Finite Connections for Supercritical Bernoulli Bond Percolation in - - PowerPoint PPT Presentation

Introduction Sketch of Proof Summary

Finite Connections for Supercritical Bernoulli Bond Percolation in 2D

• M. Campanino1
• D. Ioffe2
• O. Louidor3

1Università di Bologna (Italy) 2Technion (Israel) 3Courant (New York University)

Courant Probability Seminar, 11/6/2009

Introduction Sketch of Proof Summary

Outline

Introduction Percolation on Zd Logarithmic Asymptotics of Connectivities Sharp Asymptotics of Connectivities Sketch of Proof Setup Geometry of Finite Connections The Structure of a Cluster Asymptotics for No Intersection of Two Decorated RWs Summary

Introduction Sketch of Proof Summary

Outline

Introduction Percolation on Zd Logarithmic Asymptotics of Connectivities Sharp Asymptotics of Connectivities Sketch of Proof Setup Geometry of Finite Connections The Structure of a Cluster Asymptotics for No Intersection of Two Decorated RWs Summary

Introduction Sketch of Proof Summary

Percolation

• Take G = (Zd, Ed) - the integer lattice with nearest neighbor

edges.

• Open each edge with probability p ∈ [0, 1] independently.
• Let Bp be the underlying measure.

Theorem (Broadbent, Hammersley 1957) For all d > 2, there exists pc(d) ∈ (0, 1) such that: Bp(0 ← → ∞) = if p < pc(d) θ(p) > 0 if p > pc(d)

Introduction Sketch of Proof Summary

Basic Picture

Sub-critical density p < pc(d):

• All clusters (connected components) are finite.
• Radii of clusters have exponentially decaying distributions:

∃ξp ∈ (0, ∞) : Bp(0 ← → ∂B(R))≈e−ξpR . Russo-Menshikov (86), Barskey-Aizenman (87). Super-critical density p > pc(d):

• Unique infinite cluster Bp-a.s.
• Radii of finite clusters have exponentially decaying distributions:

∃ζp ∈ (0, ∞) : Bp(∞ 0 ↔ ∂B(R))≈e−ζpR . Chayes2-Newman (87), C2-Grimmett-Kesten-Schonmann (89).

Introduction Sketch of Proof Summary

Connectivities

The point-to-point connectivity function is defined as: τp(x, y) = Bp(x ← → y) ; x, y ∈ Zd .

Introduction Sketch of Proof Summary

Connectivities

The point-to-point connectivity function is defined as: τp(x, y) = Bp(x ← → y) ; x, y ∈ Zd . Sub-critical case: Subadditivity (via FKG of Bp) and exponential decay of cluster radius distribution imply: Theorem Assume p < pc(d). Then, for all x ∈ Rd: ξp(x) = lim

n→∞ − 1 n log τp(0, ⌊nx⌋)

is well-defined, convex and homogeneous function that is strictly positive on Rd \ {0}. In other words ξp is a norm on Rd, called the inverse correlation norm.

Introduction Sketch of Proof Summary

Connectivities - cont’d

Super-critical case: FKG gives a uniform positive lower bound for all x, y ∈ Zd: Bp(x ← → y) Bp(x ← → ∞)Bp(y ← → ∞) = θ2(p) > 0 .

Introduction Sketch of Proof Summary

Connectivities - cont’d

Super-critical case: FKG gives a uniform positive lower bound for all x, y ∈ Zd: Bp(x ← → y) Bp(x ← → ∞)Bp(y ← → ∞) = θ2(p) > 0 . Therefore, define the finite (truncated) connectivity function: τ f

p(x, y) = Bp(x f

← →y) = Bp(∞ x ↔ y) ; x, y ∈ Zd .

Introduction Sketch of Proof Summary

Connectivities - cont’d

Super-critical case: FKG gives a uniform positive lower bound for all x, y ∈ Zd: Bp(x ← → y) Bp(x ← → ∞)Bp(y ← → ∞) = θ2(p) > 0 . Therefore, define the finite (truncated) connectivity function: τ f

p(x, y) = Bp(x f

← →y) = Bp(∞ x ↔ y) ; x, y ∈ Zd . Theorem Assume p / ∈ {0, pc(d), 1}. Then, for all x ∈ Rd: ζp(x) = lim

n→∞ − 1 n log τ f p(0, ⌊nx⌋)

is well-defined, homogeneous and strictly positive on Rd \ {0}. This is the finite (truncated) inverse correlation function.

Introduction Sketch of Proof Summary

Logarithmic Scale Asymptotics

In other words: Bp(x ← → y)≈e−ξp(θ)y−x2 and Bp(x

f

← → y)≈e−ζp(θ)y−x2 for all x, y ∈ Zd as y − x → ∞, where θ = (x − y)/x − y2.

Introduction Sketch of Proof Summary

Logarithmic Scale Asymptotics

In other words: Bp(x ← → y)≈e−ξp(θ)y−x2 and Bp(x

f

← → y)≈e−ζp(θ)y−x2 for all x, y ∈ Zd as y − x → ∞, where θ = (x − y)/x − y2. Some relations:

• ξp = ξp(e1). ζp = ζp(e1).
• If d = 2, p > pc(2) = 1

2 then ζp = 2ξ1−p.

Chayes-Chayes-Grimmett-Kesten-Schonmann (89).

Introduction Sketch of Proof Summary

Logarithmic Scale Asymptotics

In other words: Bp(x ← → y)≈e−ξp(θ)y−x2 and Bp(x

f

← → y)≈e−ζp(θ)y−x2 for all x, y ∈ Zd as y − x → ∞, where θ = (x − y)/x − y2. Some relations:

• ξp = ξp(e1). ζp = ζp(e1).
• If d = 2, p > pc(2) = 1

2 then ζp = 2ξ1−p.

Chayes-Chayes-Grimmett-Kesten-Schonmann (89). Want sharp asymptotics: Bp(x ← → y)∼ ? and Bp(x

f

← → y)∼ ?

Introduction Sketch of Proof Summary

Sharp Asymptotics - Subcritical Case

For all d 2, p < pc(d), x, y ∈ Zd: Bp(x ← → y)∼Ap(θ)y − x−(d−1)/2

2

e−ξp(θ)y−x2 as y − x → ∞.

• Campanino-Chayes-Chayes (88) (y − x is on the axes).
• Campanino-Ioffe (02) (all y − x).

With the Gaussian correction this is called Ornstein-Zernike Behavior. after the work of L.Ornstein and F.Zernike.

Introduction Sketch of Proof Summary

Sharp Asymptotics - Supercritical Case

d 3: For all p > pc(d), x, y ∈ Zd, it is expected: Bp(x

f

← → y)∼ Ap(θ)y − x−(d−1)/2

2

e−ζp(θ)y−x2 as y − x → ∞. Verified for p ≫ pc(d) and y − x on axes. Braga-Procacci-Sanchis (04).

Introduction Sketch of Proof Summary

Sharp Asymptotics - Supercritical Case

d 3: For all p > pc(d), x, y ∈ Zd, it is expected: Bp(x

f

← → y)∼ Ap(θ)y − x−(d−1)/2

2

e−ζp(θ)y−x2 as y − x → ∞. Verified for p ≫ pc(d) and y − x on axes. Braga-Procacci-Sanchis (04). d = 2:

Introduction Sketch of Proof Summary

Sharp Asymptotics - Supercritical Case

d 3: For all p > pc(d), x, y ∈ Zd, it is expected: Bp(x

f

← → y)∼ Ap(θ)y − x−(d−1)/2

2

e−ζp(θ)y−x2 as y − x → ∞. Verified for p ≫ pc(d) and y − x on axes. Braga-Procacci-Sanchis (04). d = 2:

?

Introduction Sketch of Proof Summary

A Related Model - Nearest-Neighbor Ising

Introduction Sketch of Proof Summary

A Related Model - Nearest-Neighbor Ising

Exactly solvable in d = 2 (Onsager (44)). Explicit formulas for (truncated) k-point correlation functions at all temperatures (Wu, McCoy, Tracy, Potts, Ward, Montroll (’70)). Theorem (Cheng-Wu, Wu) If β < βc(2) then σx; σyβ σxσyβ ∼ Aβ(θ)y − x−1/2

2

e−ξβ(θ)y−x2 and if β > βc(2) then: σx; σyT

β σxσyβ − σxβ σyβ ∼

Aβ(θ)y − x−2

2 e−ζβ(θ)y−x2

for all x, y ∈ Z2 as y − x → ∞. No OZ Behavior in d = 2 below the critical temperature!

Introduction Sketch of Proof Summary

Sharp Asymptotics - Supercritical Case

d 3: For all p > pc(d), x, y ∈ Zd, it is expected: Bp(x

f

← → y)∼ Ap(θ)y − x−(d−1)/2

2

e−ζp(θ)y−x2 as y − x → ∞. Verified for p ≫ pc(d) and y − x on axes. Braga-Procacci-Sanchis (04). d = 2:

?

Introduction Sketch of Proof Summary

Sharp Asymptotics - Supercritical Case

d 3: For all p > pc(d), x, y ∈ Zd, it is expected: Bp(x

f

← → y)∼ Ap(θ)y − x−(d−1)/2

2

e−ζp(θ)y−x2 as y − x → ∞. Verified for p ≫ pc(d) and y − x on axes. Braga-Procacci-Sanchis (04). d = 2: Theorem (Campanino, Ioffe, L. (09)) For all p > pc(2) = 1/2, x, y ∈ Z2: Bp(x

f

← → y)∼ Ap(θ)y − x−2

2 e−ζp(θ)y−x2

as y − x → ∞.

Introduction Sketch of Proof Summary

Outline

Introduction Percolation on Zd Logarithmic Asymptotics of Connectivities Sharp Asymptotics of Connectivities Sketch of Proof Setup Geometry of Finite Connections The Structure of a Cluster Asymptotics for No Intersection of Two Decorated RWs Summary

Introduction Sketch of Proof Summary

Setup

Dual lattice.

• In d = 2 there is an isomorphic dual Z2

∗.

• Set: b∗ is open

⇐ ⇒ b is close .

• The dual model is Percolation with p∗ = 1 − p.
• We assume p < pc(2) and find Bp(x∗

f

← → y∗).

• However, we’ll express this event mainly using direct bonds.
• For simplicity: x∗ = 0∗,

y∗ = 0∗ + (N, 0) N∗.

Introduction Sketch of Proof Summary

Setup

Dual lattice.

• In d = 2 there is an isomorphic dual Z2

∗.

• Set: b∗ is open

⇐ ⇒ b is close .

• The dual model is Percolation with p∗ = 1 − p.
• We assume p < pc(2) and find Bp(x∗

f

← → y∗).

• However, we’ll express this event mainly using direct bonds.
• For simplicity: x∗ = 0∗,

y∗ = 0∗ + (N, 0) N∗. A bit of notation.

• Clm,r(x, y) - The (possibly empty) cluster that contains x, y and

uses only edges in the strip [m, r] × Z.

Introduction Sketch of Proof Summary

Geometry of Finite Connections

• 0∗

f

← → x∗

N

• = {0∗ ←

→ x∗

N} ∩ {∃ direct loop CN around 0∗ and x∗ N} .

0∗ N∗ γ∗ CN

Introduction Sketch of Proof Summary

Decomposition of Finite Connection Event

Idea: Find a geometric decomposition which is both unique and the sets of bonds that define each pieces are disjoint.

Introduction Sketch of Proof Summary

Decomposition of Finite Connection Event

Idea: Find a geometric decomposition which is both unique and the sets of bonds that define each pieces are disjoint. In our case: Let CN be the inner most loop that contains Cl(0∗, N∗). Cut along the left-most line Hm and right-most line HN−r that intersect CN exactly twice: CN ∩ Hm = {x, v} ; CN ∩ HN−r = {u, y} Get 3 pieces: L([v, x]), A([v, x], [u, y]), R([u, y]).

Introduction Sketch of Proof Summary

Decomposition - cont’d

0∗ N∗ γ∗ CN Bp

• 0∗

f

← → x∗

N

Introduction Sketch of Proof Summary

Decomposition - cont’d

0∗ N∗ γ∗ CN x y v u ℓ N − r Bp

• 0∗

f

← → x∗

N

• =
• x,v,y,u

Bp (I([v, x], [u, y])) + Bp (I(∅))

Introduction Sketch of Proof Summary

Decomposition - cont’d

0∗ N∗ x x y y v v u u Bp

• 0∗

f

← → x∗

N

• =
• x,v,y,u

Bp (I([v, x], [u, y])) + Bp (I(∅)) =

• x,v,y,u

Bp (L([v, x])) Bp (A([v, x], [u, y])) Bp (R([u, y])) + Bp (I(∅))

Introduction Sketch of Proof Summary

Decomposition - cont’d

In fact: Bp

• 0∗

f

← → x∗

N

• ∼
• |x|,|v|log N

|N−y|,|N−u| log N

Bp (L([v, x])) Bp (A([v, x], [u, y]))Bp (R([u, y])) and, modulo the exponential decay, the asymptotics will come from Bp (A([v, x], [u, y]))

Introduction Sketch of Proof Summary

Two Disjoint Boundary Clusters

A([v, x], [u, y]) = {. . . , x ← → y, v ← → u, Clm,N−r (v, u) ∩ Clm,N−r (x, y) = ∅}

v u x y Hm HN−r Clm,N−r(x, y) Clm,N−r(v, u)

Introduction Sketch of Proof Summary

Two Disjoint Boundary Clusters

A([v, x], [u, y]) = {. . . , x ← → y, v ← → u, Clm,N−r (v, u) ∩ Clm,N−r (x, y) = ∅} A([v, x], [u, y]) = {. . . , x ← → y, v ← → u, Clm,N−r (v, u) ∩ γup(Clm,N−r(x, y)) = ∅}

v u x γup y Hm HN−r Clm,N−r(x, y) Clm,N−r(v, u)

Introduction Sketch of Proof Summary

Two Disjoint Boundary Clusters

A([v, x], [u, y]) = {. . . , x ← → y, v ← → u, Clm,N−r (v, u) ∩ Clm,N−r (x, y) = ∅} A([v, x], [u, y]) = {. . . , x ← → y, v ← → u, Clm,N−r (v, u) ∩ γup(Clm,N−r(x, y)) = ∅}

Exploration of Clm,N−r(v, u) and γup(Clm,N−r(x, y)) uses different bonds. v u x γup y Hm HN−r Clm,N−r(x, y) Clm,N−r(v, u)

Introduction Sketch of Proof Summary

Two Disjoint Boundary Clusters

A([v, x], [u, y]) = {. . . , x ← → y, v ← → u, Clm,N−r (v, u) ∩ Clm,N−r (x, y) = ∅} A([v, x], [u, y]) = {. . . , x ← → y, v ← → u, Clm,N−r (v, u) ∩ γup(Clm,N−r(x, y)) = ∅}

Exploration of Clm,N−r(v, u) and γup(Clm,N−r(x, y)) uses different bonds. = ⇒ We can sample the clusters independently: Bp(A(. . . )) = ⊗Bp(A(. . . )) v u x γup y Hm HN−r Clm,N−r(x, y) Clm,N−r(v, u)

Introduction Sketch of Proof Summary

The Structure of One Cluster

We would like to have some geometric decomposition of a cluster Cl(x, y).

Introduction Sketch of Proof Summary

The Structure of One Cluster

We would like to have some geometric decomposition of a cluster Cl(x, y). Assume x = 0, y = (N, 0).

Introduction Sketch of Proof Summary

The Structure of One Cluster

We would like to have some geometric decomposition of a cluster Cl(x, y). Assume x = 0, y = (N, 0). Definition: Hm is an α-cone-cut-line and z ∈ Hm is an α-cone-cut-point of Cl(x, y) if Cl(x, y) ⊆ (z − Cα) ∪ (z + Cα) . where: Cα = {x = (t, x) : |x| αt}. x y z z + Cα z − Cα Hm

Introduction Sketch of Proof Summary

The Structure of One Cluster

We would like to have some geometric decomposition of a cluster Cl(x, y). Assume x = 0, y = (N, 0). Definition: Hm is an α-cone-cut-line and z ∈ Hm is an α-cone-cut-point of Cl(x, y) if Cl(x, y) ⊆ (z − Cα) ∪ (z + Cα) . where: Cα = {x = (t, x) : |x| αt}. x y z z + Cα z − Cα Hm There is a well-defined irreducible decomposition of Cl(x, y) along cone-cut-lines:

Introduction Sketch of Proof Summary

Cluster Decomposition - An Illustration

x = 0 y = (N, 0)

Introduction Sketch of Proof Summary

Cluster Decomposition - An Illustration

x = 0 y = (N, 0)

Introduction Sketch of Proof Summary

Cluster Decomposition - An Illustration

x = 0 y = (N, 0) Γf Γ1 Γ2 Γ3 Γb Cbf

3

Introduction Sketch of Proof Summary

Cluster Decomposition - An Illustration

x = 0 y = (N, 0) σf Γf σ1 Γ1 σ2 Γ2 σ3 Γ3 σb Γb Sbf

3

Cbf

3

Introduction Sketch of Proof Summary

Cluster Decomposition - An Illustration

x = 0 y = (N, 0) σf Γf σ1 Γ1 σ2 Γ2 σ3 Γ3 σb Γb Sbf

3

Cbf

3

Introduction Sketch of Proof Summary

Cluster Decomposition - cont’d

Let F be the set of all pieces that can appear between any two succeeding cone-cut-points in any decomposition.

Introduction Sketch of Proof Summary

Cluster Decomposition - cont’d

Let F be the set of all pieces that can appear between any two succeeding cone-cut-points in any decomposition. A piece Γ ∈ F comes with an offset vector σ = σ(Γ), which is the difference between its left and right cut-points.

Introduction Sketch of Proof Summary

Cluster Decomposition - cont’d

Let F be the set of all pieces that can appear between any two succeeding cone-cut-points in any decomposition. A piece Γ ∈ F comes with an offset vector σ = σ(Γ), which is the difference between its left and right cut-points. Similarly, define Fb and Ff for all possible initial and final pieces. σ Γ σf Γf σb Γb Fb = Ff = F =

Introduction Sketch of Proof Summary

Cluster Decomposition - cont’d

Then, Bp(Cl(x, y) = ∅) = Bp(no cone-cut-lines) +

• Γ•

Bp({Γb})Bp({Γ1}) · · · · · Bp({Γn})Bp({Γf}) where the sum is over all:

• Γb ∈ Fb,
• Γi ∈ F for i = 1, . . . , n and all n,
• Γf ∈ Ff,

such that: y = x + σb + σ1 + · · · + σn + σf .

Introduction Sketch of Proof Summary

Cluster Decomposition - cont’d

Then, Bp(Cl(x, y) = ∅) = Bp(no cone-cut-lines) +

• Γ•

Bp({Γb})Bp({Γ1}) · · · · · Bp({Γn})Bp({Γf}) where the sum is over all:

• Γb ∈ Fb,
• Γi ∈ F for i = 1, . . . , n and all n,
• Γf ∈ Ff,

such that: y = x + σb + σ1 + · · · + σn + σf . In fact, Bp(Cl(x, y) = ∅)∼

• Γ•

Bp({Γb})Bp({Γ1}) · · · · · Bp({Γn})Bp({Γf }) .

Introduction Sketch of Proof Summary

Growing a cluster

We can grow Cl(x, y) iteratively:

• 1. Draw Γb ∈ Fb w.p. Bp({Γb}).

Draw Γf ∈ Ff w.p. Bp({Γf}).

• 2. Set C0 = ∅, S0 = 0.
• 3. At each step m:

3.1 Draw Γm ∈ F w.p. Bp({Γm}). 3.2 Cm = Cm−1 ∨ Γm ; Sm = Sm−1 + σm. 3.3 Cbf

m = {x} ∨ Γb ∨ Cm ∨ Γf.

Sbf

m = x + σb + Sm + σf.

Introduction Sketch of Proof Summary

Growing a cluster

We can grow Cl(x, y) iteratively:

• 1. Draw Γb ∈ Fb w.p. Bp({Γb}).

Draw Γf ∈ Ff w.p. Bp({Γf}).

• 2. Set C0 = ∅, S0 = 0.
• 3. At each step m:

3.1 Draw Γm ∈ F w.p. Bp({Γm}). 3.2 Cm = Cm−1 ∨ Γm ; Sm = Sm−1 + σm. 3.3 Cbf

m = {x} ∨ Γb ∨ Cm ∨ Γf.

Sbf

m = x + σb + Sm + σf.

If Px is the measure for this decorated RW, then Bp(Cl(x, y) = ∅, ∃cone-cut-lines) = Px(∃n s.t Sbf

n = y).

Introduction Sketch of Proof Summary

Growing a Cluster - Illustration

x = 0 y = (N, 0) σf Γf σb Γb Sbf Cbf

Introduction Sketch of Proof Summary

Growing a Cluster - Illustration

x = 0 y = (N, 0) σf Γf σ1 Γ1 σb Γb Sbf

1

Cbf

1

Introduction Sketch of Proof Summary

Growing a Cluster - Illustration

x = 0 y = (N, 0) σf Γf σ1 Γ1 σ2 Γ2 σb Γb Sbf

2

Cbf

2

Introduction Sketch of Proof Summary

Growing a Cluster - Illustration

x = 0 y = (N, 0) σf Γf σ1 Γ1 σ2 Γ2 σ3 Γ3 σb Γb Sbf

3

Cbf

3

Introduction Sketch of Proof Summary

Growing a Cluster - Illustration

x = 0 y = (N, 0) σf Γf σ1 Γ1 σ2 Γ2 σ3 Γ3 σb Γb Sbf

3

Cbf

3

x = 0 y = (N, 0) σf Γf σb Γb Sbf Cbf

Introduction Sketch of Proof Summary

Growing a Cluster - Illustration

x = 0 y = (N, 0) σf Γf σ1 Γ1 σ2 Γ2 σ3 Γ3 σb Γb Sbf

3

Cbf

3

x = 0 y = (N, 0) σf Γf σ1 Γ1 σb Γb Sbf

1

Cbf

1

Introduction Sketch of Proof Summary

Growing a Cluster - Illustration

x = 0 y = (N, 0) σf Γf σ1 Γ1 σ2 Γ2 σ3 Γ3 σb Γb Sbf

3

Cbf

3

x = 0 y = (N, 0) σf Γf σ1 Γ1 σ2 Γ2 σb Γb Sbf

2

Cbf

2

Introduction Sketch of Proof Summary

Growing a Cluster - Illustration

x = 0 y = (N, 0) σf Γf σ1 Γ1 σ2 Γ2 σ3 Γ3 σb Γb Sbf

3

Cbf

3

x = 0 y = (N, 0) σf Γf σ1 Γ1 σ2 Γ2 σ3 Γ3 σb Γb Sbf

3

Cbf

3

Introduction Sketch of Proof Summary

Normalizing Steps

Γ• are not properly defined:

Γ∈F• Bp({Γ}) < 1.

i.e. the random walk can die.

Introduction Sketch of Proof Summary

Normalizing Steps

Γ• are not properly defined:

Γ∈F• Bp({Γ}) < 1.

i.e. the random walk can die. Tilt to make proper: P(Γ• = γ) = eKσ(γ),e1 P(Γ• = γ). P(Γb,f = γ) = k1eKσ(γ),e1 P(Γb,f = γ).

Introduction Sketch of Proof Summary

Normalizing Steps

Γ• are not properly defined:

Γ∈F• Bp({Γ}) < 1.

i.e. the random walk can die. Tilt to make proper: P(Γ• = γ) = eKσ(γ),e1 P(Γ• = γ). P(Γb,f = γ) = k1eKσ(γ),e1 P(Γb,f = γ). K = ξp(e1) !!

Introduction Sketch of Proof Summary

Normalizing Steps

Γ• are not properly defined:

Γ∈F• Bp({Γ}) < 1.

i.e. the random walk can die. Tilt to make proper: P(Γ• = γ) = eKσ(γ),e1 P(Γ• = γ). P(Γb,f = γ) = k1eKσ(γ),e1 P(Γb,f = γ). K = ξp(e1) !! Then: Bp(Cl(x, y) = ∅) ∼ Ke−ξp(e1)y−x,e1Px(∃n s.t Sbf

n = y).

Introduction Sketch of Proof Summary

Back to the Two Boundary Clusters

Thus, ⊗ Bp(A([v, x], [u, y])) ∼ Ke−2ξp(e1)y−x,e1 × ⊗Px,v

• ∃n1, n2 s.t Sbf,1

n1

= y, Sbf,2

n2

= u and Cbf,1

n1

∩ Cbf,2

n2

= ∅

• ,

where under ⊗Px,v two decorated RWs: Cbf,1

• , Cbf,1
• evolve

independently starting from (x, v). The prefactor will come, therefore, from the asymptotics of the probability of the RW event above.

Introduction Sketch of Proof Summary

Asymptotics of No Intersection

Let N = y − x, e1 = u − v, e1. Then Lemma As N → ∞, uniformly in x, y, u, v ∈ D(N): ⊗ Px,v

• ∃n1, n2 s.t Sbf,1

n1

= y, Sbf,2

n2

= u and Cbf,1

n1

∩ Cbf,2

n2

= ∅

• ∼ U(x − v)U(y − u) 1

N2 . U(·) has polynomial growth.

Introduction Sketch of Proof Summary

Proof of Intersection Asymptotics

• 1. One (unbiased) RW staying positive:

P0(Sn = y, Sk > 0 ∀k) ∼ U(y) n P0(Sn = y) ; y ≪ n−1/2

Combinatorial proof by Alili-Doney (97).

Introduction Sketch of Proof Summary

Proof of Intersection Asymptotics

• 1. One (unbiased) RW staying positive:

P0(Sn = y, Sk > 0 ∀k) ∼ U(y) n P0(Sn = y) ; y ≪ n−1/2

Combinatorial proof by Alili-Doney (97).

• 2. Starting from x > 0:

Px(Sn = y, Sk > 0 ∀k) ∼ U(x)U(y) n Px(Sn = y) ; x, y ≪ n−1/2

Introduction Sketch of Proof Summary

Proof of Intersection Asymptotics

• 1. One (unbiased) RW staying positive:

P0(Sn = y, Sk > 0 ∀k) ∼ U(y) n P0(Sn = y) ; y ≪ n−1/2

Combinatorial proof by Alili-Doney (97).

• 2. Starting from x > 0:

Px(Sn = y, Sk > 0 ∀k) ∼ U(x)U(y) n Px(Sn = y) ; x, y ≪ n−1/2

• 3. Two independent RWs not intersecting:

⊗Px,v(S1

n = y, S2 n = u, S1 k > S2 k ∀k)

∼ U(x − v)U(y − u) n Px(Sn = y)Pv(Sn = u) ; x, v, y, u ≪ n−1/2

Introduction Sketch of Proof Summary

Proof of Intersection Asymptotics - cont’d

4 Add random time steps:

⊗ P(0,x),(0,v)

• ∃n1, n2 s.t S1

n1 = (N, y), S2 n2 = (N, u), S1

• ∩ S2
• = ∅
• ∼ U(x − v)U(y − u)

N 1 √ N 1 √ N ; x, v, y, u ≪ N−1/2

Introduction Sketch of Proof Summary

Proof of Intersection Asymptotics - cont’d

4 Add random time steps:

⊗ P(0,x),(0,v)

• ∃n1, n2 s.t S1

n1 = (N, y), S2 n2 = (N, u), S1

• ∩ S2
• = ∅
• ∼ U(x − v)U(y − u)

N 1 √ N 1 √ N ; x, v, y, u ≪ N−1/2

5 By entropic repulsion: P(0,x),(0,v)

• C1

n1 ∩ C2 n2 = ∅

• S1
• ∩ S2
• = ∅, . . .
• → const

Introduction Sketch of Proof Summary

Proof of Intersection Asymptotics - cont’d

4 Add random time steps:

⊗ P(0,x),(0,v)

• ∃n1, n2 s.t S1

n1 = (N, y), S2 n2 = (N, u), S1

• ∩ S2
• = ∅
• ∼ U(x − v)U(y − u)

N 1 √ N 1 √ N ; x, v, y, u ≪ N−1/2

5 By entropic repulsion: P(0,x),(0,v)

• C1

n1 ∩ C2 n2 = ∅

• S1
• ∩ S2
• = ∅, . . .
• → const

6 Initial and final piece only change the constant:

⊗ P(0,x),(0,v)

• ∃n1, n2 s.t S1,bf

n1

= (N, y), S2,bf

n2

= (N, u), C1,bf

n1

∩ C2,bf

n2

= ∅

• ∼ U(x − v)U(y − u)

N2 ; x, v, y, u ≪ N−1/2

Introduction Sketch of Proof Summary

Outline

Introduction Percolation on Zd Logarithmic Asymptotics of Connectivities Sharp Asymptotics of Connectivities Sketch of Proof Setup Geometry of Finite Connections The Structure of a Cluster Asymptotics for No Intersection of Two Decorated RWs Summary

Introduction Sketch of Proof Summary

Open Questions

• Do the same for 2D Random Cluster (FK) model (q = 2).
• Supercritical percolation on Zd for d 3

Show OZ behavior for finite connectivities in all directions and all p > pc(d).

Introduction Sketch of Proof Summary

Thank You

Thank you.