with: r) . ofstad - staff.uni-mainz.de fileof rm s ˆτ n (k = X x ∈ Z d τ n (x) e ik · x k ∈...

25
The super-process limit of oriented percolation above 4+1 dimensions Remco van der Hofstad Joint work with: Gordon Slade (UBC, Vancouver) Frank den Hollander (Eurandom).

Transcript of with: r) . ofstad - staff.uni-mainz.de fileof rm s ˆτ n (k = X x ∈ Z d τ n (x) e ik · x k ∈...

The

super-

pro

cess

lim

itoforiente

dperc

ola

tion

above

4+

1

dim

ensions

Rem

co

van

der

Hofs

tad

Join

twork

with:

Gord

on

Sla

de

(UBC,Vancouver)

Fra

nk

den

Hollander

(Eura

ndom

).

Oriente

dperc

ola

tion

Oriente

dbonds

join

(x,n

)to

(y,n

+1)

for

n≥

0and

x,y

∈Zd

.

Make

bond

((x,n

),(y

,n+

1))

independently

occupie

dw

ith

pro

bability

pD

(y−

x),

vacant

with

pro

bability

1−

pD

(y−

x).

Here

,p∈

[0,1

/‖D

‖ ∞]is

the

perc

ola

tion

para

mete

r.

Goals

Invest

igate

scaling

behavio

urcriticaloriente

dperc

ola

tion

for

d>

4.

-1-

Oriente

dperc

ola

tion

Fig

ure

by

B.Cass

elm

an

and

G.Sla

de

(02).

-2-

Key

quantities

Surv

ivalpro

bability θ n

=P p

(∃x∈

Zd:(0

,0)→

(x,n

)).

Two-p

oin

tfu

nction

τ n(x

)=

P p((

0,0

)→

(x,n

)).

Hig

her-

poin

tfu

nctions

Let

(~x,~n

)=

((x1,n

1),

...,

(xr−1,n

r−1))

,

τ(r

)~n

(~x)=

P p((

0,0

)→

(~x,~n

)).

Main

quest

ion

How

do

θ n’s

and

τ(r

)n

’sbehave

as

n→

∞?

-3-

Pre

vio

us

resu

lts

OP

has

aphase

transition,i.e,th

ere

isa

criticalpro

bability

pc=

pc(

d,L

)∈

(0,1

),su

ch

that

•fo

rp≤

pc,

there

isa.s.no

infinite

clu

ster.

•fo

rp

>pc,

there

isa.s.uniq

ue

infinite

clu

ster.

Resu

ltfo

rp=

pc:Bezuid

enhout

and

Grim

mett

(90).

Applies

also

toconta

ct

pro

cess

,a

rela

tive

oforiente

dperc

ola

tion.

Implies

that

θ n→

0as

n→

∞.

-4-

Pre

vio

us

resu

lts

Inte

rms

ofFourier

transf

orm

τ̂ n(k

)=

∑ x∈

Zdτ n

(x)e

ik·x

,k∈

[−π,π

]d.

For

p<

pc,

τ̂ n(0

)is

exponentially

small.

For

p>

pc,

τ̂ n(0

)in

cre

ase

sas

nd.

For

p=

pc,

behavio

ur

τ̂ n(0

)not

unders

tood.

Goal

Invest

igate

θ n,

τ̂(r

)n

(k)

at

pc

for

d>

4,

where

dc=

4is

the

criticaldim

ension.

Expect

Behavio

ur

sim

ilar

as

criticalbra

nchin

gra

ndom

walk

.

Meth

od

Lace

expansion.

(Pre

vio

us

resu

lts:

Nguye

n&

Yang

(93,95)

fortw

o-p

oin

tfu

nction.)

-5-

Mom

ent

measu

res

ofcanonic

alm

easu

resu

per-

Bro

wnia

nm

otion

{Xt}

t≥0

super-

pro

cess

=m

easu

revalu

ed

diff

usion.

Fourier

transf

orm

ofjo

int

mom

ent

measu

reis

M̂(l)

~ t(~ k

)=

E µ[ ∫ R

ldei

k1·x

1X

t 1(d

x1)···e

ikl·x

l Xt l(d

xl)

] .

{Xt}

t≥0

issu

per-

Bro

wnia

nm

otion.

Identify

mom

ent-

measu

resofSBM

recurs

ively

by

M̂(1

)t

(~ k)=

e−|k|2

t/2d,

M̂(l)

~ t(~ k

)=

∫ t 0dt

M̂(1

)t

(k1+···+

kl)

∑I⊂

J\{

1}:|I|≥

1

M̂(|

I|)

~ t I−

t(~ k

I)M̂

(l−|I|)

~ t J\I−

t(~ k

J\I

),

where

J={1

,...

,l},

t=

min

it i

,~ t I

=(t

i)i∈

I,and

~ t I−

t=

(ti−

t)i∈

I.

-6-

Resu

lts

OP

Theore

m1.

Fix

d>

4,

p=

pc,

r≥

2and

δ∈

(0,1

∧ε∧

d−4

2)

and

~ t=

(t1,.

..,t

r−1)∈

(0,∞

)r−1,~ k

=(k

1,.

..,k

r−1)∈

R(r−1)d

.T

hen

there

exist

L0

=L

0(d

)and

finite

positive

const

ants

A=

A(d

,L),

v=

v(d

,L),

V=

V(d

,L)

such

that

for

L≥

L0,

τ̂(r

)

bn~ tc(~ k

/

√ vσ2n)=

A2r−3V

r−2n

r−2[M̂

(r−1)

~ t(~ k

)+O

(n−

δ)]

.

Extr

are

sult

for

r=

2:

sup

xτ n

(x)=O

(L−

dn−

d/2).

Implies

Converg

ence

finite

dim

ensionaldistr

ibutionsofO

Pto

SBM

.

Miss

Tig

htn

ess

.

-7-

Resu

lts

OP

Theore

m2.

Under

the

above

conditio

ns

θ n=

1 Bn[1

+o(1

)].

Also,

B=

AV

/2.

Implies

that

conditio

nally

on

0→

m,

m−1N

m=⇒

Exp(λ

)w

ith

λ=

2/(A

2V

),

and

Nn

=#{y

∈Zd

:(0

,0)→

(y,n

)}.

-8-

Incip

ient

Infinite

Clu

ster

Kest

en

(1986)

has

const

ructe

dth

ein

cip

ient

infinite

clu

ster

(IIC

)

for

perc

ola

tion

on

Z2.

IIC

desc

ribes

localst

ructu

reofla

rge

criticalclu

sters

.

Will

pro

vid

ea

const

ruction

for

oriente

dperc

ola

tion

indim

ension

d>

4.

For

cylinder

events

E,define

Qn(E

)=

P pc(E|(0,0

)→

n)

and

(IIC

)Q∞(E

)=

lim

n→∞

Qn(E

).

-9-

Existe

nce

IIC

Theore

m3.

Let

d>

4.T

hen

there

isan

L0

=L

0(d

)su

ch

that

for

all

L≥

L0

the

follow

ing

hold

s:If

(ASY

)lim

n→∞

nθ n

=1/B∈

(0,∞

),

then

the

lim

itin

(IIC

)exists

for

every

cylinder

event

E.M

ore

over,

Q∞

exte

nds

toa

pro

bability

measu

reon

the

full

sigm

a-a

lgebra

of

events

.

The

diffi

culty

com

pare

dto

earlie

rwork

(NY

(93,95),

HS

(2001))

isth

at

the

lace

expansion

needs

tobe

adapte

dto

dealw

ith

poin

t-

to-p

lane

inst

ead

ofpoin

t-to

-poin

tconnections.

-10-

The

lace

expansion

for

OP

two-p

oin

tfu

nction

The

lace

expansion

giv

es

are

curs

ion

form

ula

τ̂ n+

1(k

)=

pD̂

(k)( τ̂ n

(k)+

n+

1 ∑m

=2

π̂m(k

)τ̂n−

m(k

)) +π̂

n+

1(k

),

where

πm(x

)has

dia

gra

mm

atic

expre

ssio

n

πm(x

)=

∞ ∑N

=0

(−1)N

π(N

)m

(x),

with

π(N

)m

(x)=

P((0

,0)⇒

(x,m

)),and

π(2

)m

(x)

=�

��

@@ @

�� �

@@@

J J

(x,m

)(0

,0)

+�

��

@@ @

�� �

@@@

(x,m

)(0

,0)

Rem

ark

Critical

BRW

satisfi

es

sam

ere

curr

ence

rela

tion

with

p=

1,π̂

n(k

)≡

0.

-11-

Pro

of

Induction

on

nShow

that

πm(x

)sm

all

for

mand

Lla

rge.

Induction

hypoth

ese

son

τ̂ n(k

).Turn

soutth

atwe

can

bound

πm(x

)

inte

rms

of

τ̂ j(k

)fo

rj

<m

.Exam

ple

:

π(0

)m

(x)=

P p((

0,0

)doubly

connecte

dto

(x,m

)).

BK

-inequality

:

π(0

)m

(x)≤

τ m(x

)2≤

(pD∗

τ m−1)(

x)2

.

Induction

hypoth

ese

sgiv

ebounds

for

all

m≤

n

sup

xτ m

(x)≤

K

Ldm

d/2,

∑ xτ m

(x)≤

K.

Key

toin

duction

Bounds

τ mfo

rm≤

nim

ply

bounds

πm

for

m≤

n+

1,th

at

intu

rn

again

imply

bounds

τ n+

1.Loop

clo

sed.

-12-

Inte

rmezzo:

the

lace

expansion

for

perc

ola

tion

1

Path

isst

ring

ofsa

usa

ges

and

piv

ota

lbonds.

0

x

0x

Want:

show

OP

islike

SRW

with

bubble

distr

ibution.

-13-

Inte

rmezzo:

the

lace

expansion

for

perc

ola

tion

2

Use

nota

tion

u=

(u,m

),v

=(v

,m+

1),

x=

(x,n

),0

=(0

,0).

Recall

π(0

) (x)=

P(0⇒

x).

Then

τ(x

)=

δ 0,x

(0) (

x)+

∑(u

,v)

P(0⇒

u,(

u,v

)open

and

piv

ota

lfo

r0→

x).

Naiv

ely,

P(0⇒

u,(

u,v

)open

and

piv

ota

lfo

r0→

x)

≈P(

0⇒

u)P

((u

,v)

open)P

(v→

x)

=(δ

0,u

(0) (

u))

pD

(v−

u)τ

(x−

v).

Then

we

would

be

done

with

π(x

)=

π(0

) (x).

-14-

Inte

rmezzo:

the

lace

expansion

for

perc

ola

tion

3

Let

C̃(u

,v) (

0)

be

those

poin

tsy

s.t.

0→

yw

ithout

using

(u,v

).

τA(x

,y)=

P(x→

yin

Zd×

Z +\A

).

Independence

ofperc

ola

tion:

P(0⇒

u,(

u,v

)occ.and

piv

ota

lfo

r0→

x)

=pD

(v−

u)〈

I[0⇒

u]τ

C̃(u

,v) (

0) (

v,x

)〉.

Rew

rite

τC̃

(u,v

) (0) (

v,x

)=

τ(x

−v)−

P(v

C̃(u

,v) (

0)

−−−−−−→

x).

Giv

es τ(x

)=

[δ0,x

(0) (

x)]

+∑

(u,v

)

pD

(v−

u)[

δ 0,u

(0) (

u)]

τ(x

−v)

−∑

(u,v

)

pD

(v−

u)〈

I[0⇒

u]P

(vC̃

(u,v

) (0)

−−−−−−→

x)〉

.

-15-

Inte

rmezzo:

the

lace

expansion

for

perc

ola

tion

4

Gra

phic

ally:

〈I[0⇒

u]P

(vC̃

(u,v

) (0)

−−−−−−→

x)〉

=

Fin

dfirs

tpiv

ota

lfo

rv→

xaft

er

connection

thro

ugh

C̃(u

,v) (

0).

Denote

π(1

) (x)

contr

ibution

where

such

piv

ota

ldoes

not

exist.

Gra

phic

ally:

π(1

) (x)=

Cut

aft

er

this

bond

and

repeat

itera

tively

!

-16-

Hig

her

poin

tfu

nctions

For

hig

her

poin

tfu

nctions,

need

double

expansion.

xx

x1

23

00

x

x

x

x

x y -- ---

-

12

3

4

5

Use

induction

on

r.

Initia

lization

are

the

resu

lts

two-p

oin

tfu

nction.

-17-

Hig

hdim

ensionalperc

ola

tion

and

super-

pro

cess

es:

The

sequel!

Past

years

:in

vest

igation

bra

nchin

gst

ructu

reperc

ola

tion

clu

ster:

Hara

and

Sla

de

(00a

and

00b):

Invest

igate

τ̂ pc,

h(k

),w

hic

his

Fourier

transf

orm

of

τ pc,

h(x

)=

∞ ∑ n=

0

e−nhP(

0→

x,|

C(0

)|=

n).

Pro

ve

τ p(x

)≈

C

k2+√

h.

Resu

ltwell-k

now

nfo

rbra

nchin

gra

ndom

walk

.Suggest

sth

at

larg

e

clu

sters

behave

like

super-

Bro

wnia

nm

otion,w

hic

his

lim

itofBRW

.

-18-

The

conta

ct

pro

cess

Sim

ilarre

sultsare

expecte

dfo

rconta

ctpro

cess

,w

hic

his

continuous

tim

evers

ion

oforiente

dperc

ola

tion.

Infe

cte

dsite

xin

fects

yw

ith

rate

λD

(y−

x),

xbecom

es

healthy

with

rate

1.

Again

,th

ere

isa

criticalin

fection

rate

λc

above

whic

hth

edisease

surv

ives

with

positive

pro

bability,

and

belo

ww

hic

hit

die

sout

with

pro

bability

one.

Expectsa

me

resu

ltsasfo

rO

Pfo

rd

>4

(Join

twork

with

A.Sakai).

Impro

ves

upon

resu

lts

by

Durr

ett

and

Perk

ins

(99).

-19-

Conta

ct

pro

cess

Fig

ure

by

B.Cass

elm

an

and

G.Sla

de

(02).

-20-

Tig

htn

ess

Inte

rest

ing

quest

ion

isw

heth

erO

Pconverg

es

toSBM

as

apro

cess

,

i.e,is

criticaloriente

dperc

ola

tion

tight.

Theore

m1

show

sth

atth

efinite

dim

ensionaldistr

ibutions

converg

e

toth

ose

ofSBM

,w

hic

his

subst

antially

weaker

than

weak

conver-

gence.

Missing

ingre

die

nt

Tig

htn

ess

.

Quest

ion

What

are

handy

crite

ria

for

tightn

ess

of

measu

reval-

ued

pro

cess

es?

-21-

Tig

htn

ess

for

stochast

icpro

cess

es

Let

ωn(t

)be

ast

ochast

icpro

cess

.T

hen

we

have

the

follow

ing

tightn

ess

resu

lt:

Theore

mA.T

he

pro

cess{ω

n(t

)}t≥

0is

tight,

when

forevery

0≤

t 1≤

t 2≤

t 3,

En(t

1,t

2,t

3)=

E[ |ωn(t

1)−

ωn(t

2)|

2|ω

n(t

2)−

ωn(t

3)|

2] ≤

C|t 1

−t 2||t

2−

t 3|.

Exam

ple

1.

Let{S

n} n≥0

be

sim

ple

random

walk

on

Zd,and

write

ωn(t

)=

1 √n

Sdt

ne.

Then,

En(t

1,t

2,t

3)=

n−2|d

t 1ne−dt

2ne||d

t 2ne−dt

3ne|≈|t 1

−t 2||t

2−

t 3|.

-22-

Tig

htn

ess

for

stochast

icpro

cess

es

Exam

ple

2.

Let{S

i}n i=

0be

neare

st-n

eig

hbourse

lf-a

void

ing

walk

on

Zdw

ith

dla

rge,and

write

for

t∈

[0,1

]

ωn(t

)=

1 √n

Sdt

ne.

Denote

measu

reof{S

i}n i=

0by

P n,and

expecta

tion

w.r.t.

P nby

E n.

Then,w

ith

c nth

enum

ber

ofSAW

ofle

ngth

n,

En(t

1,t

2,t

3)≤

c dt 1

necd(

t 2−

t 1)necd(

t 3−

t 2)nec

n−dt

3ne

c n

×n−2E d

(t2−

t 1)ne[ |S

d(t 2−

t 1)ne|

2] E d

(t3−

t 2)ne[ |S

d(t 3−

t 2)ne|

2] .

Thus,

tightn

ess

isim

plied

by

c n≈

nand

E n[|S

n|2

]≤

Dn...

-23-

Tig

htn

ess

for

super-

pro

cess

es

For

B⊂

Rd,define

the

random

measu

re

µn t(B

)=

1 n

∑x∈√

nB

1{(

0,0

)→(x

,dnte

)},

soth

at

〈µn t,f〉=

1 n

∑ x∈

Zd1{(

0,0

)→(x

,nt)}f

(x √n)

isth

eexpecta

tion

with

resp

ect

toth

era

ndom

measu

reµ

n t.

Pro

positio

nA.W

eak

converg

ence

of{µ

n t} t≥0

inth

evague

topol-

ogy

isequiv

ale

nt

to

(i)W

eak

converg

ence

offd

d’s

(〈µ

n t 1,f

1〉,

...,〈µ

n t N,f

N〉)

forevery

t 1,.

..,t

N≥

0

and

f1,.

..,f

Ncom

pactly

support

ed,bounded

continuous.

(ii)

Tig

htn

ess

of

stochast

icpro

cess

{〈µ

n t,f〉}

t≥0

on

Rfo

revery

bounded

continuous

function

fw

ith

com

pact

support

.

-24-