A new approach to the long-time behavior of self-avoiding random walks

ANNALS

OF PHYSICS

217,

142-169 (1992)

A New Approach to the Long-Time Behavior of Self-Avoiding Random Walks* STEVEN GOLOWICH~ Department

of Physics,

Harvard

University,

Cambridge,

Massachusetts

02138

AND

JOHN Z. IMBRIE Department

of Mathematics,

University

of Virginia,

Charlottesville,

Virginia

22903

Received January 23, 1992

We study a T step weakly self-avoiding random walk in d > 4 dimensions by treating it as a one-dimensional statistical mechanical system. We use the polymer expansion along with the lace expansion to arrive at an expression for the Fourier-transformed propagator of the self-avoiding walk, valid in some neighborhood of the origin, that up to small edge effects is equal to that of some simple random walk. This yields simple proofs that, as T+ 00, the variance of the endpoint is asymptotically linear in T and the scaling limit of the endpoint is gaussian. 0 1992 Academic Press, Inc.

1. INTRODUCTION

AND RESULTS

We study a model for weakly self-avoiding random walks on a d-dimensional hypercubic lattice. The model is defined by the two-point function c n Cl- nL@,) 0:0+x O
(1.1)

Iw( = T

which is proportional to the probability of travelling from 0 to x in T steps. Here d is the dimension of the lattice, o: [0, T] + Zd any path on the lattice beginning at the origin with nearest neighbor steps, and 1 the strength of the self-interaction. We also define the Fourier-transformed propagator by C(k, T) = 1 eik’xC(x, T) x

(1.2)

* Research supported in part by the National Science Foundation under Grants DMS 91-20962 and DMS 91-96161. t Supported in part by a National Science Foundation Graduate Fellowship.

142 0003-4916/92 $9.00 Copyright 0 1992 by Academic Press, Inc. All rights of reproduction in any form reserved.

SELF-AVOIDING

RANDOM

WALKS

143

with k E [ -rc, rrId and, here and throughout, the arguments will serve to distinguish functions from their Fourier transforms. The above model, when A = 0, reduces to the simple random walk with nearest neighbor steps; i.e., the probability distribution p(x, y) governing a single step is proportional to the nearest neighbor coupling matrix JXY- 1 ,x-y, = 1. In general, the distribution of the endpoint of a simple random walk after T steps (which is a sum of displacements of single steps) is equal to the convolution P*~(O, x), which in Fourier space is just p(k)? So in the nearest neighbor case, we have C,,,(k, T) = D(k)=, where D(k) =;

i: cos(k,).

(1.3)

p=l

We will exhibit an exact expression for C(k, T) in some open neighborhood of the origin. Specifically, for any T’, we will define a function T,.(k) such that for T6 T’, C(k, T) = (D(k)e rT’(k))T. (edge effects).

(1.4)

We will show that for k in some neighborhood of the origin (which depends on T’) T,.(k) and its first two derivatives are 0(A) when d> 5 and that the edge effects are negligible (these statements will be made more precise below). This is a particularly revealing form since, for k small, it mimics the form of the propagator for some simple random walk (up to edge effects) and hence makes explicit the fact that selfavoiding random walks behave like simple random walks above four dimensions. It also allows us to easily compute quantities of interest; for example, we find the mean square end-to-end distance:

(N72) = i, a$k(k;;’ 3 /k=O = T(l + O(I));

(1.5) (1.6)

i.e., the effect of self-avoidance is to add a small correction to the diffusion constant of the free walk. This model has previously been studied by Brydges and Spencer in [l], who introduced the lace expansion and used it to prove diffusive behavior of the endpoint of a T-step walk and the Gaussian nature of the scaling limit. They studied the Laplace-transformed two-point function C(k, z) (the generating function of C(k, T)), and used Cauchy’s theorem to extract fixed-T information. Their results were extended to, and other results proven about, the case of the standard self-avoiding walk above some high dimension do in [9-11,7], and in live or more dimensions in [6]. These works also used the lace expansion and generating functions, but in different ways. Related works [3-5, 81 used similar methods to 595/217/l-10

144

GOLOWICH

AND

IMBRIE

prove results on percolation and lattice trees and animals. In this paper we also use the lace expansion, but with a view towards other models (e.g., branched polymers) we dispense with generating functions and work directly with the propagator in (k, T)-space using techniques drawn from statistical mechanics. The principal technique we use is the polymer expansion. To see how such an expansion can arise, we rewrite the two-point function in a more suggestive form. We expand the product n (1 - AS,,,) into a sum of graphs on the interval [0, T]. These graphs split into connected components, where a graph that spans an interval Z is considered connected if every point in the interior of Z is spanned by some bond in the graph. For an interval Z we define the activity

c c n (-~~oJ$J~

0:0+x 14 = III

(1.7)

GC (s.t)eGc

where G’ are connected graphs consisting of bonds (s, t) on [0, 7’1. Then it is easy to see that (1.8)

where the {Ii} are nonoverlapping intervals in [0, T]. This is a form amenable to polymer expansion, but we cannot prove its convergence in this form. Instead, following [l] and using the idea of the renormalization group, we first consider walks that self-avoid only on short lengthscales and gradually work our way up to fully self-avoiding walks. Roughly, we define a sequence of times T, indexed by a number Z, and write as C,(k, T) the propagator for walks that self-avoid only on times shorter than T,. Then, defining Z(T) by T,, TJ= T, we write

1... C!(T) Cl(T).-Cl(T)-= co Cl(T)1 Cl(T)--2

(1.9)

and find an expansion for C,/C,- I similar to (1.8). For an appropriate choice of T, we can prove convergence of such an expansion. The structure of the rest of the paper is as follows. In Section 2 we will exhibit the expansion for C,(k, T), the convergence of which will be proven via an induction on timescales in Sections 3-6. In Section 7 we use the expansion to prove the main theorems, which we now state. 1.1. For any T’ > 2 ther e exist real-valued functions T,,(k), Y$@(k, T) and T’-independent constants A0 > 0, f > 0, E’ > 0, and W > 0 such that for any T< T’, with O TIPfIT’ THEOREM

C(k, T) = (D(k)err’(k))T.

&“(k,T)

(1.10)

SELF-AVOIDING

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145

WALKS

and

where u is a multi-index for k-derivatives, and Iuj d 2. There is also convergence of T,.(k) in an appropriate T’ --) co. This is used in the following theorem. THEOREM

sense near k =0

as

1.2.

(ri~(T)~)=llT(l

+10(T-“4)),

(1.13)

where D = 1 + %?,I,with ‘%:independent of T. Also, the scaling limit of the endpoint is gaussian in the sense that ,im

C(kl&,

S“X

st)

C(O>

=

e - Dr(k*/2d)

(1.14)

St)

for any real t and k E Rd, with the same D us before.

2. THE EXPANSION We first define the propagator smaller than T,:

for the model that self-avoids only on scales

This gives rise to an activity ZZ,(k, I), just as in (1.7), except now the bonds (s, t) in the connected graphs are restricted in length, Is - tl d TI, We then define 617,(k, T) = Zi=,(k, T) - Z7-,(k, and it is not hard to see that

T)

(2.2)

146

GOLOWICH

AND

IMBRIE

Here {Ii} are again nonoverlapping intervals on [0, T], and (Kj} are the complementary intervals; e.g., see Fig. 1. The last term here acts as an interaction between the Ii, so we must do a further expansion to obtain a form amenable to polymer expansion. First, we make an inductive hypothesis about the result of the expansion:

C,Ll(k T) = exp(TWCI-2(k 0

,(k) + 2Wf-

,(k) + W:P ,(k, T)).

(2.4)

Note that this implies CL- ,(k T) = @kFexp(TT,-

,(k) + 2ff?- ,@I + r:T ,(k, T)),

(2.5)

where Izy!,(-)=

1

1 j=l

ay.).

(2.6)

In the above, exp(Z,) is the multiplicative correction to D(k), and Zy and Z:” are edge effects arising from interactions with just one and both edges of [0, T], respectively. This form holds trivially for Tl < 2, since this case reduces to simple random walk (the shortest subwalk that can intersect itself is two units long). In the remainder of this section, we will define ST,, Mp, and 6ZjF so that CJC,-i is in the form of (2.4). We will then be able to prove estimates on these quantities in the following sections. We begin by substituting (2.5) into (2.3), to find

CA T) C,-l(k,

T)=e

-rjFlw,T) C 1 n=O

x fi

ii (6WYk {It) i=l

(1 + dZZ:*(k, Z$)),

Ii)) (2.7)

j=O

where we have defined (2.8)

(2.9)

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147

WALKS

and (Ii> are nonoverlapping intervals with integer endpoints in [0, T], while {Z$} are the intervals between the Z;s, as in (2.3). Now, we see that the interactions between the intervals Zj are the terms 1 + 6ZZp*, so we must expand the product overj. The resulting sum of graphs splits into connected components of two types: those connected to the edges by a factor of sZZy* and those that are not. We define activities for both types of connected components. The activity of an interval not connected to an edge is given by

sfm, 4 = c 1 fi (sn:(k,ZJm:*(k,Ki))m:(k,I,), n=O

(I,)

(2.10)

i=l

where the {Ii} are nonoverlapping intervals on .Z such that the left and right endpoints of Z, and Z,, respectively, coincide with those of J. The activity of an interval connected to an edge is

“;tk> J, = 1 n=l

C fi (6n:*(k, Ki_ 1)dIly(k, Ii)), {I,}

(2.11)

i=l

where the {Ii} are nonoverlapping intervals on .Z such that the right endpoint of Z, coincides with that of J. The expression for C,/C,-, now splits into three parts: C,(k T) = {no interaction with edges} Cl-1% T) + (interaction with one edge only} + (interaction

with both edges}.

(2.12)

In a moment, we will apply the polymer expansion to the piece that does not interact with the edges; for now, define

P(k I)= 1 1 fi (dfi,(k, Ii)), n=O

where the {Ii} are nonoverlapping allows us to write

CAk T) C/- ,(k, T) = e

-rjF1(k.T)

+

c

KL,KR

{I,)

(2.13)

i=l

intervals on Z, otherwise unrestricted.

This

P(k, T) + 2 1 Gif;(k, KJ P(k, I) KL ~&‘(k, KL) W,

1) Gif;(k, KR)

(2.14)

KL~K~=(~

where K,, K, are intervals contained in [0, T] with left and right endpoints at the left and right edges, respectively, of [0, T], and Z is [0, T]\{KL u KR}.

148

GOLOWICH

AND

IMBRIE

We now must recast this into the form of the inductive assumption. The first step is clearly to perform a polymer expansion on P(k, I), as follows (see, e.g., [2]), W,

0

=exp

fi

(6iI,(k,

Ji))

C

n

A(J,,

(s,r)eGc

J,)

,

(2.15)

>

on I

where (Jr, ...) denotes an ordered collection of intervals on Z (otherwise unrestricted), G’ are connected graphs on { 1 ... n}, and A is defined by -1 NJ,, J,) = o

if J,n J,# 12/ otherwise.

(2.16)

Now, (2.15) becomes P(k, I) = exp( TGZ,(k) + 2 6Z;7l(k) + 6ZIF’(k, I)) if we make the appropriate

(2.17)

definitions. Let

where the meaning of C’ depends on the quantity being defined: W,(k) = s,

(2.19)

where we require inf(J, u ... u J,} to be the left endpoint of Z, 2w3k)

(2.20)

= -s,

where we require (J1 u ... u J,} to intersect the right endpoint of Z, and GzyF’(k) E s,

(2.21)

where we require (J1 u ... u Jn} to intersect both endpoints of I. Note that we have begun to recover the inductive form, as 6Z, is one of the three functions we need. The other two will include contributions from the edge terms in (2.14) as well as 6Zd1 and 6Z IF1. To see what they are, we rewrite (2.14) as

C,(k T) = exp(TW,(k) C,-l(k T) x cl +2&(k) where

+ 2 W:‘(k)

- rj’,(k,

T))

+ A,(k) + B,(k, T) + 2B,(k, T) + B,(k, T)],

(2.22)

SELF-AVOIDING

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WALKS

(2.24) IK! = 1

B,(k, T) s -

f

aji;(k, Qe-

IKI = T+

x [Sli;(k,

1x1afr(k)

1

K,)e-‘KRJ sn(k)]

(2.27)

with K,., I, iu, defined as in (2.14), and kri, rR intervals on ( - cy3,CO) that intersect the left and right endpoints of [O, T], respectively, as well as each other. So we are led to define 26r~2(k)=ln(l

+2A2+A3)

=21n(l

(2.28)

SA2) (1 -t B, + 2B, f B3) G4, + A,)(B, + 2& + 4)

‘--(t+2A,+A~)(1+B,+2B2+B,)

)I

(2.29)

and we can finish recovering (2.4) by defining W;(k) W;F(k,

= W;‘(k)

T) = 6rf=(k,

-t- C?(k) T) - r;”

(2.30) ,(k, T)

(2.31)

which tells us that rfF(k,

T) = WfF2(k, T).

(2.32)

150

GOLOWICHANDIMBRIE

3. INDUCTION STARTED; MAIN ESTIMATES We now begin the proof of convergence of the expansion described in the previous section. We achieve this with an induction on T,, the scale on which walks self-avoid, with the choice of T/ E f’*.

(3.1)

Note that for T, < 2 the model becomes the simple random walk, so for I< 1 we have Zi.‘( .) E 0. We make the following inductive assumptions: 6) (ii) (iii)

Id; W,(k)1 < CIllTj”“2T;~~‘2+”

(3.2)

18; W;i(k)l

(3.3)

la~r~‘(k,

< G&lTj”“2T;~~‘2+E

T)I <

1 (iv)

(‘G&A) T~‘/‘Tf:fl*+’ for T< T I _ 1 (%YllIz)rT’T’lTjui’2TZ-d’2+E for T,- 1 < TdN3 6% 1) rmi Tjui/*T,-a3T/TI for T, N, T,

TI

(3.4)

Ila;C,( ., T)lli‘Space< WI2T~“~‘2-d’2pexp( T W,(O) + 2Wf(O) + bI’:F(O, T)) for

T> TI+1, 1
(3.5)

where [cl1 denotes the smallest integer greater than or equal to u. Throughout this paper we will use the symbol g to denote positive real constants independent of induction step, k, and T (they may depend on the dimension d). V’ will denote a constant that is independent of $i, while 9? may not be. (i)-(iii) are valid for ke UI, where U, = (k: ID(k)1 a S,}

(3.6)

and for for

lcl Ia 1;

(3.7)

f, N,, and a3 are constants to be determined in the course of the proof, and we

choose E E $. 1 is fixed smaller than some I,, and d> 5; u is a multi-index for derivatives with respect to ki, and I uI = C ui G 2. A comment on the origin of the restrictions on k in (i)-(iii) is in order. We see that D(k)“’ appears in the denominator of (2.8), which must be cancelled by the decay of 6ZZ,(k, I) with I. Clearly this decay becomes slower as T, increases, so we can expect the aforementioned cancellation to occur only for JD(k)J close to one; i.e., for any given value of k we can perform the expansion only out to some finite value of T, (which depends on k). Since the long-distance behavior of the walk is reflected in the small-k region of C(k, T), the fact that our expansion will only be valid in this region need not concern us. In proving the iteration of (iv) we will need

SELF-AVOIDING

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RANDOM WALKS

to control the contributions to the norms from the large-k region, but this is easily accomplished (Section 6). At the beginning of the induction, I< 1 and all of the 6ri’ are zero, so (ik(iii) hold trivially. Also, it is easy to see that (iv) holds for C,(k, T) = D(k)? So we must show that the hypotheses iterate. PROPOSITION

3.1.

Zf (i t(iv)

hold for I - 1 then they hold for I, any I k 1.

We proceed in three stages. First we consider the contributions to (i)-(iii) arising from the polymer expansion (2.17), then the corrections due to edge effects (2.28) and (2.29). Once the proofs of (i)-(iii) are complete, we can prove the iteration of (iv). Since the polymer expansion is in terms of the quantity Sfi,, we prove estimates on it first. The main result of this section is LEMMA

18;: dii,(k,

3.2.

T)I <

For any 6’ > 0 there exists a choice of gII so that 0 G’(WIik;,A)‘=‘=”Tj”“2T”-d’2 S’WI, 2)

YT/T/l

T~U~/~T;"~(T/TI)

for for for

T< T,-,/6 T,- 1/6 < T< N, T, T,N, T,,

(3.8)

where a,, N, are constants to be determined and k E U,.

Since i?, is defined in terms of 617* and 6ZI** in (2.10) we prove estimates on these first. Now, 6I7** is easy, since from its definition we see Idn;“*(k,

K)I = Ier:rl(k,K)-

11


(3.9)

and we can use the inductive assumption (similarly for its derivatives). SZ7T will require more work. We prove LEMMA

3.3.

18;: UZ,?(k, Z)I <

;%?‘I)rl”/TJ1 T1”“” IZl”-d’2

where kE U,, fixing a, 6 i(d/4$xing N, such that a,N, >d/2-

for for for

1) - (%f + WA) -6,

111< T,_,/6 TIP 1/6 < IZI d N, T, 111> N, T,,

(3.10)

with any choice of 6 > 0, and

E.

Prooj First consider the case of u = 0. We begin by obtaining bounds on hII,. Using the lace expansion of [ 1 ] (see the appendix for a proof of the first inequality below), we see that

152

GOLOWICH

AND

IMBRIE

Zm-1 W,(k

01


c

A”

c

m=l

“Ix,

n

{PI,,,‘,;’

(wrn)m

(I,

IIG(n,)-l(.?

njNk;space

J= ’

(‘PI’

P*)

2m-1 xexp

jCI

Cnjrl(nj)-

Ito)

+

2rf’,n,)-

Ito)

+

r&-

,Cnj,

(3.11)

O))),

(

where Z(n) is defined by TIC,) = n, {q} satisfies 2 nj= 1Z1,at least one ni> T,-,/6, all nj < T,, and * = 2 or 1, with exactly one 1 in the product. In the second line we have substituted the inductive assumptions. We can now substitute this into (2.g), which yields

The important this in LEMMA

x exp -III [ (

rl-,(k)+Cnjrl(,)-,(0)

-2c

l(k) +I

)I

i

))I1

l(O)+ r&

W(,,-

i

,(nj, 0)

. (3.12)

term here is t,, while t, and I, are just small corrections. We prove

3.4. sup t,t, G exp(%Ak2 [I( + mVA), In!)

(3.13)

where k E Ul. Proof. Fix a choice of (ni}. Using (i), along with the fact that f, is even in k, we see that Ir,- ,(k)l < r,- I(O) + %?Ak’.

(3.14)

Inserting this into our expression for t2, we find It21 Gexp

(

%X2 111+C nj i

I-

1 i=I(nj)

1

dI’i(O) >

@?:lk’ 111+ c nj%?AT~$;:~‘* , > i

(3.15)

SELF-AVOIDING

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WALKS

153

where we have used (i) and performed the sum in the second line. Now, T&:~” < VT,T,:,, so njT&:“12 < %, and we have It*1 < eV’lk2“‘(ebb)?

(3.16)

The third term t, is even simpler as no cancellations need occur; we just directly substitute the inductive hypotheses to see that sup t, 6 (eMA)m :%I and we are done.

(3.17)

1

Next we must handle the main term, t,. We first need LEMMA

3.5.

d/2* <

0

for

111< T,- ,/6

cg’m 111-42

for

T,~,/6~14~N,T,

C&WW#~1~1/T/7-(1/*)(1 --d/4)(lWT/) I

for

14> N, T,,

(3.18)

where *, {ni} are defined as in (3.11). Proof: For 111< TIP ,/6, we observe that at least one ni must be longer than T,- ,/6. For (T,- ,/6) < 111< N, T,, we note that at least one of the ni must be longer than lZl/(2m - 1). We use the condition C ni = 111 to fix the length of the longest interval given the other 2m - 2. Inserting a combinatorial factor of (2m - 1) for which interval is long and taking * = 1 on the long interval, we obtain

For IZ( > N, T,, we set * = 2 on all nj as an upper bound. We note that at least rjZl/2T,J of the n, must be longer than lZl/2(2m - 1) in order to satisfy C ni= 111, subject to the constraint that ni < Tl. Next insert a combinatorial factor of ([fi;;l:l,) for which intervals are longest, and on these intervals use the bound

(3.20)

154

GOLOWICH

The short intervals contribute

AND

a constant factor each. So

Next use the general result (j) 6 (U’(i/j))j

< ~‘~~‘lW’/~jl/W

which completes the proof.

IMBRIE

to see that

-44)(14/T/)

(3.22)

1

Finally, we put the results on t,, for k near 0,

t,,

and

t,

together to bound sI7:. We note that, (3.23)

and when ID(k)/ > 6, along with k near 0, we have (3.24) We require A small and fix f so that N,(gf+ %A) -CE, and noting that the series in 1 starts at some m 2 rpp,l, we obtain the result for u = 0 (the 6 in the definition of a, comes from the derivatives, which we consider next). To handle the derivatives, we must consider their action on each of the four terms that constitute 6I7: (and Leibnitz’s rule gives us a factor in front that is dominated by the powers of A). We easily see

ia; cml(k, Z)I < (%FIZl’““‘) c (WAy c (‘yj l ny*) Wl=l

in,)

i=

(3.25)

1

*I?-1 x exp

jTl

tnjrl(nj)-

l(O)

+

2rf(nj)-

l(O)

+

rig,)-

ltnj3

0)) >

Now, use lZl* In T, <%‘Tf+6(11i’T’),

any6>0.

(3.26)

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WALKS

This allows us to absorb the extra factors of 111 that the derivatives give into the exponential, which makes subsequent bounding of derivatives convenient: the action of a derivative on our estimates is just to bring down a factor of T:‘*. Note that the same is true for 6I7p*. 1 We can now combine the results on 6l7: and 8Z7:*. We will need LEMMA 3.6.

Define T-l F(T)

=

where k E U, and I = [0, T]\K.

c IKl=O

Sn:(k,

I)

NIy*(k,

K),

(3.27)

Then for any 6’ > 0 there exists a choice of y1 so that

Proof. Fix 6’ > 0 and for now take IuI= 0. For T< TI- ,/6, we have &ll:(k, T) =O. We separate the /I dependence from the T dependence by noting

that if we divide each term in the sum by {~‘~)ri~i’~’ ~~~~~)r’~“~-ll we obtain, for (T,-,/6)GT<2N,T,-,, E-d/2 CR’

z

NjT/-1 ~ii’“‘+‘6’(~2+i-d’2(,=~,,0~.-d-2+~~,T,T;~,laT,,)

13

c K=l

g$$‘T3+“-d’2T”-d/2 f-1

3

(3.29)

where we have used (3.10) and (3.9), and for Ta 2N3 TII 1,

where we require a3, N3 to satisfy a3 N3 > d/2 - 2 - E. Now we bound the sum by multiplying this result by the leading order of %A, which we find by noting that the bonds in MI,? are T, units long while those in iSI:* are only TI- 1, meaning that at most one power of Iz can come from the 617:* at leading order. Hence the leading A dependence is, for T d T,,

and for T > TI it becomes (3.32)

We see that by taking $$r, large enough we obtain the result.

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GOLOWICH

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For IuI > 0, the derivatives bring down the desired factors of T,, along with a harmless factor from the Leibnitz rule. 1 We are now in a position to prove Lemma 3.2: Proof Fix 6’ > 0. For the n = 1 term, we have S&‘, = sZZ:, and so the result follows from Lemma 3.3 as long as we require a2 N,, with %n large enough. Now consider the n > 1 terms. First let 1~1= 0. For T-c TIP ,/6, all of the subintervals in the partition of [0, T] are shorter than T,- 1/6 and hence the CUE,? terms are zero. For T-c N, T,, we take {Ji} a partition of [0, T] and note that the longest interval has length at least T/n. So with %Tre,, large enough we have

E - d/2 (T;f:“-“)“-I

+I&

r TIT~ T” - 42.

(3.33)

For T > N, T,, we look at Iafi,(k

T)In>

lterms T?‘r’T” n-l

< 1 1 n n>l {lj} j=l

(IhZZ:(k, Zj) T:2”‘/“T”1 IH7:*(k,

KJ Ty”q”T’)))

x IcU7:(k, Z,) Ty”‘n”T”I (3.34)

where we have analyzed the 1 dependence as in the proof of Lemma 3.6, inserted our bounds for HI: and 6I7,?*, and defined s,, s2 as NITI s1 E

1 III=

gt

[q-d/2

~;12(lIt/T/)

+

1


c

cT&T~~W)(~~~/~l)

III z=-NIT,

(3.35)

7

where we fix a2 such that a2 N, < 1 - 3.5 and (a, - a2) N1 > d/2 - 1 + 2~. Also, s2=

NJT/-I c V’ (KI 2+ &- 42 T;2(lKI/T/) + IKI = 1

<%T-,

3 t E ~ d/2

c

f ~rlKI/TI-tlT--o3(1K1/~/-I)T;IZ(IKK(/~/) l-1 QT/J

IKI=-N~T/LI .

(3.36)

SELF-AVOIDING

157

RANDOM WALKS

Note that we have used the fact that H7:* contains bonds of length at most TIP I, so after we pull out a factor of ~r)KI’T’l we are still left with a factor of JVrJKIITI-” for T, > 2. So the n > 1 terms give us

For 1~1> 0, again the derivatives bring down the expected factors of T,, along with harmless factors from the Leibnitz rule. 1

4. CONVERGENCE OF POLYMER EXPANSION LEMMA

4.1.

For any 6’ > 0 there exists a choice of $, such that

where k E U, and the sum is over all intervals Ic (-co, interval J. Proo$

co) that intersect some

We write

IJI ,p.)=~ow~,

c inf{l)

I

(.)

(4.2)

= j

= ,,;, IZlj+’ I~fi,(k 4 + IJI ,f

III’ lSil,k 111,

(4.3)

‘I’ = 1

where we have defined the origin to be at the left endpoint of J. Apply Lemma 3.2, fixing N, so that a2 Nz > d/2 -E. m We are now in a position to prove convergence of the polymer expansion. We begin with Sr,, defined in (2.19). We overcount configurations of intervals by requiring only J, to satisfy the condition inf{ J1 } = 0 and inserting a combinatorial factor of n for which interval has its endpoint fixed. We do not restrict the other intervals. We can now use standard techniques to bound the sum (we only sketch the argument here; see [2] for details): first note that, given a connected graph G’ and a compatible configuration of intervals { Ji}, fl,,, 1jEGCA(J,, J,) = ( - 1)b(GC), where b(G’) is the number of bonds in G’. Then we have

158

GOLOWICH

AND

IMBRIE

i.e., for an upper bound we can discard bonds (1 + A) until the intervals are connected in a tree structure (see [2] for proof). Recalling that we are summing over ordered collections of intervals, we choose .Z, to be the root by thinking of it as attached to an additional vertex with coordination number fixed at one. We then form a tree of n + 1 vertices with coordination numbers (1, d, + 1, .... d, + l), defining K to be the number of vertices with coordination number greater than 1. We choose J2, .... J, +d, to be the intervals connected to J1, then to be those connected to J2, etc., and multiply by a J 1 + d, + 1 2 .--7 J l+dl+dz combinatorial factor of the number of trees with this structure (n - l)!/(d,! . ..d.!) (given by Cayley’s formula). Summing over all trees of this type, we arrive at n=l u

lC=l

JI

4 = 1 Wd,

inf(/l}

XC c ... 4= 1 (J)d2

=0

c d,c=

c 1

(J)d,

x fi ~JI, Jl+s,) fi A(J,,J,+,,+,,b s, = 1

s* = I

fi ISfi,(Ji)l,

(4.5)

i=l

where we have defined

I-

c

+d,-,+I..-rJl+d,+

(Jl+dl+-

(4.6) Cd;-,+d,)

and the {di} satisfy f

di=n-1.

(4.7)

i= 1

Now fix n, rc, and the {di} and perform the sums on the (Ji}. Apply Lemma 4.1 in the following manner: the terminal intervals (those furthest down the tree) give factors of G~?‘;lTif;-~‘~ 111, where Z is the node they are forced to intersect. So if dJ terminal intervals overlap some J, and J is forced to intersect some J’, the CJnJ, gives J;J, IS&(/t, J)I (WAT;‘;-d’2

IJI)dJ<(W;l)dJ+l

(d,!) T;‘;-d’2

IJ’I,

(4.8)

where we have used T~ff-d12T,<%“. Note this gives the same power of T,-, multiplying the length of the interval IJ’I as did the terminal intervals. So we can proceed in this manner up the entire tree until we get to the root, which requires special treatment as it is a restricted sum: for d, = 0, 1 we can choose WI1 so that for any 6’ > 0 f’

VII= 1

I&+,(/k, Z,)I (T;-‘;-“”

IZII)d’~6’~~;1~T:t;-d’2

(4.9)

SELF-AVOIDING

RANDOM

1.59

WALKS

while for d, 2 2 we find $,

Icmr(k, I,)/ (Tpy”‘2

(z,/)““~s’~;,~T:~:+“-d’2’T:+&+d~--lf/* GY%&AT;_+,“-@.

(4.10)

So (4.5) becomes

(4.11)

Now, there are ( ;I:) Ck (;I;) = 2”-2, so

solutions

to

the

equation

1sr,(k)l GY~~JT;);-“/2

Cf=, d, = n - 1, and (4.12)

which is the desired result for lul= 0. Derivatives act on the Sfi, terms to bring down a factor of Tj”“” along with harmless factors from the Leibnitz rule, so finally for %‘rI large enough, any 6’ > 0, Ii?;: W,(k)1 < i?‘%?f,tZTj”“2T;_+,“-d’2.

(4.13)

Clearly, by looking at (2.20) we see that 6r;j’ goes exactly like Jr,, except instead of requiring the root sum to have one endpoint fixed we only require it to intersect a given point. So the root sum gives a factor of Tf_‘,Ewdi2, and we can obtain for any 6’ > 0 Id; cWf’(k)l ~~‘~~,1T1”“2T:+Ic-d/2.

(4.14)

We must treat Sf fF1 a little more carefully because of its T dependence. con~gurations by only requiring For T< TI-,, we overcount contributing (.z, v . .. u Jn) to intersect 0 instead of both 0 and T. This reduces it to the case of 6rF’, so that result applies. For Tt- r < T< N3 T,, we note that at least one of the intervals is longer than T/n, so we require the root to be at least this long and put in a combinatorial factor of n. Now, every configuration that contributes must have at least rT/T,J powers of I, since they must all have connected sets of intervals that stretch from 0 to T. We overcount configurations by allowing some that do not stretch all the way, but we multiply these by enough powers of A so the total is rT/T,l. We then proceed as before, and here the root sum yields, for d, = 0, any 6’ > 0 with VI1 large enough,

160

GOLOWICH

AND

IMBRIE

and for d, > 1

(4.16) which is the desired result. For T > N, T,, we show that for any 6’ > 0 with %?rk;, large enough, l~W’~‘(k, T)I TF’T’T” < S’(%&l)rT’T”

(4.17)

which implies the desired result. To show this we will need LEMMA

4.2. ,&

For any 6’ > 0 there exists a choice of WI1 such that IcSil,(k, Z)I III’ Ty(““T’) <~‘~,,~“~j!

TjT;f;-“‘+‘I”

IJI

for

IJI a-

T,-I

6 ’

(4.18)

where k E U,. Proof We fix Lemma 4.1. 1

a3 so that

a,N, < $. Then

We now proceed as in the proofs of ST, and 6rf’, lGT’F’(k, T)J T:3’T’T”
#(%‘,,A)

everything

proceeds

as in

finding

max(n,r~/~h)

2nT2+E-d/2i1/4 I-

1

n <

6’(‘%;,

(4.19)

pT”.

Once again, derivatives yield a factor of T, lui”. So we have shown, for any 6’>0 and %‘n large enough, ,3’~$&74/27--42 18; wFi(k,

T)I G

d’(Q?Ill)‘T’T” ~‘(~,I 1)

rT/T/l

TjU”2T2+“-d’2 T~I/~T,-W(T/T/) /

5.

EDGE

for for for

T< T,-, T,-, < T6 N3 Tl T> N, TI.

(4.20)

EFFECTS

In this section we obtain estimates on the edge effects (2.28) and (2.29), which allows us to complete the proof that (ik(iii) iterate. It is clear from Iln( 1 +x)1 < %Z’1x1 for small x that we need only bound the Ai and Bj and the results for Srf and 6rfF2 will follow.

SELF-AVOIDING LEMMA

RANDOM

WALKS

161

5.1. For any 6’ > 0 there exists a choice of %& such that Ii?;: Sf ;‘z(k)i d6’Ce,,/Tj”‘f2Tff-d12,

(5.1)

where k E UI. Proof Fix 6’ > 0. Since A 3 = A:, we need only prove the result for A *. Recalling our definition of F in (3.27), we see that (5.2)

where {Ji> is a partition of K. Using the same methods as in Lemma 3.2, we find for any 6” > 0 and Q?nlarge enough, 0 ia; cm;(k,

K)j

G

syge,,A)

for

rwm

qmp+;.-

d12pq:1”-42

rIRI/Th Tj”l/2T:f,E-d/2T;“z”Kl/T/) 1 w’%;, A)

for for

li(l<

T,-

1/6

r,-,/6< lkll
i.e., Sir; obeys the same bounds as Sfi, times a factor of Tf?f-“*. So immediately all the results we proved involving Si?, carry over, with this extra factor. Derivatives of A2 can act on the exponential term as well as the Sfi; term. In this case, they yield an extra term of at most ;i(U( lKl’/T,)1”1’2 + V /Kl Iu1’2). Applying (5.3) to bound the sum, and using smallness of a to ensure (a2 - %‘n)N, > d/2 - E,

Proof. We examine each of the B, in succession. The result is obvious for B, from (4.20). Fix 6’ > 0. We write B, = V, + Q, where the ui are the two sums in (2.25). First look at vi : for T< T{- i, apply the result from A, an as upper bound. For TIP i < T< N, T,, an application of (5.3) yields the result. For T> N3 T,, we note

162

GOLOWICH

AND

IMBRIE

two requirements on N, (which we will use below when we fix N,): N, > N, and (aI - a3 - %?,I)N, > 4 + E - d/2. Applying (5.3) once again gives the result. For u2 and T < Tl- r, we note that Gjf,‘(K) = 0 unless /RI B T,- r/6, so we assume this. Then one of K and Z is longer than TIP r/12. We sum the shorter interval, and use the fact that the size of the longer interval is fixed by that of the shorter. When Z is the longer interval, we use lGT;F1(k, Z)l &Y9?I,AT;t;Pd’2

(5.6)

to bound the longer interval, along with i

161?;(k, K)e-‘“I

lKl=

6fi(k)l 6 c?‘%?~,,J.T~‘~-~

(5.7)

1

to bound the sum on the shorter interval. When K is the longer interval, we use I~~;W, K)I exp(VATjTJd’2

IK() 6 &(%,,A) Tf?fPd”

IKl”-d’2

<~‘((e,,~)T;‘~-d

(5.8)

to bound the longer interval, and the sum on the shorter intervals is bounded by

Multiplying these bounds, we obtain the result. A similar argument gives the result for T,-, N3 T,, we examine T-1 lu21

T$‘T’T”

<

1

V

[IGir;(k,

K)I T~“K”T”exp(V~T:,“-d’2

lK()]

IKI=O

. [ jGZ-;F1(k, Z)I T;13(““T’)]

(5.9)

but we can bound the last term by &(%‘r;,n) rl’l’T’l for 111> N, T, (see (4.20)), and for any 111< N, T, we have

<

sp,,,

2)

rW21

(5.10)

TajN)+2+~-d/2 I

so we see that Iv,1 T$‘T’T” < cY2@&A) rTlT,7 N2 T/ c

X

Ta3N)+2+~~d/2T:f~-dd/2 I lKl”-d/2

~y(lKl/T/)+

T/L1/6

~~‘2wI;,~) c, [

IKI =

rTlT/l

+T,-I

C JKI > NzT,

(a2-o,-~AKIKlIT/) T’

1

Ta3N’+2+E~d/2T:-t:E-dda3N2

3+E-ddiZT3+E-d/2+a3Nj+ajNZ--(a2--IZ)NZ I

I.

(5.11)

SELF-AVOIDING

So we fix N3, subject to the above conditions to obtain lu21 ~y7-f)

163

RANDOM WALKS

along with a3 N3
6 syigI, pTf1.

(5.12)

So we have completed B, for IuI = 0. The derivatives give the usual factor of Tp”’ by the same mechanisms as in Lemma 5.1. B, can be handled in exactly the same manner as B,. 1 The proofs that (i)-(iii) iterate are now complete if we choose 6’ small enough in the various lemmas and %i, large enough.

6. INDUCTION COMPLETED We now close the induction by showing that (iv) holds for 1. In this section we fix T3 T,,,. We estimate the L, norm by splitting k-space into [1] + 1 disjoint regions:

U,r {k:ID(k)1 2 S,} u,f {k:P(k)1 EC&Y &+d)

for

y=Z-

[Z], .... I- 1.

(6.1)

We first handle U,, looking at k in the neighborhood of the origin (analogous results hold for the rest of U,). For these values of k we have the expression Cl(k, T) = D(k)Texp(TT,(k)

+ 2r;‘(k)

+ rfF(k,

T))

(6.2)

which, since we have proven (i)-(iii) iterate, satisfies lC[(k, T)l Q ID(k)1 ‘exp( Tr,(O) + 2f f’(0) + TiF(O, T) + %?ATk*).

Also, from (i)-(iii)

(6.3)

it is not hard to see that when Jul = 1,

18; exp(TT,(k)

+ 2ry(k)

6 +?AT Ik( exp( Tr,(k)

+ I’fF(k, T))l + 2ry(k)

+ riF(k,

T)),

(6.4)

and when 1~1= 2 we have

Ia: exp(TT,(k) + 2rf(k) + T:F(k, T))I < A(VT2k2 + VT) exp( TT,(k) + 2r;‘(k ) + I-fF(k, T)).

(6.5)

Using these bounds, it is easy to see that rr

1 llP

1J

la;C,(k,

T)IP ddk

6 %?Tf”“2Pd’2pexp(Tf,(O)

+ 2f f(O) + rfF(O, T)).

ke (/I

(6.6) NOW, the constant here is some universal constant plus corrections

of o(n), and

164

GOLOWICH

AND

IMBRIE

only the corrections depend on ‘ik;, and ST&.By taking lambda small and choosing G& approp~ately, we can arrange for the %?in (6.6) to be $&. Next examine the integral over U,, y d I- 1. For now take y = I- 1 and /u[ = 0, and work in the region of U, closest to the origin (analogous arguments hold for the rest of U,). Recalling (2.3), we write C,(k, T) exp( - TT,(O) - 2Zy(O) - Z!F(O, T)) =exp(-W(O)X fj i=

([dZZ,(k,

fiF(O, T)) C

C CC,_ ,(k, Ko)e-tKo’ Q(O)]

n=O {n;}

Zi)e-tl,tfico)][C

,-., (k, Ki)e-‘Ktfi(0)]).

(6.7)

1

Now, in U,- 1 we can apply our expansion to Cf- r, so it is not hard to see that IC,- ,(k, K)l e-

If4 r/(o)

< ~(D(k)e~~kZ+~fi”T:-:-di2)IRl <

(e(-~i-~~)k2+Yl~~_;E-di2~IRI.

(6.8)

Also, we can use (3.11) to see that pn,(k,

I)] e-“l fi(O)<

0 (~~p/fil WA) riwh

111-dl2 T,-mm

for for for

1116 7’!- i/6 T,.-r/6< II) N, TI.

(6.9)

i Next we use the fact that the longest interval of {Zi, Ki} is longer than T/(2n + l), and its length is fixed by those of the 2n shorter intervals. So as an upper bound we let the sums over the shorter intervals be unrestricted, (6.10) c IClvl(k, K)I e-IKtficot< ‘K’=O

[Wk2-%?AT~~;-d/2]--) (6.11)

where in the last line we have used k* aW(ln T,/T,). Inserting these into (6.7), we find ICAk, T)I exp(-- TI;(O) - 2rf(O) - f:F(O, T))

(6.12)

SELF-AVOIDING

RANDOM

165

WALKS

where the upper limit on the sum arose, since 617,(lz, I) = 0 if 1 is short enough. We bound the sum by (number of terms). (maximum value): IC,(k, T)I exp( - Tf,(O) - 2r;1(0) - rfF(O, T)) 112


-%ln(V1.)$ln

*,

I

1

.

(6.13)

In the case (~1> 0, we can use (i)-(iii) to bound derivatives of all of the terms on the right-hand side of (6.7). It is evident that all factors due to these derivatives are dominated by the exponential in (6.13). Since TZ T,, ,, with A small and an appropriate choice of %Yi2,we can arrange for the contribution from U,_ I to the norm to be less than any inverse power of T times any inverse power of I, so in particular, flP

l&‘C,(k,

T)l” ddk

1

<;~12Tiy’2-d/~P evW,(O)

+ 2rf(O) + f:F(O, T))

(6.14)

We can handle the other regions U, in the same fashion: e.g., for y = I- 2 we write C,(k T) = c 1 C,-,(k n=O (I,}

Ko)

X i (CSn,(k 1,)+ hfl,- ,(k, ‘iI1 C,-,(k, Ki)) i=l

(6.15)

and proceed as before to find an expression like (6.14). We sum these terms, along with (6.6), to see that (iv) holds for 1, and Proposition 3.1 is proven.

7.

RESULTS

The proofs of Theorems 1.1 and 1.2 are now almost trivial. With the completion of our induction we have an expression for the full propagator valid for T< T’, C(k T) = C,(,,,(k

T)

= (D(k)e r’(~‘)(k))rexp(2r~T,,(k)

+ f &,,(k, T))

(7.1)

with ke U,(,,, along with bounds for the quantities in the last line. This proves Theorem 1.1 if we choose @yk,

T) E 2r&(k)

+ r:,&(k,

T).

(7.2)

166

GOLOWICH

AND

IMBRIE

We can also compute the mean square end-to-end distance (given by (1.5)). We find d

We have bounds on the edge terms, for any ju/ = 2 la;(zr~,,(k)+r:,F,,(k,

T))Ik=O~~(&IZTmax(3+2&-d’2,0)

(7.4)

along with, for T,< T’, 2 + 2~ -- d/2

la;r,,,.,(k)-a~r,,,(k)l,=o~V~T

which tells us that ~~~~~~~(~)l~=oconverges to a limit aif, (which is zero for the mixed partials), and hence that Q-l(T)Wlk=O

=

sf?A +

of order ,I as T+ 00

2+2e-d/2 WT

(7.5)

(7.6)

)-

Putting these results together yields the first result of Theorem 1.2. To prove the second, we note that for any fixed values of k, t there exists an so such that we can apply our expansion to the terms in the scaling limit of the endpoint for every sas,. So the limit becomes lifr,,(k,‘fi)

lim (WQ’fi)e s-+00

SI 1

(e

l-/($,)(O)

)

st

e

e

I-$@fk/fi,sr)

(7.7)

lIRd*ec(O,st) 31

Now, the edge terms cancel - in the limit s + co (using Taylor’s theorem along with Theorem 1.1), and @k/J s ),, =t=-exp(k2~/2d). Also, by Taylor’s theorem, (7.8

where qs is on the line segment stretching from 0 to k/G.

We can see that (7.9

by using the fact that 18; W,(k)1 is continuous in k for kE UI and exponentially small in 1, along with the comment following (7.5). This proves the second part of Theorem 1.2. 1

SELF-AVOIDING

RANDOM

167

WALKS

A

APPENDIX

In this appendix we derive the bounds on 8; 6ZZ,(k, T) used in (3.11). We closely follow a similar derivation (of the Laplace-transformed quantity) in [ 11. We begin by bounding Z7,(k, T), defined by n,(k

T)=

1 T j-j ,o,=T c eik.w’T’ c ( > CC

n

( -/Id,&),

(A.11

(s,I)EG~

where G’ are connected graphs consisting of bonds (s, t) on [0, T] satisfying 1s- tI < T,. Next we use the fact that every connected graph can be uniquely decomposed into a lace Y and a set of bonds X(9) compatible with the lace. Here a lace is defined to be a minimally connected subgraph, i.e., a graph of the form shown in Fig. 2, and an m-lace is a lace consisting of m bonds. Bonds (i, j) compatible with a lace $P are those such that the lace of 9 u (i, j) is equal to Y (see [ 1 ] for a more precise definition). Hence, we can split the sum over connected graphs into a sum over laces followed by a sum over bonds compatible with the lace, which we then resum:

= c

c

n

M P 1 m-laces iy

WktJ~,)

(i, j) E 9

l-l

u-RWJ.

64.2)

(S.f)EX(Y)

Note that an m-lace is uniquely specified by a set of 2m - 1 integers {nj} subject to the restrictions for all i

n,3 1 zm-1 JI,

n,

ni=T

T,

+n,d

(A.3) for i = 2, 4, .... 2m - 4

n,+ni+l+n;+z
FIG.

2.

A 3-lace.

168

GOLOWICH

AND

IMBRIE

As an upper bound, we discard those factors of (1 - As,,,) that connect different intervals ni. The effect of this is to break up our walk o into 2m - 1 independent subwalks, each of which propagates according to C,, and with restrictions on the various endpoints given by the lace:

The self-intersections of the walk, governed by the lace, occur in a simple pattern of which we can take advantage. Define the operators which act as multiplication and convolution by a propagator,

G:f(x)-C

CAx-.h n)f(y)

(A.61

IIC,(., nj)ll~2p~~rr,oneco~

(A.7)

which give us

6% 1 2” C ml21

n

in,1 i= 1

(In the last line we have used Lemma 5.8 in [l], an application of Holder’s inequality). The norms here are all L2 except for one which is L, (we have the freedom to choose which one). Next use C,(x, n) < C,.(x, n) for all x, n, and 12 I’. This, along with the fact that ni < T, and the Hausdorff Young theorem, yields

m,l

is)

.i= 1

Next consider 617,. The terms in n, and Z7_ I in which all bonds are of length Is- tl < T,-,/2 exactly cancel each other. Hence each term in IT,, l7-, that contributes must have at least one long bond IS- tl > T,- ,/2. Denote by n; and IT-, the sum of such terms, and use the bound IsZ7,j < IZ7;l + IZ7-,l. Next we note that the above condition implies that at least one ni > T,- i/6, so we require this to obtain an upper bound. We see that

fH31

(n,)

j=

1

.

SELF-AVOIDING

RANDOM WALKS

169

where, as an upper bound and for later convenience, we relax somewhat the conditions (A.3) on the fni> and only require all FZ~ECl, T,], Z: n,= T, and at least one ni > T,- ,/6. This bound can easily be modified to allow k-derivatives. We write the dispIacement of the full walk o(T) in exp(ik . o(T)) as a sum of displacements of the subwalks, and act on this product according to the Leibnitz rule. We find

Gq c 1” c x ma1

inIl

furI

fl

II~K,,,z,,-I(->q,ll:-:pEr ,. one,, (A.101

i=l

where the sum over (ui>, with C jail = 1~1,is over the possible ways of distributing the derivatives in applying the Leibnitz rule.

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8.

D. C. BRYDGE~AND T. SPENCER,Co~un. Muth. Phys. 97 (1985), 125. J. GLIMM AND A. JAFFE,“Quantum Physics,” Springer-Verlag, New York, 1987. T. HARA, Probab. Theory Rel. Fields 86 (1990), 337. T. HARA AND G. SLADE, Commun. Math. Phys. 128 (1990), 333. T. HARA AND G. SLADE, J. Star. Phys. 59 (1990). 1469. T. HARA AND G. SLADE, Self-avoiding walk in five or more dimensions, I, II, Preprint, 1991. G. LAWLER, Ann. Probub. 17 (1989), 1367. B. G. NGUYEN AND W.-S. YANG, Triangle condition for oriented percolation in high dimensions, preprint, 1991. 9. G. SLADE, Commun. Math. Phys. 110 (1987), 661. 10. G. SLADE, f. Phys. A 21 (1988), L417. 11. G. SLADE, Ann. Probab. 17 (1989), 91.