An introduction to random interlacements Artem Sapozhnikov - - PowerPoint PPT Presentation

An introduction to random interlacements

Artem Sapozhnikov

University of Leipzig

23-25 May 2016

Example: Random walk on 3-dimensional torus

◮ (Xk)k≥0, a nearest neighbor random walk on Z3 n = (Z/nZ)3, ◮ Vu n = Z3 n \ {X0, . . . , Xun3}, u > 0, ◮ red = the largest connected component of Vu n , ◮ blue = the second largest connected component of Vu n ,

Simulation by D. Windisch

Example: Random walk on 3-dimensional torus

◮ (Xk)k≥0, a nearest neighbor random walk on Z3 n = (Z/nZ)3, ◮ Vu n = Z3 n \ {X0, . . . , Xun3}, u > 0, ◮ red = the largest connected component of Vu n , ◮ blue = the second largest connected component of Vu n ,

Simulation by D. Windisch

Example: Random walk on 3-dimensional torus

◮ (Xk)k≥0, a nearest neighbor random walk on Z3 n = (Z/nZ)3, ◮ Vu n = Z3 n \ {X0, . . . , Xun3}, u > 0, ◮ red = the largest connected component of Vu n , ◮ blue = the second largest connected component of Vu n ,

Simulation by D. Windisch

Example: Random walk on 3-dimensional torus

◮ (Xk)k≥0, a nearest neighbor random walk on Z3 n = (Z/nZ)3, ◮ Vu n = Z3 n \ {X0, . . . , Xun3}, u > 0, ◮ red = the largest connected component of Vu n , ◮ blue = the second largest connected component of Vu n ,

Simulation by D. Windisch

Example: Random walk on 3-dimensional torus

◮ (Xk)k≥0, a nearest neighbor random walk on Z3 n = (Z/nZ)3, ◮ Vu n = Z3 n \ {X0, . . . , Xun3}, u > 0, ◮ red = the largest connected component of Vu n , ◮ blue = the second largest connected component of Vu n ,

Simulation by D. Windisch

Example: Random walk on 3-dimensional torus

◮ (Xk)k≥0, a nearest neighbor random walk on Z3 n = (Z/nZ)3, ◮ Vu n = Z3 n \ {X0, . . . , Xun3}, u > 0, ◮ red = the largest connected component of Vu n , ◮ blue = the second largest connected component of Vu n ,

Simulation by D. Windisch

Example: Random walk on 3-dimensional torus

◮ (Xk)k≥0, a nearest neighbor random walk on Z3 n = (Z/nZ)3, ◮ Vu n = Z3 n \ {X0, . . . , Xun3}, u > 0, ◮ red = the largest connected component of Vu n , ◮ blue = the second largest connected component of Vu n ,

Simulation by D. Windisch

Example: Random walk on 3-dimensional torus

◮ (Xk)k≥0, a nearest neighbor random walk on Z3 n = (Z/nZ)3, ◮ Vu n = Z3 n \ {X0, . . . , Xun3}, u > 0, ◮ red = the largest connected component of Vu n , ◮ blue = the second largest connected component of Vu n ,

Simulation by D. Windisch

Questions:

Vu

n = Z3 n \ {X0, . . . , Xun3}, u > 0, ◮ Structural phase transition: ∃uc ∈ (0, ∞) such that

◮ if u < uc, then

lim

n→∞ P

• |Cmax(Vu

n )| > cn3

= 1,

◮ if u > uc, then

lim

n→∞ P

• |Cmax(Vu

n )| ≪ n3

= 1.

◮ The second largest connected component of Vu n is small:

lim

n→∞ P

• |C(2)(Vu

n )| ≪ n3

= 1.

◮ Geometric properties of Cmax(Vu n ), for u < uc.

First results:

Let d ≥ 3, u > 0, X0 ∼ U(Zd

n), and

Vu

n = Zd n \ {X0, . . . , Xund}.

Then:

◮ There exist c1 = c1(d) > 0, c2 = c2(d) > 0 such that

e−c1u ≤ lim inf

n

P [0 ∈ Vu

n ] ≤ lim sup n

P [0 ∈ Vu

n ] ≤ e−c2u. ◮ If d is large enough and u small enough then there exists

c = c(d, u) > 0 such that lim

n→∞ P

• |Cmax(Vu

n )| > cnd

= 1.

Benjamini-Sznitman (JEMS, ’08)

Main obstruction:

Complexity of self intersections in the random walk trace.

Random interlacements as local limit:

Let X[a,b] = {Xa, Xa+1, . . . , Xb}.

• 1. Random interlacements at level u > 0, Iu, is a random

subgraph of Zd such that for each finite K ′ ⊆ K ⊂ Zd, lim

n→∞ P

• X[0,und] ∩ K = K ′

= P

• Iu ∩ K = K ′

.

Sznitman (AM, ’10), Windisch (ECP, ’11)

• 2. For any u > 0, ε ∈ (0, 1), δ ∈ (0, 1), and λ, there exists a

coupling of the random walk (Xk)k≥0 and the random interlacements Iu(1−ε), Iu(1+ε), so that P

• Iu(1−ε) ∩ [0, Nδ]d ⊆ X[0,uNd] ∩ [0, Nδ]d ⊆ Iu(1+ε)

≥ 1 − N−λ.

Teixeira-Windisch (CPAM, ’11)

Some results using coupling:

• 1. For d ≥ 3, there exist u1 ≤ u2 such that

◮ if u > u2 then for some λ = λ(u),

lim

n→∞ P

• |Cmax(Vu

n )| ≤ logλ n

• = 1,

◮ if u < u1 then there exists c = c(u) such that

lim

n→∞ P

• |Cmax(Vu

n )| ≥ cnd

= 1.

• 2. For d ≥ 5, there exist u3 > 0 such that for all u < u3,

◮ there exists η(u) > 0 such that for all ε > 0,

lim

n→∞ P

• |Cmax(Vu

n )|

nd − η(u)

• > ε
• = 0,

◮ there exists λ(u) such that

lim

n→∞ P

• |C(2)(Vu

n )| ≤ logλ n

• = 1.

Teixeira-Windisch (CPAM, ’11)

Understanding local picture, I

Basic facts about random walk:

• 1. Random walk on Zd is transient iff d ≥ 3. P´
• lya (MA, ’21)

= ⇒ Random walk is locally transient on Zd

n, d ≥ 3.

• 2. Lazy random walk on Zd

n is rapidly mixing,

• y∈Zd

n

• P [Xk = y] − 1

nd

• ≤ C exp
• −c k

n2

• .

see, e.g., Saloff-Coste, ’97 or Levin-Peres-Wilmer, ’09

= ⇒ The mixing time is of order n2.

• 3. Exit time from a ball of radius r is of order r2,
• Prob. to visit center before leaving ball ∼ cd(x2−d − r 2−d),

see, e.g., Lawler, ’91

= ⇒ Random walk on Zd

n started far away from 0

will need about nd steps to hit 0.

Understanding local picture, II

K finite subset of Zd How does the random walk (re)visit K?

• 1. if X0 ∼ U(Zd

n), then X0 is likely to be far from K

= ⇒ average time to visit K is of order nd,

• 2. after visiting K it takes about n2 steps to move far from K
• 3. after n2 steps, the random walk “forgets” where it started.

Thus,

• 1. locally transient

= ⇒ returns to K are by means of rare excursions,

• 2. rapid mixing

= ⇒ excursions are almost i.i.d., entrance points ∼ eK(·),

• 3. hit K in n2 steps with prob. ∼ n2−dcap(K)

= ⇒ number of excursions up to und is almost Poi(u cap(K)).

Understanding local picture, III

◮ Above heurisitcs suggests that as n → ∞, all the “almost”s

can be neglected.

◮ By doing this, we will define the random interlacements:

• 1. For each finite K, define N ∼ Poi(u cap(K)).
• 2. Take X (1)

0 , . . . , X (N)

i.i.d., ∝ P·[Xn / ∈ K for n ≥ 1].

• 3. Consider independent simple random walks (X (i)

k )k≥0.

• 4. Define Iu

K as the vertices in K visited by the walks.

• 5. Consistency: for K1 ⊂ K2, Iu

K1 d

= Iu

K2 ∩ K1.

= ⇒ there exists Iu such that Iu

K d

= Iu ∩ K.

Preliminaries on random walks:

◮ Zd, d ≥ 3,

y ∼ x if y − x = 1

◮ (Xn)n≥0 nearest neighbor random walk on Zd, ◮ Green function, G(x, y) = ∞ n=0 Px[Xn = y],

◮ G(x, y) = G(y, x), ◮ G(x, y) = G(0, y − x) =: g(y − x), ◮ g is harmonic on Zd \ {0}:

g(x) = 1 2d

• y∼x

g(y) + δ0x, x ∈ Zd.

◮ g(x) < ∞ for some/all x ∈ Zd iff d ≥ 3 (transience), ◮ for any d ≥ 3 there exist c1 = c1(d) and c2 = c2(d) such that

c1(1 + x)2−d ≤ g(x) ≤ c2(1 + x)2−d, x ∈ Zd.

Green function and hitting probabilities:

◮ For K ⊂ Zd, define

◮ first entrance time to K: HK = inf{n ≥ 0 : Xn ∈ K}, ◮ first hitting time of K:

HK = inf{n ≥ 1 : Xn ∈ K}.

◮ g(0) = 1 P0[ H0=∞].

(Proof: Mean of a geometric random variable.)

◮ For all finite K ⊂ Zd, d ≥ 3, and all x ∈ Zd,

Px[HK < ∞] =

• y∈K

G(x, y) Py[ HK = ∞]. (Proof: Last exit time decomposition.)

Equilibrium measure and capacity:

◮ K finite subset of Zd, ◮ Equilibrium measure of K:

eK(x) =

• Px[

HK = ∞] x ∈ K x / ∈ K.

◮ Capacity of K:

cap(K) =

• x

eK(x).

◮ Capacity measures “hittability” of sets by random walk:

min

y∈K G(x, y) cap(K) ≤ Px[HK < ∞] ≤ max y∈K G(x, y) cap(K). ◮ Capacity of a ball B(n) = {x ∈ Zd : x ≤ n}:

C1 nd−2 ≤ cap(B(n)) ≤ C2 nd−2.

Properties of capacity:

◮ Monotonicity:

cap(K1) ≤ cap(K2), K1 ≤ K2

cap(K1) =

• x∈K1

Px [ HK1 = ∞] =

• x∈K1
• y∈K2

Py [HK1 < ∞, XHK1 = x] Py [ HK2 = ∞] =

• y∈K2

Py [HK1 < ∞] eK2 (y) ≤

• y∈K2

eK2 (y) = cap(K2).

◮ Subadditivity:

cap(K1 ∪ K2) ≤ cap(K1) + cap(K2), K1, K2 ∈ Zd

cap(K1 ∪ K2) =

• x∈K1∪K2

Px [ HK1∪K2 = ∞] ≤

• x∈K1

Px [ HK1 = ∞] +

• x∈K2

Px [ HK2 = ∞] = cap(K1) + cap(K2).

◮ cap({x}) = 1 g(0), cap({x, y}) = 2 g(0)+g(y−x).

Random interlacements, I

Let u > 0 and K a finite subset of Zd, d ≥ 3. Consider independent random variables (and processes):

• 1. NK ∼ Poi(u cap(K))
• 2. X (1)

0 , . . . , X (N)

, P[X (i) = x] =

eK (x) cap(K).

• 3. (X (i)

k )k≥0, simple random walks.

Define

Iu

K = N

• i=1

X (i)

[0,∞) ∩ K

Consistency: for K1 ⊂ K2, Iu

K1 d

= Iu

K2 ∩ K1.

• y∈K2

eK2(y)Py[HK1 < ∞, XHK1 = x] = eK1(x).

= ⇒ there exists Iu such that Iu

K d

= Iu ∩ K.

Random interlacements, II

Iu is called random interlacements at level u Remarks:

◮ The law of Iu ∩ K is explicit, while Iu is defined implicitly

using Kolmogorov extension theorem

◮ It follows from the definition that for each finite K ⊂ Zd,

P[Iu ∩ K = ∅] = P[Iu

K = ∅] = P[NK = 0] = e−u cap(K) ◮ By inclusion-excusion and Dynkin’s π − λ lemma, there exists

at most one random subset S of Zd satisfying the equations P[S ∩ K = ∅] = e−u cap(K), K ⊂ Zd

◮ Therefore, the random interlacements at level u can be

defined as the unique random subset of Zd satisfying P[Iu ∩ K = ∅] = e−u cap(K), K ⊂ Zd.

Basic properties of random interlacements:

◮ Long-range correlations:

Cov(1x∈Iu, 1y∈Iu) ∼ 2u g(0)2 g(y − x) exp

• − 2u

g(0)

• P[x, y /

∈ Iu] − P[x / ∈ Iu] P[y / ∈ Iu] = e−u cap({x,y}) − e−u cap({x}) e−u cap({y}) = exp

• −

2u g(0) + g(y − x)

• − exp
• −

2u g(0)

• .

◮ Shift invariance: for all x ∈ Zd,

(x + Iu) d = Iu

P[(x + Iu) ∩ K = ∅] = P[Iu ∩ (K − x) = ∅] = e−u cap(K−x) = e−u cap(K) = P[Iu ∩ K = ∅].

◮ Ergodicity: For any measurable function F defined on

sugbraphs of Zd and invariant under shifts, F(S) = F(x + S), F(Iu) is almost surely a constant

Poisson point processes: Poisson distribution

◮ X ∼ Poi(λ) if P[X = k] = λk k! e−λ ◮ X ∼ Poi(λ), then E

• zX

= eλ(z−1)

◮ X1, . . . , Xn, . . . indep., Xi ∼ Poi(λi) =

⇒

i Xi ∼ Poi( i λi) ◮ if X ∼ Poi(λ) and Y1, . . . , Yn, . . . indep., P[Yi = yk] = pk,

then Zk = X

i=1 1Yi=yk are indep. and Zk ∼ Poi(λ pk)

Poission point processes: Definition

◮ (Ω, F) measurable space ◮ µ = i δωi is a point measure on Ω, µ[A] = i 1ωi∈A ◮ For sigma finite measure λ on (Ω, F),

random point measure µ is PPP with intensity measure λ if

• 1. for all A ∈ F, µ[A] ∼ Poi(λ[A])
• 2. for A1, . . . , An pairwise disjoint, µ[A1], . . . , µ[An] are indep.

◮ if λ[Ω] < ∞, then

µ =

X

• i=1

δYi, where X ∼ Poi(λ[Ω]), Y1, . . . are indep., P[Yi ∈ ·] = λ[·]

λ[Ω].

Example: Random interlacements

Random interlacements at level u in K is generated as follows:

• 1. NK ∼ Poi(u cap(K))
• 2. X (1)

0 , . . . , X (NK )

, P[X (i) = x] =

eK (x) cap(K).

• 3. X (i) = (X (i)

k )k≥0, simple random walks,

• 4. all random variables and processes are independent,
• 5. Iu

K = N i=1 X (i) [0,∞) ∩ K.

◮ (W+, W+),

space of infinite nearest neighbor paths in Zd

◮ PeK , measure on W+ with PeK [W+] =

x eK(x) Px[W+] = cap(K)

◮ µK,u = NK i=1 δX (i) is PPP on W+ with intensity measure u PeK ◮ Iu K = w∈supp(µK,u) Range(w) ∩ K.

Poission point processes: Properties

µ =

i δwi,

PPP on (Ω, F) with intensity measure λ

◮ A ∈ F, then

1A µ =

• i:wi∈A

δwi is a PPP on Ω with intensity measure 1A λ = λ[· ∩ A]

◮ A1, . . . ∈ F pairwise disjoint, then 1A1µ, . . . are indep. PPP ◮ if ϕ : (Ω, F) → (Ω′, F′) is measurable, then

ϕ ◦ µ =

• i

δϕ(wi) is a PPP on Ω′ with intensity measure ϕ ◦ λ = λ[ϕ−1(·)]

◮ µ1, . . . indep. PPP with intensity measures λi, then

• i µi is PPP with intensity measure

i λi.

Example: Consistency for random interlacements

Consistency: for K1 ⊂ K2, Iu

K1 d

= Iu

K2 ∩ K1.

Let µKi := µKi ,u be a PPP on W+ with intensity measure u PeKi . Then Iu

Ki =

• w∈supp(µKi )

Range(w) ∩ Ki and Iu

K2 ∩ K1 =

• w∈supp(µK2 )

Range(w) ∩ K1 =

• w∈supp(1W1 µK2 )

Range(w) ∩ K1 where W1 is the set of paths from W+ that intersect K1.

◮ 1W1 µK2 is a PPP on W+ with intensity u PeK2 [·, HK1 < ∞] ◮ if ϕ : W1 → W1 such that ϕ(w)(k) = w(HK1 + k) then by

• y∈K2

eK2(y) Py[HK1 < ∞, XHK1 = x] = eK1(x),

ϕ ◦ (u PeK2 [·, HK1 < ∞]) = u PeK1 , hence ϕ ◦ (1W1 µK2) is a PPP on W+ with intensity u PeK1

Example: Mixing

K1, K2 finite subsets of Zd,

• P
• Iu ∩ K1 = K ′

1, Iu ∩ K2 = K ′ 2

• − P
• Iu ∩ K1 = K ′

1

• P
• Iu ∩ K2 = K ′

2

• ≤ C u cap(K1) cap(K2)

dist(K1, K2)d−2 Let K = K1 ∪ K2 and µK := µK,u a PPP with intensity u PeK . Consider

◮ µ1, restriction of µK to paths that only intersect K1 ◮ µ2, restriction of µK to paths that only intersect K2 ◮ µ12, restriction of µK to paths that intersect K1 and K2

µ1, µ2, µ12 are indep. PPP’s and µK = µ1 + µ2 + µ12 = ⇒ RHS ≤ 3 P[µ12[W+] = 0] = 3

• 1 − e−u PeK [HK1 <∞, HK2 <∞]

≤ 3u PeK [HK1 < ∞, HK2 < ∞] ≤ C u cap(K1) cap(K2) dist(K1, K2)d−2

Example: “Infinite divisibility” of random interlacements

u1, u2 > 0, if Iu1 and Iu2 independent, then Iu1 ∪ Iu2 d = Iu1+u2 For each finite K ⊂ Zd, Iui ∩ K

d

=

• w∈supp(µK,ui )

Range(w) ∩ K and µK,u1+u2 = µK,u1 + µK,u2 is PPP on W+ with intensity (u1 + u2) PeK

◮ Elementary (but less intuitive) way to see this:

P [(Iu1 ∪ Iu2) ∩ K = ∅] = P [Iu1 ∩ K = ∅] P [Iu2 ∩ K = ∅] = e−u1 cap(K) e−u2 cap(K) = e−(u1+u2) cap(K) = P

• Iu1+u2 ∩ K = ∅
• ◮ Stochastic domination: for u1 ≤ u2, Iu1 ≤st Iu2

More insight about Iu?

Recall:

◮ Iu ∩ K is explicit, but Iu is implicit

Can we get more insight about Iu?

◮ Can we define Iu explicitly? ◮ Can we couple all Iu’s so that Iu1 ⊆ Iu2 for u1 ≤ u2?

PPP of doubly infinite paths: Spaces

◮ Nearest neighbor doubly infinite paths:

W =

• w : Z → Zd
• w(n) − w(n + 1) = 1, lim

n→∞ w(n) = ∞

• ◮ Time shift: θk : W → W ,

θk(w)(n) = w(n + k)

◮ Equivalence of paths: w1, w2 ∈ W ,

w1 ∼ w2 iff w2 = θk(w1) for some k ∈ Z

◮ Doubly infinite paths modulo time shift:

W ∗ = W / ∼

◮ Coordinate maps: Xn : W → W , Xn(w) = w(n) ◮ Sigma algebras:

W = σ(Xn : n ∈ Z), W∗ = π∗(W)

(π∗ : W → W ∗ – projection)

PPP of doubly infinite paths: Measure on W

Measure on (W , W):

◮ K finite subset of Zd ◮ Measure on paths intersecting K: A, B ∈ W+, x ∈ Zd,

QK [(Xn)n≤0 ∈ A, X0 = x, (Xn)n≥0 ∈ B] = Px

• A |

HK = ∞

• · eK(x) · Px [B]

◮ Note:

◮ QK is uniquely extended to a measure on W, ◮ QK is finite: QK[W ] =

x eK(x) = cap(K) < ∞

= ⇒

1 cap(K)QK is a probability measure on W

PPP of doubly infinite paths: Measure on W ∗

◮ WK – paths in W intersecting K ◮ W ∗

K = π∗(WK) – equivalence classes in W ∗ intersecting K

Theorem (Sznitman, AM ’10)

There exists a unique σ-finite measure ν on (W ∗, W∗) such that ∀A ∈ W∗, A ⊆ W ∗

K :

ν[A] = QK[(π∗)−1(A)]

◮ Equivalent formulation: for all finite K ⊂ Zd,

1W ∗

K ν = π∗ ◦ QK

PPP of doubly infinite paths: Measure on W ∗

Theorem (Sznitman, AM ’10)

There exists a unique σ-finite measure ν on (W ∗, W∗) such that ∀A ∈ W∗, A ⊆ W ∗

K :

ν[A] = QK[(π∗)−1(A)]

Proof:

◮ Consistency: for all K1 ⊂ K2 and F ∈ W∗, F ⊂ W ∗

K1 ⊆ W ∗ K2,

QK2[(π∗)−1(F)] = QK1[(π∗)−1(F)] Equivalently: for all K1 ⊂ K2 and A, B ∈ W+, QK2

• HK1 < ∞, (XHK1+n)n≤0 ∈ A, (XHK1+n)n≥0 ∈ B
• = QK1 [(Xn)n≤0 ∈ A, (Xn)n≥0 ∈ B]

◮ Given Kn ↑ Zd, ν[F] =

n QKn

• (π∗)−1

F ∩ (W ∗

Kn \ W ∗ Kn−1)

PPP of doubly infinite paths: Definition

◮ W ∗ × R+, space of marked doubly infinite paths (mod shift) ◮ ν ⊗ λ, sigma-finite measure on W ∗ × R+

ν ⊗ λ [W ∗

K × [0, u]] = cap(K) · u < ∞

◮ ω = n δ(w∗

n ,un), PPP on W ∗ × R+ with intensity ν ⊗ λ

◮ Random interlacements at level u:

Iu(ω) =

• un<u

Range(w ∗

n ),

if ω =

• n

δ(w ∗

n ,un)

Note:

◮ All Iu’s are defined on the same probability space (space of

point measures on W ∗ × R+)

◮ Iu1 ⊆ Iu2 if u1 ≤ u2 ◮ P [Iu ∩ K = ∅] = P [ω [W ∗

K × [0, u)] = 0] = e−ν⊗λ[W ∗

K ×[0,u)] = e−u cap(K)

(Further) properties of random interlacements:

◮ Shift invariance: for all x ∈ Zd, (x + Iu) d

= Iu

◮ Ergodicity: Any shift invariant property of subsets of Zd

either a.s. holds for Iu or a.s. does not hold for Iu

◮ Connectedness: for each u > 0, Iu is a.s. connected

◮ immediate on Z3 and Z4, since two random walk ranges a.s.

intersect, and Iu is the range of countably many random walks

◮ for general d ≥ 3, using Burton-Keane type argument or a

more refined analysis of intersections of random walks

◮ any x, y ∈ Iu can be connected in Iu through at most ⌈ d

2 ⌉

interlacement trajectories R´

ath-S (ALEA, ’12), Procaccia-Rosenthal (ECP, ’11)

◮ No finite energy property:

0 < P [x ∈ Iu | σ(1y∈Iu : y = x)] < 1, P-a.s.

◮ Long-range correlations: Cov(1x∈Iu, 1y∈Iu) ≍ x − y2−d

Comparison with Bernoulli percolation:

Bernoulli percolation, Bp,

◮ p ∈ [0, 1] ◮ P [Bp ∩ K = K1] = p|K1| (1 − p)|K\K1| ◮ Iu does not stochastically dominate Bp:

P

• Bp ∩ [1, n]d = ∅
• = (1 − p)nd ≪ e−u cap([1,n]d ) = P
• Iu ∩ [1, n]d = ∅
• ◮ Bp does not stochastically dominate Iu:

P

• Bp ⊃ [1, n]d

= pnd ≪ 1 2 e−(ln n)2 nd−2 ≤ P

• Iu ⊃ [1, n]d

P

• Iu ⊃ [1, n]d

≥ P

• Iu ⊃ [1, n]d
• N = Cnd−2 log n
• · P
• N = Cnd−2 log n
• ≥

  1 −

• x∈[1,n]d

P

• x /

∈ Iu

• N = Cnd−2 log n
• 

  · e−(log n)2nd−2 =   1 −

• x∈[1,n]d
• 1 −

cap(0) cap([1, n]d ) Cnd−2 log n    · e−(log n)2nd−2 ≥ 1 2 e−(log n)2nd−2 .

Vacant set of random interlacements:

Vu = Zd \ Iu, u > 0

◮ P [K ⊂ Vu] = e−u cap(K),

for all finite K ⊂ Zd

◮ coupling =

⇒ Vu2 ⊆ Vu1 for u1 ≤ u2

◮ view Vu as random subgraph of Zd with edges between

nearest neighbor vertices of Vu

◮ shift-invariance, ergodicity, long-range correlations ◮ Vu is not connected almost surely ◮ connected component of x in Vu – cluster of x, Cu(x)

Percolation of Vu:

◮ Percolation probability:

η(u) = P[|Cu(0)| = ∞]

◮ Percolation threshold:

u∗ = sup {u : η(u) > 0}

◮ u < u∗ =

⇒ η(u) > 0

(ergodicity)

= ⇒ P[∃ infinite cluster in Vu] = 1

◮ u > u∗ =

⇒ P[∃ infinite cluster in Vu] ≤

x P[|Cu(x)| = ∞] = 0 ◮ for all u, P[∃ at most one infinite cluster in Vu] = 1

Note: no finite energy property!

Percolation phase transition:

Theorem (Sznitman, AM ’10, Sidoravicius-Sznitman, CPAM ’09)

For all d ≥ 3, u∗ ∈ (0, ∞).

◮ Main obstruction: strong (polynomial) correlations ◮ Original proofs rely on so-called decoupling inequalities ◮ We follow the recent short proof of (R´

ath, ECP ’15), based on multiscale analysis from (Sznitman, IM ’12)

Proof of u∗ < ∞, I:

u∗ < ∞ ⇐ ⇒ ∃u < ∞ η(u) = 0 ⇐ = lim inf

n

P

• S(0, n)

Vu

← → S(0, 2n)

• = 0

where S(x, k) = {y ∈ Zd : |y − x| = k}

Proof of u∗ < ∞, II: Multiscale analysis

◮ Scales:

◮ L0 ≥ 1,

Ln = 6 · Ln−1 = 6n · L0

◮ Ln = Ln · Zd

◮ Trees:

◮ T(n) = {1, 2}n, n ≥ 0

(T(0) = ∅)

◮ Tn = ∪n

k=0T(k), diadic tree of depth n

◮ two children of m = (ξ1, . . . , ξk) ∈ T(k): mi = (ξ1, . . . , ξk, i)

◮ Embedding of trees in Zd:

◮ ϕ : Tn → Zd is a proper embedding with root x ∈ Ln if ◮ ϕ(∅) = x ◮ ∀ 0 ≤ k ≤ n, m ∈ T(k),

ϕ(m) ∈ Ln−k

◮ ∀ 0 ≤ k < n, m ∈ T(k), i ∈ {1, 2},

ϕ(mi) ∈ S(ϕ(m), i Ln−k)

◮ Λn,x,

set of proper embeddings with root x

◮ |Λn,x| = C 2n

d

Proof of u∗ < ∞, III:

P

• S(0, Ln)

Vu

← → S(0, 2Ln)

• L0=1

≤ P

• ∃ϕ ∈ Λn,0 : ∀m ∈ T(n) ϕ(m) ∈ Vu

≤

• ϕ∈Λn,0

P

• ∀m ∈ T(n) ϕ(m) ∈ Vu

≤ C 2n

d · max ϕ∈Λn,0 P

• ∀m ∈ T(n) ϕ(m) ∈ Vu

◮ ∀ϕ ∈ Λn,0, Sϕ :=

m∈T(n) ϕ(m) is (uniformly) spread-out

= ⇒ cap(Sϕ) ≥ cd · 2n

◮ P

• ∀m ∈ T(n) ϕ(m) ∈ Vu

= e−u cap(Sϕ) ≤ e−u cd 2n

◮ P

• S(0, Ln)

Vu

← → S(0, 2Ln)

• ≤ C 2n

d · e−u cd 2n −

− − − − − − − →

n→∞,u large 0

Proof of u∗ > 0, I:

u∗ > 0 ⇐ ⇒ ∃u > 0 η(u) > 0 ⇐ = P

• 0 Vu∩F

← → ∞

• > 0

where F = Z2 × {0}d−2 — plane in Zd

Duality: P

• 0 Vu∩F

← → ∞

• > 0

⇐ ⇒ P [∃ ∗-circuit around 0 in Iu ∩ F] < 1 ⇐ = P

• S(0, Ln)

∗-path in Iu∩F

← → S(0, 2Ln)

• ≤ 2−2n

Proof of u∗ > 0, II:

◮ Multiscale analysis in the plane F (proper embedding of trees in F)

P

• S(0, Ln)

∗-path in Iu∩F

← → S(0, 2Ln)

• ≤

P

• ∃ϕ ∈ Λn,0 : ∀m ∈ T(n) S(ϕ(m), L0) ∩ Iu = ∅
• ≤
• ϕ∈Λn,0

P

• ∀m ∈ T(n) S(ϕ(m), L0) ∩ Iu = ∅
• ≤

C 2n

2 · max ϕ∈Λn,0 P

• ∀m ∈ T(n) S(ϕ(m), L0) ∩ Iu = ∅

Proof of u∗ > 0, III:

◮ Union of frames:

Sϕ =

• m∈T(n)

S(ϕ(m), L0)

◮ Random interlacements inside the frames:

Iu ∩ Sϕ =

N

• i=1

X (i)

[0,∞) ∩ Sϕ

where

◮ N ∼ Poi(u cap(Sϕ)) ◮ X (i) independent SRW started from eSϕ cap(Sϕ)

◮ Zi — number of frames visited by X (i),

i.i.d.

◮ N

i=1 Zi ≥ number of frames intersected by Iu,

compound Poisson

◮ cap(S(0, L0)) ≤

CdL0 d ≥ 4 Cd

L0 log L0

d = 3 ≪ cap(B(0, L0))

(L0 large)

= ⇒ Zi ≤st geometric r.v. with small parameter

(u small)

= ⇒ P N

i=1 Zi ≥ 2n

≤ (2 C2)−2n

Further properties of Iu:

Strong-connectivity: for all d ≥ 3 and u > 0, P

• ∀x, y ∈ Iu ∩ B(0, n),

x

Iu∩B(0,2n)

← → y

• ≥ 1 − C exp
• −n

1 6

• R´

ath-S (ECP ’11) = ⇒ P [Iu is transient] = 1 = ⇒ ∀ǫ > 0, a.e.-ω, pn(0, 0) ≤ C n− d

2 +ǫ

(pn is the transition density of random walk on Iu)

= ⇒ ∀ǫ > 0, P

• du(x, y) ≤ |x − y|1+ǫ

x, y ∈ Iu ≥ 1 − C e−|x−y|δ

(du is the graph distance in Iu)

Further properties of Iu:

Decoupling inequalities for monotone events:

◮ Let Ai (Bi) be decreasing (increasing) events in B(xi, 10L) ◮ R large, u ≥ (1 + R−ǫ)ˆ

u, |x1 − x2| ≥ RL,

◮ then

Pu[A1 ∩ A2] ≤ Pˆ

u[A1] Pˆ u[A2] + e−(log L)2

Pˆ

u[B1 ∩ B2] ≤ Pu[B1] Pu[B2] + e−(log L)2

Sznitman (AM ’10)

= ⇒ quenched Gaussian heat kernel bounds: ∀ n large, a.e.-ω, x, y ∈ Iu, c n− d

2 e−C |x−y|2 n

≤ pn(x, y) + pn−1(x, y) ≤ C n− d

2 e−c |x−y|2 n

= ⇒ quenched invariance principle: For a.e.-ω,

Xn2t n

⇒ Bt,

(Bt isotropic BM with deterministic covariance)

= ⇒ quenched local CLT, Harnack inequalities, isoperimetric inequalities, etc...

Further properties of Vu: Conjectures

◮ u > u∗ (subcritical regime):

P

• Vu

← → S(0, n)

• ≍

e−cn d ≥ 4 e−c

n log n

d = 3

Note that P [[0, n] ⊂ Vu] = e−u cap([0,n]) ≍      e−cn d ≥ 4 e

−c n log n

d = 3

◮ u < u∗ (supercritical regime):

◮ Unique infinite cluster: quenched invariance principle, Gaussian

heat kernel bounds, etc.

◮ Finite clusters: P

• Vu

← → S(0, n), |Cu(0)| < ∞

• ≤ C e−nδ

Further properties of Vu: Subcritical regime

Auxiliary threshold: u∗∗ = inf

• u > 0 : lim inf

n

P

• S(0, n)

Vu

← → S(0, 2n)

• = 0
• Note: (a) u∗∗ ≥ u∗,

(b) We’ve proved u∗∗ < ∞.

Theorem (Popov-Teixeira (EJM ’15))

For all u > u∗∗,

C1 e−c1n d ≥ 4 C1 e−c1

n log n

d = 3 ≤ P

• Vu

← → S(0, n)

• ≤

C2 e−c2n d ≥ 4 C2 e−c2

n log n

d = 3

Conjecture: u∗ = u∗∗.

Further properties of Vu: Supercritical regime

Regime of local uniqueness: u ∈ U iff (a) P Vu ∩ B(0, n) contains connected component of diam ≥ n

• ≥ 1 − e−(log n)2

(b) P all connected components of Vu ∩ B(0, n)

• f diam ≥

n 10 are connected in B(0, 2n)

• ≥ 1 − e−(log n)2

Note: U ⊂ (0, u∗].

Theorem (Teixeira (PTRF’11), Drewitz-R´

ath-S (AIHP’14)) There exists u1 > 0 such that (0, u1] ⊂ U.

Theorem (Procaccia-Rosenthal-S (PTRF’16), S (AP’16))

For all u ∈ U, for the unique infinite cluster of Vu, the quenched invariance principle, quenched Gaussian heat kernel bounds, local CLT etc hold. Conjecture: U = (0, u∗).