ioan manolescu joint work with demeter kiss and vladas … · introduction result result theorem...

Planar lattices do not recover from forest fires

Ioan ManolescuJoint work with Demeter Kiss and Vladas Sidoravicius

University of Geneva

31 March 2014

Ioan Manolescu (University of Geneva) Subcritical phase of 2D SDP 31 March 2014 1 / 8

Introduction The model

What is self-destructive percolation?

Let p, δ ∈ [0, 1].Two percolation configurations:

ω - intensity p (measure Pp).

σ - intensity δ (small).

ωclose infinite cluster−−−−−−−−−−−→ ω

enhancement−−−−−−−→ ωδ.

ωδ = ω ∨ σ is an enhancement of ω.

Call Pp,δ the measure for ω, σ, ωδ, . . . .

δc(p) = sup{δ : Pp,δ(0ωδ

←→∞) = 0}.

Question: δc(p)→ 0 as p ↘ pc?

←→∞) = 0}.

ω - intensity p (measure Pp).σ - intensity δ (small).

←→∞) = 0}.

Introduction Result

Result

Theorem (Kiss, M., Sidoravicius)

There exists δ > 0 such that, for all p > pc ,

Pp,δ(infinite cluster in ωδ) = 0.

So limp→pc δc(p) > 0

Corollary

Infinite parameter forest fires are not defined on planar lattices.

See Van den Berg and Brouwer 2004 for proof.

Introduction Result

Result

Corollary

Introduction Result

Result

Corollary

Reminders on planar percolation

There exists pc ∈ [0, 1] such that

p < pc ⇒ Pp(infinite cluster) = 0,

p > pc ⇒ Pp(infinite cluster) = 1.

At pc ..

n[ ]≥ ε∀n,

At pc ..

n[ ]≥ ε∀n,

At pc ..

n[ ]≥ ε∀n,

At pc ..

n[ ]≥ ε∀n,

At pc ..

n[ ]≥ ε∀n,

At pc ..

n[ ]≥ ε Ppc2n

n[ ]≥ ε∀n,

At pc ..

n[ ]≥ ε Ppc2n

n[ ]≥ ε∀n,

[ ]≤ n−α1 Ppc

[ ]≤ n−α5

A crossing probability estimate

ω containing crossing

delete crossing cluster−−−−−−−−−−−−→ ω̃enhancement−−−−−−−→ ω̃δ

Proposition

For δ > 0 small enough, as n→∞,

Ppc,δ ( )4n nn

nω ω̃δand → 0

ω containing crossingdelete crossing cluster−−−−−−−−−−−−→ ω̃

enhancement−−−−−−−→ ω̃δ

Proposition

Ppc,δ ( )4n nn

nω ω̃δand → 0

Proposition

Ppc,δ ( )4n nn

nω ω̃δand → 0

Proposition

Ppc,δ ( )4n nn

nω ω̃δand → 0

Proposition

Ppc,δ ( )4n nn

nω ω̃δand → 0

Proof of crossing estimate ⇒ Theorem

For δ

> 0 small, Ppc,δ ( )4n nn

nω ω̃δand → 0

For δ

> 0 small, Ppc,δ ( )4n nn

nω ω̃δand → 0

For p1 ≥ p2 and δ1, δ2 such that p1 + (1− p1)δ1 ≤ p2 + (1− p2)δ2,

Pp1,δ1 ≤st Pp2,δ2 .

For δ′> 0 small, ( )4n nn

nω ω̃δand → 0Pp,δ′ , for p ≥ pc .

For p1 ≥ p2 and δ1, δ2 such that p1 + (1− p1)δ1 ≤ p2 + (1− p2)δ2,

Pp1,δ1 ≤st Pp2,δ2 .

In both ω̃δ and ωδ!

Proof of crossing probability estimate

γ - vertical crossing with minimal number of enhanced points.

X = {enhanced points used by γ}. If no crossing X = ∅.

Ppc ,δ(vertical crossing in ω̃δ) =∑

X 6=∅Ppc ,δ(X = X ).

Annulus surrounding passage points but not containing passage points:6 arms or 4 half-plane arms in ω (possibly with one defect).

Ppc ( )R

≤(rR

)2+λr Ppc ( )R ≤

)2+λr

For a set X , what is Ppc ,δ(X = X )?

Pp,δ(X = X ) ≤ ckn−2−λ∏

d−2−λj × δk ,

where d1, . . . , dk are the merger times of X .

#{X with merger times d1, . . . , dk} ≤ C kn2∏

P(vetical crossing in ω̃δ) ≤ n−λ∑

k≥1d1,...,dk

(δkck

d−1−λk

= n−λ∑

d−1−λ

→ 0,

for δ > 0 small.

P(vetical crossing in ω̃δ) ≤ n−λ∑

k≥1d1,...,dk

(δkck

d−1−λk

= n−λ∑

d−1−λ

→ 0,

for δ > 0 small.

Thank you!

ioan manolescu joint work with demeter kiss and vladas … · introduction result result theorem...

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