On exponential growth for a certain class oflinear systems
Ryoki Fukushima(Tokyo Institute of Technology)
Joint work with Nobuo Yoshida (Kyoto university)

Universality and Scaling Limits inProbability and Statistical Mechanics
September 1, 2010

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Simplest Example(1+1)-dimensional oriented percolation.

Q. Asymptotics of the number of open paths of length n?
(We regard it as the “population”.)
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Related models
I Homogeneous branching random walks: each particle
splits and moves independently.
I Branching random walks in space time random

environment: the branching law depends on sites.
I Path counting on the oriented percolation: every particle

at the same site splits and moves in the same manner.

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Trivial halfp ≤ pc =⇒ extinct a.s.
(p = pc : Grimmett-Hiemer, Progr. Probab., 2002.)
Let p > pc . How fast the population grows on the event ofsurvival?

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Trivial halfp ≤ pc =⇒ extinct a.s.
(p = pc : Grimmett-Hiemer, Progr. Probab., 2002.)
Let p > pc . How fast the population grows on the event ofsurvival?
Branching process analogyFor the supercritical Galton-Watson process,a.s.

survival ⇐⇒ exponential growth

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Remark
Let |Nn | denote the total population at time n.
I For the supercritical Galton-Watson process with theKesten-Stigum condition,
()
|Nn |
P lim
> 0 > 0.
n→∞ E [|Nn |]5 / 21

Remark
Let |Nn | denote the total population at time n.
I For the supercritical Galton-Watson process with theKesten-Stigum condition,
()
|Nn |
P lim
> 0 > 0.
n→∞ E [|Nn |]I Without the Kesten-Stigum condition, we still have(|Nn |P lim>0n→∞ Cn)> 0.

for some Cn ≈ E [|Nn |].

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I N. Yoshida (J. Stat. Phys., 2008) proved that(P)|Nn |> 0 > 0.
n→∞ E [|Nn |]limfor the (1 + d)-dimensional oriented percolation if d ≥ 3 and p
is suﬃciently large. (suﬃcient: p >return probability of SRW)
However, he also proved that it fails for d = 1, 2.

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Simple proof for the (1+1)-dimensional oriented percolation
Consider the rightmost inﬁnite open path γ.
1. It changes the direction “many times”.
2.

are open with probability p 2independently.

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Simple proof for the (1+1)-dimensional oriented percolation
Consider the rightmost inﬁnite open path γ.
1. It changes the direction “many times”.
2.

are open with probability p 2
independently.

The law of large numbers shows that γ has
O(n)-bypasses until n.
Since the population doubles wheneverit has a bypass,...
=⇒ exponential growth along γ¤

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Remark
Kesten-Nazarov-Peres-Sidoravicius have recently studied
def“maximal paths” = paths going through maximal
number of open sites.
Their reselt says:
For any p ∈ (0, 1) and any space dimension,
the number of maximal paths of length ngrows exponentially.

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We are interested in another generalization.
(1) General space dimension(2) Non nearest neighbor immigration(3) Spatial correlation
The “simple proof” breaks down since(1) or (2) =⇒ no rightmost path,(3) in addition =⇒ (possibly) nobut-bypass-bypass.

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Example (a discrete version of contact process)(1)

is open with probability q.

(2) For non-vertical bonds, choose one nearest neighborsite at the previous level uniformly and

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General model{Nn,x : the population at (n, x) ∈ Z+ × Zd ,Nn = (Nn,x )x∈Zd
The evolution rule of the oriented percolation is∑
Nn+1,x =Nn,y · ηn+1,y ,x · 1{|x−y |=1}
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We can rewrite the equation as
Nn+1 = Nn An+1 ,where An,x,y = ηn,x,y · 1{|x−y |=1} .
Key properties of {An }n∈N(1) Z+ -valued elements,(2) I.I.D. in n ∈ N,law(3) (An,x,y )x,y ∈Zd = (An,x+z,y +z )x,y ∈Zd for any z ∈ Zd ,(4) For some m ∈ N and x ∈ Zd ,P ((A1 · · · Am )0,x ≥ 2) > 0.

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The last condition (4) ensures the existence of a bypass.
Indeed,∑(A1 · · · Am )0,x =A1,0,x1 · · · Am,xm−1 ,x .
x1 ,...,xm−1

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The last condition (4) ensures the existence of a bypass.
Indeed,∑(A1 · · · Am )0,x =A1,0,x1 · · · Am,xm−1 ,x .
x1 ,...,xm−1

If {An }n∈N are of ﬁnite range, then (4) is necessary forexponential growth since(4) fails =⇒ Nn,x ∈ {0, 1},
ﬁnite range =⇒ #{occupied sites} = O(nd ).

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Viewing A1 · · · Am as one step, we replace (4) by(5) P(A1,0,x ≥ 2) > 0 for some x ∈ Zd .

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Viewing A1 · · · Am as one step, we replace (4) by(5) P(A1,0,x ≥ 2) > 0 for some x ∈ Zd .
Deﬁnition
Nn = (δ0,x )x∈Zd A1 · · · An .
Theorem (F. and Yoshida)
Suppose (1)–(3) and (5) hold and the process survives withpositive probability. Then,a.s.

survival ⇐⇒ exponential growth

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Idea of the proof
We are going to ﬁnd an open path on whichAn,γ(n−1),γ(n) ≥ 2 many times.
However, for an inﬁnite open path, it is diﬃcult
to establish “independence” required to use thelaw of large numbers.
Thus we ﬁrst construct a path which is notopen in general.

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Let x ∈ Zd be such that P(A1,0,x ≥ 2) > 0.
Set γ(0) = 0 and given γ(n),(1) γ(n + 1) = γ(n) + x if(2) γ(n + 1) = γ(n) + y (y ̸= x) if(3) If (n, γ(n)) is a dead end, then...

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Properties of γ
I prefers to choose x-direction,
I does not refer to the future,
I does not refer to the past beyond a percolation point,I ∃ open path going through all percolation points on γ.

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Deﬁne the good events by{}
Gn = An,γ(n−1),γ(n−1)+x ≥ 2 and (n, γ(n)) Ã ∞(1) ∃ open path going through all sites where Gn occurs,(2) {Gn }n∈N is stationary,(3) P(Gn ) = P(An,γ(n−1),γ(n−1)+x ≥ 2)P((n, γ(n)) Ã ∞)= P(A1,0,x ≥ 2)P(survival)> 0,(4) {Gn }n∈N is mixing.

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P(Gm ∩ Gn )= P(Am,γ(m−1),γ(m−1)+x ≥ 2, (m, γ(m)) Ã ∞,An,γ(n−1),γ(n−1)+x ≥ 2, (n, γ(n)) Ã ∞)= P(Am,—. ≥ 2, (m, γ(m)) Ã (n − 1, γ(n − 1)))× P(An,—. ≥ 2, (n, γ(n)) Ã ∞)∼ P(Gm )P(Gn ) as n − m → ∞.

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The law of large numbers shows that1∑
1G −→ P(A1,0,x ≥ 2)P(survival).
n k=1 kn

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The law of large numbers shows that1∑
1G −→ P(A1,0,x ≥ 2)P(survival).
n k=1 kn

By (1), there exists an open path going throughall sites where Gn occurs. Therefore we have anopen path with many bypasses.
=⇒ exponential growth.

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Thank you!

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