Random walks on the East model
Oriane Blondel
CNRS; Université Lyon 1 Claude Bernard
Warwick, January 2016
j.w. L. Avena and A. Faggionato
O. Blondel
RW on East
East model
Markov process
(ξt )t∈R+ on {0, 1}Z . ρ ∈ (0, 1) density parameter.
(
ρ
0 −→ 1
At site x,
if x + 1 (the East neighbor of x) is empty.
1−ρ
1 −→ 0
LE f (ξ) =
X
(1 − ξ(x + 1)) ρ(1 − ξ(x)) + (1 − ρ)ξ(x) f (ξ x ) − f (ξ)
x
ρ = 1/2; space horizontal, time goes down, 1 = •, 0 = •.
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RW on East
A few properties of the East model
I Reversible w.r.t. µ := Ber(ρ)⊗Z .
I Mixing in L2 (µ) [Aldous-Diaconis ’00]:
∀ρ ∈ (0, 1), ∃γ > 0 s.t. ∀f , g ∈ L2 (µ)
|Eµ [f (η0 )g (ηt )] − µ(f )µ(g )| ≤ C (f , g )e −γt .
O. Blondel
RW on East
A few properties of the East model
I Reversible w.r.t. µ := Ber(ρ)⊗Z .
I Mixing in L2 (µ) [Aldous-Diaconis ’00]:
∀ρ ∈ (0, 1), ∃γ > 0 s.t. ∀f , g ∈ L2 (µ)
|Eµ [f (η0 )g (ηt )] − µ(f )µ(g )| ≤ C (f , g )e −γt .
Some non-properties of the East model
Not attractive.
No uniform or cone-mixing property.
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RW on East
RW in (dynamic) random environment
(Zt )t≥0 process on Zd such that
P(Zt+dt = x + y |Zt = x) = r (y , τx ξt )dt,
d
where (ξt )t≥0 is the random environment, with values in {0, 1}Z .
Today: ξ is the East model.
O. Blondel
RW on East
RW in (dynamic) random environment
(Zt )t≥0 process on Zd such that
P(Zt+dt = x + y |Zt = x) = r (y , τx ξt )dt,
d
where (ξt )t≥0 is the random environment, with values in {0, 1}Z .
Today: ξ is the East model.
Environment seen from the walker
ηt := τZt ξt .
has generator
Lf (η) = LE f (η) +
X
y
r (y , η) [f (τy η) − f (η)]
Jumps of (Xt )t≥0 ←→ spatial shifts for (ηt )t≥0 .
Zt depends on (ξs )s≤t .
O. Blondel
RW on East
Random walks
X (unperturbed walker) and X () (perturbed walker) with continuous rates
r (y , η), r (y , η) ≥ 0 such that
I
r (y , η) = r (y , η) + r̂ (y , η).
I
ηt := τXt ξt has invariant measure µ.
In particular: (τXt ξt )t≥0 has same invariant measure and mixing property
as (ξt )t≥0 .
I
2 = kL̂ k < γ.
P
4
y |y | supη (r (y , η) + |r̂ (y , η)|) < ∞.
r (y , η) > 0 =⇒ r (y , η).
I
I
where
L f (η)
=
LE f (η) +
X
y
Lf (η)
=
LE f (η) +
X
y
L̂ f (η)
=
X
y
r (y , η) [f (τy η) − f (η)]
r (y , η) [f (τy η) − f (η)]
r̂ (y , η) [f (τy η) − f (η)]
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RW on East
Examples
Driven ghost probe [Jack-Kelsey-Garrahan-Chandler PRE 2008]
r (±1, η)
=
r (±1, η)
=
(1 − η(0))(1 − η(±1))
(1 − η(0))(1 − η(±1))r̃ (±1),
where r̃ (1) = 2/(1 + e − ) = e r̃ (−1).
O. Blondel
RW on East
Examples
-RW
For ∈ (0, 1/2), jump rates given by
1/2 − 1/2 + 1/2 − 1/2 + 1
0
N.B.: In the pictures, p = 1/2 + .
p = 0.2
p = 0.8
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RW on East
General results
Theorem
()
1. The process seen from the walker ηt = τX () ξt , started from ν µ,
t
converges to an ergodic invariant measure µ such that µ ∼ µ and for all
f ∈ L2 (µ)
∞ Z ∞
X
µ (f ) = µ(f ) +
µ L̂ S (n) (s)f ds,
n=0
where S (0) (t) = e tL , S n+1 (t) =
Rt
∞
2. For all f , g ∈ L (µ)
0
0
S (0) (t − s)L̂ S (n) (s) ds.
|Eµ [f (η0 )g (ηt )] − µ (f )µ (g )| ≤ C (f , g )e −
3. If r have finite range and bounded support, ∃δ > 0 s.t.
|µ (η(x)) − ρ| ≤ Ce −δ|x| .
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RW on East
γ−2
2 t
.
General results
Theorem
()
1. The process seen from the walker ηt = τX () ξt , started from ν µ,
t
converges to an ergodic invariant measure µ such that µ ∼ µ and for all
f ∈ L2 (µ)
∞ Z ∞
X
µ (f ) = µ(f ) +
µ L̂ S (n) (s)f ds,
n=0
where S (0) (t) = e tL , S n+1 (t) =
Rt
∞
2. For all f , g ∈ L (µ)
0
0
S (0) (t − s)L̂ S (n) (s) ds.
|Eµ [f (η0 )g (ηt )] − µ (f )µ (g )| ≤ C (f , g )e −
γ−2
2 t
.
3. If r have finite range and bounded support, ∃δ > 0 s.t.
|µ (η(x)) − ρ| ≤ Ce −δ|x| .
N.B.: Point 1 proved in [Komorowski-Olla ’05] when d ν/d µ ∈ L2 (µ), via a
series expansion of the densities rather than semigroups. Perturbation L̂ can be
unbounded, should satisfy a sector condition.
O. Blondel
RW on East
General results
Theorem
1. LLN.
1 ()
X
−→ v ()
t t t→∞
2. Invariance principle.
()
p
Xnt − v ()nt
√
→ 2D Bt ,
n
3. If the environment seen from the unperturbed walker (ηt )t≥0 is reversible
w.r.t. µ, D > 0 for small enough.
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RW on East
More on the -RW
Theorem
1. If > 0 small enough, µ (η(1)) < ρ < µ (η(−1)) (and the reverse if > 0).
int
0.44
0.46
0.48
0.50
0.52
0.54
0.56
Env. seen from the particle at p: .6 rho: 0.5
-10
-5
0
5
10
place
Numerics by P. Thomann.
3
2. v () = 2(2ρ − 1) + κρ + O(5 ).
In particular, if ρ 6= 1/2, || =
6 0 small enough, sgn(v ()) = sgn((2ρ − 1))
(the sign of the speed is given by the dominating type of site in the
environment).
What if ρ = 1/2?
O. Blondel
RW on East
Speed as function of p in east and rho 0.5
0.00
−0.05
−0.10
speed
0.05
0.10
r: 0.5 N: 12 g: 1
0.0
0.2
0.4
0.6
p
Numerics by P. Thomann.
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RW on East
0.8
1.0
More on the speed v ()
Conjecture
If ρ = 1/2, · v () < 0.
(Weaker: κ1/2 < 0.)
A consequence of the orientation choice?
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RW on East
More on the speed v ()
Conjecture
If ρ = 1/2, · v () < 0.
(Weaker: κ1/2 < 0.)
A consequence of the orientation choice?
Theorem
If 2 < γ,
v (−) = −v ().
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RW on East
More on the speed v ()
Conjecture
If ρ = 1/2, · v () < 0.
(Weaker: κ1/2 < 0.)
A consequence of the orientation choice?
Theorem
If 2 < γ,
v (−) = −v ().
In particular,
vEast () = vWest ().
Rather, a signature of the space-time correlated bubbles.
N.B.: The antisymmetry holds for any reversible environment with L2
exponential mixing.
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RW on East
A degenerate version of the -RW
"Infinite rates".
(Yt )t≥0 lives on edges of Z of type • − •.
When the particle on the left is erased, jumps to the right-most such edge.
When the hole on the right is filled, jumps to the left-most such edge (one step
to the right).
lim sup Yt /t < 0
O. Blondel
RW on East
A degenerate version of the -RW
"Infinite rates".
(Yt )t≥0 lives on edges of Z of type • − •.
When the particle on the left is erased, jumps to the right-most such edge.
When the hole on the right is filled, jumps to the left-most such edge (one step
to the right).
lim sup Yt /t < 0
Three-point correlation function
Proposition
If for any y ≥ 1, for any s, t ≥ 0,
EEast
ξ0 (0)(2ξt (y ) − 1)ξt+s (0) > 0,
ρ
then κρ < 0.
O. Blondel
RW on East
Thank you for your attention.
O. Blondel
RW on East