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

  • T h e

    su p e r-

    p ro

    c e ss

    li m

    it o f o ri e n te

    d p e rc

    o la

    ti o n

    a b o v e

    4 +

    1

    d im

    e n si o n s

    R e m

    c o

    v a n

    d e r

    H o fs

    ta d

    J o in

    t w o rk

    w it h :

    G o rd

    o n

    S la

    d e

    (U B C , V a n c o u v e r)

    F ra

    n k

    d e n

    H o ll a n d e r

    (E u ra

    n d o m

    ).

  • O ri e n te

    d p e rc

    o la

    ti o n

    O ri e n te

    d b o n d s

    jo in

    (x ,n

    ) to

    (y ,n

    + 1 )

    fo r

    n ≥

    0 a n d

    x ,y

    ∈ Zd

    .

    M a k e

    b o n d

    (( x ,n

    ), (y

    ,n +

    1 ))

    in d e p e n d e n tl y

    o c c u p ie

    d w

    it h

    p ro

    b a b il it y

    p D

    (y −

    x ),

    v a c a n t

    w it h

    p ro

    b a b il it y

    1 −

    p D

    (y −

    x ).

    H e re

    , p ∈

    [0 ,1

    / ‖D

    ‖ ∞ ] is

    th e

    p e rc

    o la

    ti o n

    p a ra

    m e te

    r.

    G o a ls

    In v e st

    ig a te

    sc a li n g

    b e h a v io

    u r c ri ti c a l o ri e n te

    d p e rc

    o la

    ti o n

    fo r

    d >

    4 .

    -1 -

  • O ri e n te

    d p e rc

    o la

    ti o n

    F ig

    u re

    b y

    B . C a ss

    e lm

    a n

    a n d

    G . S la

    d e

    (0 2 ).

    -2 -

  • K e y

    q u a n ti ti e s

    S u rv

    iv a l p ro

    b a b il it y θ n

    = P p

    (∃ x ∈

    Zd : (0

    ,0 ) →

    (x ,n

    )) .

    T w o -p

    o in

    t fu

    n c ti o n

    τ n (x

    ) =

    P p ((

    0 ,0

    ) →

    (x ,n

    )) .

    H ig

    h e r-

    p o in

    t fu

    n c ti o n s

    L e t

    (~x ,~n

    ) =

    (( x 1 ,n

    1 ),

    .. .,

    (x r − 1 ,n

    r − 1 ))

    ,

    τ (r

    ) ~n

    (~x ) =

    P p ((

    0 ,0

    ) →

    (~x ,~n

    )) .

    M a in

    q u e st

    io n

    H o w

    d o

    θ n ’s

    a n d

    τ (r

    ) n

    ’s b e h a v e

    a s

    n →

    ∞ ?

    -3 -

  • P re

    v io

    u s

    re su

    lt s

    O P

    h a s

    a p h a se

    tr a n si ti o n , i. e , th

    e re

    is a

    c ri ti c a l p ro

    b a b il it y

    p c =

    p c(

    d ,L

    ) ∈

    (0 ,1

    ), su

    c h

    th a t

    • fo

    r p ≤

    p c,

    th e re

    is a .s . n o

    in fi n it e

    c lu

    st e r.

    • fo

    r p

    > p c,

    th e re

    is a .s . u n iq

    u e

    in fi n it e

    c lu

    st e r.

    R e su

    lt fo

    r p =

    p c : B e z u id

    e n h o u t

    a n d

    G ri m

    m e tt

    (9 0 ).

    A p p li e s

    a ls o

    to c o n ta

    c t

    p ro

    c e ss

    , a

    re la

    ti v e

    o f o ri e n te

    d p e rc

    o la

    ti o n .

    Im p li e s

    th a t

    θ n →

    0 a s

    n →

    ∞ .

    -4 -

  • P re

    v io

    u s

    re su

    lt s

    In te

    rm s

    o f F o u ri e r

    tr a n sf

    o rm

    τ̂ n (k

    ) =

    ∑ x∈Z d

    τ n (x

    )e ik ·x

    , k ∈

    [− π ,π

    ]d .

    F o r

    p <

    p c,

    τ̂ n (0

    ) is

    e x p o n e n ti a ll y

    sm a ll .

    F o r

    p >

    p c,

    τ̂ n (0

    ) in

    c re

    a se

    s a s

    n d .

    F o r

    p =

    p c,

    b e h a v io

    u r

    τ̂ n (0

    ) n o t

    u n d e rs

    to o d .

    G o a l

    In v e st

    ig a te

    θ n ,

    τ̂ (r

    ) n

    (k )

    a t

    p c

    fo r

    d >

    4 ,

    w h e re

    d c =

    4 is

    th e

    c ri ti c a l d im

    e n si o n .

    E x p e c t

    B e h a v io

    u r

    si m

    il a r

    a s

    c ri ti c a l b ra

    n c h in

    g ra

    n d o m

    w a lk

    .

    M e th

    o d

    L a c e

    e x p a n si o n .

    (P re

    v io

    u s

    re su

    lt s:

    N g u ye

    n &

    Y a n g

    (9 3 , 9 5 )

    fo r tw

    o -p

    o in

    t fu

    n c ti o n .)

    -5 -

  • M o m

    e n t

    m e a su

    re s

    o f c a n o n ic

    a l m

    e a su

    re su

    p e r-

    B ro

    w n ia

    n m

    o ti o n

    {X t}

    t≥ 0

    su p e r-

    p ro

    c e ss

    = m

    e a su

    re v a lu

    e d

    d iff

    u si o n .

    F o u ri e r

    tr a n sf

    o rm

    o f jo

    in t

    m o m

    e n t

    m e a su

    re is

    M̂ (l )

    ~ t (~ k

    ) =

    E µ [ ∫ Rl

    d ei

    k 1 ·x

    1 X

    t 1 (d

    x 1 ) ·· ·e

    ik l·x

    l X t l (d

    x l)

    ] . {X

    t} t≥

    0 is

    su p e r-

    B ro

    w n ia

    n m

    o ti o n .

    Id e n ti fy

    m o m

    e n t-

    m e a su

    re s o f S B M

    re c u rs

    iv e ly

    b y

    M̂ (1

    ) t

    (~ k ) =

    e− |k |2

    t/ 2 d ,

    M̂ (l )

    ~ t (~ k

    ) =

    ∫ t 0d t

    M̂ (1

    ) t

    (k 1 + ·· ·+

    k l)

    ∑ I ⊂

    J \{

    1 }: |I |≥

    1

    M̂ (|

    I |)

    ~ t I −

    t( ~ k

    I )M̂

    (l − |I |)

    ~ t J \I −

    t( ~ k

    J \I

    ),

    w h e re

    J = {1

    ,. ..

    ,l },

    t =

    m in

    i t i

    , ~ t I

    = (t

    i) i∈

    I , a n d

    ~ t I −

    t =

    (t i −

    t) i∈

    I .

    -6 -

  • R e su

    lt s

    O P

    T h e o re

    m 1 .

    F ix

    d >

    4 ,

    p =

    p c,

    r ≥

    2 a n d

    δ ∈

    (0 ,1

    ∧ � ∧

    d − 4

    2 )

    a n d

    ~ t =

    (t 1 ,.

    .. ,t

    r − 1 ) ∈

    (0 ,∞

    )r − 1 , ~ k

    = (k

    1 ,.

    .. ,k

    r − 1 ) ∈

    R (r − 1 )d

    . T

    h e n

    th e re

    e x is t

    L 0

    = L

    0 (d

    ) a n d

    fi n it e

    p o si ti v e

    c o n st

    a n ts

    A =

    A (d

    ,L ),

    v =

    v (d

    ,L ),

    V =

    V (d

    ,L )

    su c h

    th a t

    fo r

    L ≥

    L 0 ,

    τ̂ (r

    )

    bn ~ tc (~ k

    /

    √ vσ 2 n ) =

    A 2 r − 3 V

    r − 2 n

    r − 2 [M̂

    (r − 1 )

    ~ t (~ k

    ) + O

    (n −

    δ )]

    .

    E x tr

    a re

    su lt

    fo r

    r =

    2 :

    su p

    x τ n

    (x ) = O

    (L −

    d n −

    d / 2 ).

    Im p li e s

    C o n v e rg

    e n c e

    fi n it e

    d im

    e n si o n a l d is tr

    ib u ti o n s o f O

    P to

    S B M

    .

    M is s

    T ig

    h tn

    e ss

    .

    -7 -

  • R e su

    lt s

    O P

    T h e o re

    m 2 .

    U n d e r

    th e

    a b o v e

    c o n d it io

    n s

    θ n =

    1 B n [1

    + o (1

    )] .

    A ls o ,

    B =

    A V

    / 2 .

    Im p li e s

    th a t

    c o n d it io

    n a ll y

    o n

    0 →

    m ,

    m − 1 N

    m = ⇒

    E x p (λ

    ) w

    it h

    λ =

    2 / (A

    2 V

    ),

    a n d

    N n

    = # {y

    ∈ Zd

    : (0

    ,0 ) →

    (y ,n

    )} .

    -8 -

  • In c ip

    ie n t

    In fi n it e

    C lu

    st e r

    K e st

    e n

    (1 9 8 6 )

    h a s

    c o n st

    ru c te

    d th

    e in

    c ip

    ie n t

    in fi n it e

    c lu

    st e r

    (I IC

    )

    fo r

    p e rc

    o la

    ti o n

    o n

    Z2 .

    II C

    d e sc

    ri b e s

    lo c a l st

    ru c tu

    re o f la

    rg e

    c ri ti c a l c lu

    st e rs

    .

    W il l

    p ro

    v id

    e a

    c o n st

    ru c ti o n

    fo r

    o ri e n te

    d p e rc

    o la

    ti o n

    in d im

    e n si o n

    d >

    4 .

    F o r

    c y li n d e r

    e v e n ts

    E , d e fi n e

    Q n (E

    ) =

    P