. Paris-Sud) Isoperimetric inequalities in random geometry Les Diablerets 2018 21 / 24

The distribution of the forest coding layers

The distribution of the forest Fr,r′ can be described explicitly. If F is a deterministic forest with height r ′ − r, P(Fr,r ′ = F) ≈

• v vertex of F

θ(cv) where cv is the offspring number of v and θ(·) is the offspring distribution with generating function

∞

• k=0

θ(k) yk = 1 − 8

• 9−y

1−y + 2

2 − 1 .

J.F. Le Gall, T. Lehéricy (Univ. Paris-Sud) Isoperimetric inequalities in random geometry Les Diablerets 2018 22 / 24

Separating cycles from the coding forest

Cr′ Cr With each tree of the forest Fr,r′ having maximal height r ′ − r, one associates two special geodesics from Cr to Cr ′, which roughly speaking follow the right and left contour of the tree. The concatenation of these special geodesics yields a cycle that separates the hull B•

r from ∞ and (say when r ′ = 2r) has length linear

in r.

J.F. Le Gall, T. Lehéricy (Univ. Paris-Sud) Isoperimetric inequalities in random geometry Les Diablerets 2018 23 / 24

No cycle with sublinear length

One checks that, for β < 3, for every n ≥ 1, P

• ∃ a separating cycle of length ≤ r in A(nr, (n + 2)r)
• ≤ Cβ n−β.

Cnr C(n+2)r

special geodesics separating cycle of length ≤ r x1 x2 xj annulus A(nr, (n + 2)r)

Outside a set of probability less than n−β one can find nδ trees with maximal height in the forest Fnr,(n+2)r, hence nδ special geodesics from C(n+2)r to Cnr. The endpoints x1, x2, . . . of these geodesics are within distance ≤ 5r of each other in the exterior of Cnr. This contradicts results of Curien and Miermont about distances along the boundary in infinite planar quadrangulations with a boundary.

J.F. Le Gall, T. Lehéricy (Univ. Paris-Sud) Isoperimetric inequalities in random geometry Les Diablerets 2018 24 / 24