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Dependent percolation:some examples and multi-scale tools
Maria Eulalia Vares
UFRJ, Rio de Janeiro, Brasil
8th Purdue International Symposium, June 2012
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I. Motivation
Classical Ising model (spins ±1) in two dimensions
H(σ) = −∑
|x−y|=1
J{x,y}σxσy
σ ∈ {−1, +1}Λ; Λ ⊂ Z2 (finite)
J{x,y} ≡ J > 0 ferromagnetic interaction; x, y ∈ Z2
µΛ(σ) =1
ZΛ
exp {−βHΛ(σ)} (probability measure)
Phase transition at sufficiently low temperature
βc =1
2Jlog(1 +
√2)
multiple limits of µΛ as Λ → Z2 if β > βc.
Peierls (1936); Onsager (1944); Yang (1952)
quad1
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I. Motivation
Simulations by Vincent Beffara
quad2
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I. Motivation
Simulations by Vincent Beffara
quad3
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I. Motivation
McCoy and Wu (1968) investigated the effect of random impurities
quad4
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I. Motivation
McCoy and Wu (1968) investigated the effect of random impurities
quad5
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I. Motivation
McCoy and Wu (1968) investigated the effect of random impurities
Disordered ferromagnets – randomly layered environment
quad6
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I. Motivation/Plan
Phase transitions for systems in a (randomly) layered environment
• Growth processes in random environment
• Some forms of coordinate percolation. Winkler’s compatibility problem.
quad7
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I. Motivation/Plan
Phase transitions for systems in a (randomly) layered environment
• Growth processes in random environment
• Some forms of coordinate percolation. Winkler’s compatibility problem.
Common basic tool: multi-scale analysis
One “learns” with simpler hierarchical structures
quad8
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I. Motivation/Plan
Phase transitions for systems in a (randomly) layered environment
• Growth processes in random environment
• Some forms of coordinate percolation. Winkler’s compatibility problem.
Common basic tool: multi-scale analysis
One “learns” with simpler hierarchical structures
Based on joint results with H. Kesten, B. Lima, V. Sidoravicius
quad9
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II. Oriented percolation in a randomly layered environment
On Z2+ := {(x, y) ∈ Z× Z+ : x + y is even}
consider the following oriented (NW, NE) site percolation model:
quad10
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II. Oriented percolation in a randomly layered environment
On Z2+ := {(x, y) ∈ Z× Z+ : x + y is even}
consider the following oriented (NW, NE) site percolation model:
• lines Hi := {(x, y) ∈ Z2+ : y = i} are declared bad or good with probabilities δ and
1− δ respectively, independently of each other.
quad11
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II. Oriented percolation in a randomly layered environment
On Z2+ := {(x, y) ∈ Z× Z+ : x + y is even}
consider the following oriented (NW, NE) site percolation model
• lines Hi := {(x, y) ∈ Z2+ : y = i} are declared bad or good with probabilities δ and
1− δ respectively, independently of each other.
• Sites on good lines are open with probability pG and sites on bad lines are open with
probability pB, all independently of each other.
quad12
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II. Oriented percolation in a randomly layered environment
On Z2+ := {(x, y) ∈ Z× Z+ : x + y is even}
consider the following oriented (NW, NE) site percolation model
• lines Hi := {(x, y) ∈ Z2+ : y = i} are declared bad or good with probabilities δ and
1− δ respectively, independently of each other.
• Sites on good lines are open with probability pG and sites on bad lines are open with
probability pB, all independently of each other.
Regime of interest: 0 < pB < pc < pG, 0 < δ < 1.
What can we say about occurrence of percolation? (a.s. in the environment...)
quad13
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II. Oriented percolation in a randomly layered environment
On Z2+ := {(x, y) ∈ Z× Z+ : x + y is even}
consider the following oriented (NW, NE) site percolation model
• lines Hi := {(x, y) ∈ Z2+ : y = i} are declared bad or good with probabilities δ and
1− δ respectively, independently of each other.
• Sites on good lines are open with probability pG and sites on bad lines are open with
probability pB, all independently of each other.
Regime of interest: 0 < pB < pc < pG, 0 < δ < 1.
What can we say about occurrence of percolation? (a.s. in the environment...)
Natural Guess:
δ large ⇒ no percolation
δ small ⇒ percolation
quad14
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II. Oriented percolation in a randomly layered environment
On Z2+ := {(x, y) ∈ Z× Z+ : x + y is even}
consider the following oriented (NW, NE) site percolation model
• lines Hi := {(x, y) ∈ Z2+ : y = i} are declared bad or good with probabilities δ and
1− δ respectively, independently of each other.
• Sites on good lines are open with probability pG and sites on bad lines are open with
probability pB, all independently of each other.
Regime of interest: 0 < pB < pc < pG, 0 < δ < 1.
What can we say about occurrence of percolation? (a.s. in the environment...)
Natural Guess:
δ large ⇒ no percolation (easy)
δ small ⇒ percolation
quad15
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II. Oriented percolation in a randomly layered environment
quad16
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II. Oriented percolation in a randomly layered environment
quad17
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II. Oriented percolation in a randomly layered environment
C+(0) = {v : ∃ open oriented path from 0 to v}
Let Θ(pG, pB, δ) = P(C+(0) is infinite ).
Theorem (Kesten, Sidoravicius, V.)
∀pG > pc, ∀pB > 0, ∃δ0 > 0 so that Θ(pG, pB, δ) > 0 if δ ≤ δ0. In fact
P(C+(0) is infinite |ξ) > 0 a.s. in ξ (ξ configuration of lines)
.
quad18
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Basic tool: multi-scale analysis
Get started with a very simple situation:
hierarchical model, L large (depending on pG, pB),
ζj :=
{k, if Lk|j but Lk+1 - j ,
0, if L - j.
Replace each entry ζj = k by k consecutive bad lines (shift the rest to the right)
• bad walls of thickness k: k consecutive bad lines;
• such bad walls at distance of order Lk from each other.
quad19
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Basic tool: multi-scale analysis
This immediately calls to the consideration of rescaled lattices, all similar to the original
one, and adapted to deal with bad lines of thickness k.
cL
L
(c suitable, depending on pG)
quad20
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Basic tool: multi-scale analysis
This immediately calls to the consideration of rescaled lattices, all similar to the original
one, and adapted to deal with bad lines of thickness k.
cL
L
(c suitable, depending on pG)
pG > p∗ so that P(event in the picture|seed) ≥ 1− (1− pG)2
quad21
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Basic tool: multi-scale analysis
This immediately calls to the consideration of rescaled lattices, all similar to the original
one, and adapted to deal with bad lines of thickness k.
Renormalized k-sites Sk(i,j), (i, j) ∈ Z2
+.
quad22
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Recursively define the notion of a good k-site Sk being passable from a (k − 1)-seed.
quad23
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Recursively define the notion of a good k-site Sk being passable from a (k − 1)-seed.
It should:
(a) guarantee the existence of open oriented paths that cross it from bottom to top
(b) produce new (k − 1)-seeds on the top,
so that
the existence of oriented k-passable paths implies the existence of open oriented paths at
scale 0, and can essentially be compared with independent site percolation.
quad24
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Recursively define the notion of a good k-site Sk being passable from a (k − 1)-seed.
It should:
(a) guarantee the existence of open oriented paths that cross it from bottom to top
(b) produce new (k − 1)-seeds on the top,
so that
the existence of oriented k-passable paths implies the existence of open oriented paths at
scale 0, and can essentially be compared with independent site percolation. Want estimate
pk ≤ P(
Sk
is s-passable|(k − 1)− seed)
, k ≥ 1, (∗)
quad25
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Recursively define the notion of a good k-site Sk being passable from a (k − 1)-seed.
It should:
(a) guarantee the existence of open oriented paths that cross it from bottom to top
(b) produce new (k − 1)-seeds on the top,
so that
the existence of oriented k-passable paths implies the existence of open oriented paths at
scale 0, and can essentially be compared with independent site percolation. Want estimate
pk ≤ P(
Sk
is s-passable|(k − 1)− seed)
, k ≥ 1, (∗)
Proposition. Given pB > 0 and pG > p∗, there exists L large enough such that (*)
holds with pk = 1− qk and
qk ≤ q2k−1 for all k ≥ 1,
and where q0 = 1− pG.
quad26
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The above estimate clearly implies that for L large enough
P (there exists an infinite cluster starting from the origin) ≥+∞∏
k=0
p3k > 0.
• The main difficulty in pushing the estimate at each step is when one faces the bad
wall of larger mass.
• Planarity plays important role in these arguments.
• Enlarging the seeds and taking some extra care replace p∗ by pc.
quad27
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bad layer of mass k
seedS
k
(i, j)
Dr
K
Dr
Dr
SS(i+1, j-1)
k
(i-1, j-1)
k
k-1S
Ker
nel
of
Sk
quad28
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bad line of mass k-1
reverse partition
reverse partition
forward partition
S(i, j)
k-1
S(i, j+1)
k-1
Sk-1
(i, j+1)
bad line of mass k-2
bad line of mass k-2
quad29
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Dealing with random layers
• Step 1: Devise a suitable grouping procedure
• Step 2: Perform the recursive (much more involved!) estimates
quad30
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Step 1: Grouping procedure
ξ ∈ {0, 1}N sampled from the Bernoulli distribution Pδ with low density δ.
Γ = {i : ξi = 1} - correspond to the “bad lines” (of level 0):
L ≥ 3 integer.
quad31
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Step 1: Grouping procedure
ξ ∈ {0, 1}N sampled from the Bernoulli distribution Pδ with low density δ.
Γ = {i : ξi = 1} - correspond to the “bad lines” (of level 0):
L ≥ 3 integer. Start with L-runs ...
quad32
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Step 1: Grouping procedure
ξ ∈ {0, 1}N sampled from the Bernoulli distribution Pδ with low density δ.
Γ = {i : ξi = 1} - correspond to the “bad lines” (of level 0):
L ≥ 3 integer. Start with L-runs ...
quad33
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Step 1: Grouping procedure
ξ ∈ {0, 1}N sampled from the Bernoulli distribution Pδ with low density δ.
Γ = {i : ξi = 1} - correspond to the “bad lines” (of level 0):
L ≥ 3 integer. Start with L-runs ...
Assuming 64δL2 < 1 the procedure converges:
Each x ∈ Γ will be “re-incorporated” finitely many times a.s.; the final partition is
well defined.
quad34
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Step 1: Grouping procedure
On a set Ξ(δ) of full measure we can decompose Γ into sets Ci, called clusters, to which
an N-valued mass m(Ci) is attributed (m(Ci) ≤ |Ci|) in a way that
d(Ci, Cj) ≥ Lmin{m(Ci),m(Cj)}
, for all i, j
The Ci := C∞,i are obtained by the (limiting) recursive procedure outlined above. Each
constructed cluster has a level (the step when it was born!) and a mass.
m m1 2 3
d d1
M
M =
m
m1 m2 m3+ + _ _ logLd1 logL d2
C1
C2
C3
quad35
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Step 2: Multi-scale analysis for fixed realization of good/bad lines
Assume 3 ≤ L, 64δL2 < 1.
χ(ξ) := inf{k ≥ 0: d(C, 0) ≥ Mm(C)
for all C ∈ C∞ with m(C) > k}
with χ(ξ) = ∞ if the above set is empty or ξ /∈ Ξ(δ)
Then:
Pδ{ξ : χ(ξ) < ∞} = 1 and Pδ{ξ : χ(ξ) = 0} > 0.
We prove
P(
C+(0) is infinite)|χ(ξ) < ∞
)> 0.
Conceptually, the structure is similar to that in the simple hierarchical situation, but:
• rescaled lattices depend also on ξ;
• main estimate (drilling through the higher mass) within a good k-site Sk is much more
involved; our estimates require L somehow larger (L ≥ 192 suffices);
• pk ↗ 1 exponentially in k.
quad36
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III. Some related results
Bramson, Durrett, Schonmann (1991)
quad37
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III. Some related results
Bramson, Durrett, Schonmann (1991)
quad38
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IV. Coordinate percolation. Winkler’s compatibility
(η, ξ) a pair of sequences in Ξ = {0, 1}N
Allowed to: remove ones from η; remove zeros from ξ
Can one map both sequences to the same semi-infinite sequence?
If YES, say that (η, ξ) is compatible.
quad39
![Page 41: Dependent percolation: some examples and multi-scale tools · 2020. 4. 21. · II. Oriented percolation in a randomly layered environment On eZ2 +:= f(x;y) 2 Z£ Z+: x + y is eveng](https://reader035.vdocuments.pub/reader035/viewer/2022071101/5fda740c04d74024c71a52c1/html5/thumbnails/41.jpg)
IV. Coordinate percolation. Winkler’s compatibility
(η, ξ) a pair of sequences in Ξ = {0, 1}N
Allowed to: remove ones from η; remove zeros from ξ
Can one map both sequences to the same semi-infinite sequence?
If YES, say that (η, ξ) is compatible.
Winkler’s compatibility question:
Does it exist (p′, p) ∈ (0, 1)2 such that
Pp′ ⊗ Pp{(η, ξ) ∈ Ξ× Ξ: (η, ξ) are compatible} > 0 ?
P. Gacs (2004). Recent preprints: Basu, Sly (2012), Sidoravicius (2012).
quad40
![Page 42: Dependent percolation: some examples and multi-scale tools · 2020. 4. 21. · II. Oriented percolation in a randomly layered environment On eZ2 +:= f(x;y) 2 Z£ Z+: x + y is eveng](https://reader035.vdocuments.pub/reader035/viewer/2022071101/5fda740c04d74024c71a52c1/html5/thumbnails/42.jpg)
IV. Coordinate percolation. Winkler’s compatibility
(η, ξ) a pair of sequences in Ξ = {0, 1}N
Allowed to: remove ones from η; remove zeros from ξ
Can one map both sequences to the same semi-infinite sequence?
If YES, say that (η, ξ) is compatible.
Winkler’s compatibility question:
Does it exist (p′, p) ∈ (0, 1)2 such that
Pp′ ⊗ Pp{(η, ξ) ∈ Ξ× Ξ: (η, ξ) are compatible} > 0 ?
P. Gacs (2004). Recent preprints: Basu, Sly (2012), Sidoravicius (2012).
Let p ∈ (0, 1). Say that η ∈ Ξ is p-compatible if
Pp{ξ ∈ Ξ: (η, ξ) is compatible} > 0.
quad41
![Page 43: Dependent percolation: some examples and multi-scale tools · 2020. 4. 21. · II. Oriented percolation in a randomly layered environment On eZ2 +:= f(x;y) 2 Z£ Z+: x + y is eveng](https://reader035.vdocuments.pub/reader035/viewer/2022071101/5fda740c04d74024c71a52c1/html5/thumbnails/43.jpg)
IV. Coordinate percolation. Winkler’s compatibility
Theorem (Kesten, Lima, Sidoravicius, V.)
For every ε > 0 there exist 0 < pε < 1 and a binary sequence η ≡ ηε ∈ Ξ, such
that Zηε is a discrete fractal with Hausdorff dimension dH(Zη) ≥ 1− ε, and such that
Pp{ξ ∈ Ξ: (η, ξ) is compatible} > 0
for any p < pε.
Notation: Zη = {i ≥ 1: ηi = 0}
For the proof
• exploit a representation as coordinate oriented percolation;
• essential ingredient: the grouping procedure mentioned before.
Move from ξ to ψ ∈ ZN+: ψi ≥ 1 representing the length of the corresponding run of 1s.
Ψ ={
ψ ∈ ZN+ : ψi ≥ 1 implies ψi+1 = 0}
.
quad42
![Page 44: Dependent percolation: some examples and multi-scale tools · 2020. 4. 21. · II. Oriented percolation in a randomly layered environment On eZ2 +:= f(x;y) 2 Z£ Z+: x + y is eveng](https://reader035.vdocuments.pub/reader035/viewer/2022071101/5fda740c04d74024c71a52c1/html5/thumbnails/44.jpg)
IV. Coordinate percolation. Winkler’s compatibility
Coordinate percolation process.
Oriented graph G = (V,E), where
V = Z2+ E = { vertical n.n., northeast diagonals } oriented upwards
Given ζ, ψ ∈ Ψ, define the site configuration ωζ,ψ on G: for v = (v1, v2) with
v1, v2 ≥ 1
ωζ,ψ(v) =
{1 if ζv1
≥ ψv2,
0 otherwise.
v ∈ V open iff ωζ,ψ(v) = 1
(ωζ,ψ(0, 0) = 1, ωζ,ψ(v) = 0 if v1 ∧ v2 = 0)
quad43
![Page 45: Dependent percolation: some examples and multi-scale tools · 2020. 4. 21. · II. Oriented percolation in a randomly layered environment On eZ2 +:= f(x;y) 2 Z£ Z+: x + y is eveng](https://reader035.vdocuments.pub/reader035/viewer/2022071101/5fda740c04d74024c71a52c1/html5/thumbnails/45.jpg)
Simulations of coordinate percolation (by Lionel Levine)
quad44
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Simulations of coordinate percolation (by Lionel Levine)
quad45
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Simulations of coordinate percolation (by Lionel Levine)
quad46
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Simulations of coordinate percolation (by Lionel Levine)
quad47
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Simulations of coordinate percolation (by Lionel Levine)
quad48
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Simulations of coordinate percolation (by Lionel Levine)
quad49
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IV. Coordinate percolation. Winkler’s compatibility
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Figure 1: oriented cluster of the origin
For the compatibility question, we need more than an open oriented path.
A vertex v = (v1, v2) with v1, v2 ≥ 1 is heavy if ψv2≥ 1.
quad50
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IV. Coordinate percolation. Winkler’s compatibility
Permitted path: does not cross two heavy vertices with the same first coordinate.
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Figure 2: open permitted path from the origin
quad51
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IV. Coordinate percolation. Winkler’s compatibility
Lemma
Let ζ, ψ ∈ Ψ. If there exists an infinite open permitted path π starting from the origin
for the percolation configuration ωζ,ψ, then the pair (ζ, ψ) is compatible.
quad52
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IV. Coordinate percolation. Winkler’s compatibility
M -spaced sequences
M ≥ 2 integer. A sequence ψ ∈ Ψ is M -spaced if:
a) ij(ψ) ≥ M j for all j ≥ 1, where
ij(ψ) = inf{n ∈ N : ψn ≥ j} (+∞ if { } = ∅)
b) j − i ≥ Mmin{ψi,ψj}, for all 1 ≤ i < j.
ΨM := {ξ ∈ Ψ: ξ is M -spaced}
quad53
![Page 55: Dependent percolation: some examples and multi-scale tools · 2020. 4. 21. · II. Oriented percolation in a randomly layered environment On eZ2 +:= f(x;y) 2 Z£ Z+: x + y is eveng](https://reader035.vdocuments.pub/reader035/viewer/2022071101/5fda740c04d74024c71a52c1/html5/thumbnails/55.jpg)
IV. Coordinate percolation. Winkler’s compatibility
M -spaced sequences
M ≥ 2 integer. A sequence ψ ∈ Ψ is M -spaced if:
a) ij(ψ) ≥ M j for all j ≥ 1, where
ij(ψ) = inf{n ∈ N : ψn ≥ j} (+∞ if { } = ∅)
b) j − i ≥ Mmin{ψi,ψj}, for all 1 ≤ i < j.
ΨM := {ξ ∈ Ψ: ξ is M -spaced}
Theorem
Let L ≥ 2 and M ≥ 3(L + 1) be integers, ζ(L) the hierarchical sequence as before,
and ψ ∈ ΨM . Then the configuration ωζ(L),ψ has an infinite open permitted path π
starting from the origin.
Corollary
If ψ ∈ ΨM with M = 3(L + 1) the pair (ζ(L), ψ) is compatible.
quad54
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IV. Coordinate percolation. Winkler’s compatibility
Using this and the grouping lemma discussed before, one gets:
Theorem
Let L ≥ 2 and M = 3(L + 1). If p < 164M2 , ζ(L) ∈ Ψ is given by
(ζ(L))j =
{3Mk−1, if Lk|j and Lk+1 - j,0 , if L - j,
and η(L) is the corresponding binary sequence, then
Pp{ξ ∈ Ξ: (η(L), ξ) is compatible} > 0.
Remark. The statement about the zero set of η(L) is simple to verify, by classical results
(Barlow and Taylor).
quad55
![Page 57: Dependent percolation: some examples and multi-scale tools · 2020. 4. 21. · II. Oriented percolation in a randomly layered environment On eZ2 +:= f(x;y) 2 Z£ Z+: x + y is eveng](https://reader035.vdocuments.pub/reader035/viewer/2022071101/5fda740c04d74024c71a52c1/html5/thumbnails/57.jpg)
V. Related problems
Winkler’s Clairvoyant Demon problem
quad56
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V. Related problems
Winkler’s Clairvoyant Demon problem
quad57
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V. Related problems
Winkler’s Clairvoyant Demon problem
quad58
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V. Related problems
Winkler’s Clairvoyant Demon problem
quad59
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V. Related problems
Winkler’s Clairvoyant Demon problem
quad60
![Page 62: Dependent percolation: some examples and multi-scale tools · 2020. 4. 21. · II. Oriented percolation in a randomly layered environment On eZ2 +:= f(x;y) 2 Z£ Z+: x + y is eveng](https://reader035.vdocuments.pub/reader035/viewer/2022071101/5fda740c04d74024c71a52c1/html5/thumbnails/62.jpg)
V. Related problems
Winkler’s Clairvoyant Demon problem
0 r r
b
b
b
1 2 nr
1
2
m
Formulation as oriented percolation problem - Noga Alon
quad61
![Page 63: Dependent percolation: some examples and multi-scale tools · 2020. 4. 21. · II. Oriented percolation in a randomly layered environment On eZ2 +:= f(x;y) 2 Z£ Z+: x + y is eveng](https://reader035.vdocuments.pub/reader035/viewer/2022071101/5fda740c04d74024c71a52c1/html5/thumbnails/63.jpg)
V. Related problems
Winkler’s Clairvoyant Demon problem
0 r r
b
b
b
1 2 nr
1
2
m
Formulation as oriented percolation problem - Noga Alon
quad62
![Page 64: Dependent percolation: some examples and multi-scale tools · 2020. 4. 21. · II. Oriented percolation in a randomly layered environment On eZ2 +:= f(x;y) 2 Z£ Z+: x + y is eveng](https://reader035.vdocuments.pub/reader035/viewer/2022071101/5fda740c04d74024c71a52c1/html5/thumbnails/64.jpg)
V. Related problems
Winkler’s Clairvoyant Demon problem
Negative answer if n = 2 or n = 3;
n ≥ 4 - open question
Positive answer in the unoriented case for n ≥ 4: P Winkler (2000); Balister, Bollobas,
Stacey (2000)
For the oriented model: Gacs (2000): The Clairvoyant Demon has a hard task ...
quad63
![Page 65: Dependent percolation: some examples and multi-scale tools · 2020. 4. 21. · II. Oriented percolation in a randomly layered environment On eZ2 +:= f(x;y) 2 Z£ Z+: x + y is eveng](https://reader035.vdocuments.pub/reader035/viewer/2022071101/5fda740c04d74024c71a52c1/html5/thumbnails/65.jpg)
V. Related problems
- Stretched lattices: Jonasson, Mossel, Peres ; Hoffman
- Unoriented percolation; Potts model: Kesten, Lima, Sidoravicius, V.
- Percolation of words: Grimmett, Liggett, Richthammer (2008), Lima (2008, 2009)
- Rough isometries: Peled (2010), recent work: Basu, Sly, Sidoravicius.
quad64
![Page 66: Dependent percolation: some examples and multi-scale tools · 2020. 4. 21. · II. Oriented percolation in a randomly layered environment On eZ2 +:= f(x;y) 2 Z£ Z+: x + y is eveng](https://reader035.vdocuments.pub/reader035/viewer/2022071101/5fda740c04d74024c71a52c1/html5/thumbnails/66.jpg)
References
On the compatibility of binary sequences. H. Kesten, B. Lima, V. Sidoravicius, M.E.
Vares. (preprint on arXiv)
Oriented percolation in a random environment. H. Kesten, V. Sidoravicius, M.E. Vares.
(preprint on arXiv)
Dependent percolation on Z2. H. Kesten, B. Lima, V. Sidoravicius, M.E. Vares. (preprint)
Lipschitz embeddings of random sequences. R. Basu, A. Sly. (preprint on arXiv)
quad65