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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-
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
+π
(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
+π
(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
+π
(0) (
x)]
+∑
(u,v
)
pD
(v−
u)[
δ 0,u
+π
(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-
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≈
Aµ
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-