The Annals of Probability
1997, Vol. 25, No. 2, 573–597
CENTRAL LIMIT THEOREM FOR THE EDWARDS MODEL
By R. van der Hofstad, F. den Hollander and W. K
¨
onig
1
Universiteit Utrecht, Universiteit Nijmegen and TU Berlin
The Edwards model in one dimension is a transformed path measure
for standard Brownian motion discouraging self-intersections. We prove
a central limit theorem for the endpoint of the path, extending a law of
large numbers proved by Westwater. The scaled variance is characterized
in terms of the largest eigenvalue of a one-parameter family of differential
operators, introduced and analyzed by van der Hofstad and den Hollan-
der. Interestingly, the scaled variance turns out to be independent of the
strength of self-repellence and to be strictly smaller than one (the value
for free Brownian motion).
0. Introduction and main result.
0.1. The Edwards model. Let B
t
t≥0
be standard one-dimensional Brown-
ian motion starting at 0. Let P denote its distribution on path space and E the
corresponding expectation. The Edwards model is a transformed path mea-
sure discouraging self-intersections, defined by the intuitive formula
dP
β
T
dP
=
1
Z
β
T
exp
−β
T
0
ds
T
0
dt δB
s
− B
t
T≥ 0(0.1)
Here δ denotes Dirac’s function, β ∈0 ∞ is the strength of self-repellence
and Z
β
T
is the normalizing constant.
A rigorous definition of P
β
T
is given in terms of Brownian local times as fol-
lows.It is well known [see Revuz and Yor (1991),Section VI.1] that there exists
a jointly continuous version of the Brownian local time process Lt x
t≥0x∈R
satisfying the occupation time formula
t
0
fB
s
ds =
R
Lt xfxdx P-a.s. f R → R
+
Borelt≥ 0(0.2)
Think of Lt x as the amount of time the Brownian motion spends in x until
time t. The Edwards measure in (0.1) may now be defined by
dP
β
T
dP
=
1
Z
β
T
exp
−β
R
LT x
2
dx
(0.3)
where Z
β
T
= Eexp−β
R
LT x
2
dx is the normalizing constant. The ran-
dom variable
R
LT x
2
dx is called the self-intersection local time. Think of
Received June 1995; revised July 1996.
1
Supported by HCM Grant ERBCHBGCT93-0421 while spending the academic year 1994–
1995 at Nijmegen University.
AMS 1991 subject classifications. 60F05, 60J55, 60J65.
Key words and phrases. Edwards model, Ray–Knight theorems, central limit theorem.
573
574 R. VAN DER HOFSTAD, F. DEN HOLLANDER AND W. K
¨
ONIG
this as the amount of time the Brownian motion spends in self-intersection
points until time T.
The path measure P
β
T
is the continuous analogue of the self-repellent ran-
dom walk (called the Domb–Joyce model), which is a transformed path mea-
sure for the discrete simple random walk. The latter is used to study the
long-time behavior of random polymer chains. The effect of the self-repellence
is of particular interest. This effect is known to spread out the path on a linear
scale (i.e., B
T
is of order T under the law P
β
T
as T →∞). It is the aim of this
paper to study the fluctuations of B
T
around the linear asymptotics. Our main
result appears in Theorem 2.
0.2. Theorems. The starting point of our paper is the following law of large
numbers.
Theorem 1 [Westwater (1984)]. For every β ∈0 ∞ there exists a θ
∗
β∈
0 ∞ such that
lim
T→∞
P
β
T
B
T
T
− θ
∗
β
≤ ε
B
T
> 0
= 1 for every ε>0(0.4)
[By symmetry, (0.4) says that the distribution of B
T
/T under P
β
T
converges
weakly to
1
2
δ
θ
∗
β
+ δ
−θ
∗
β
as T →∞, where δ
θ
denotes the Dirac point mea-
sure at θ ∈ R.]
Theorem 1 says that the self-repellence causes the path to have a ballistic
behavior no matter how weak the interaction. Westwater (1984) proved this
result by applying the Ray–Knight representation for Brownian local times
and using large deviation arguments.
The speed θ
∗
β was characterized by Westwater in terms of the smallest
eigenvalue of a certain differential operator. In the present paper, however, we
prefer to work with a different operator, introduced and analyzed in van der
Hofstad and den Hollander (1995). For a ∈ R, define ⺛
a
L
2
R
+
0
∩C
2
R
+
0
→
CR
+
0
by
⺛
a
xu=2ux
u+2x
u+au − u
2
xu(0.5)
for u ∈ R
+
0
=0 ∞. The Sturm–Liouville operator ⺛
a
will play a key role
in the present paper. It is symmetric and has a largest eigenvalue ρa with
multiplicity 1. The map a → ρa is real-analytic, strictly convex and strictly
increasing, with ρ0 < 0, lim
a→−∞
ρa=−∞and lim
a→∞
ρa=∞. [The
operator ⺛
a
is a scaled version of the operator ⺜
a
originally analyzed in
van der Hofstad and den Hollander (1995), Section 5, namely ⺛
a
xu=
⺜
a
xu/2 where
xu=x2u.]
Define a
∗
, b
∗
, c
∗
∈0 ∞ by
ρa
∗
=0b
∗
=
1
ρ
a
∗
c
∗2
=
ρ
a
∗
ρ
a
∗
3
(0.6)
Our main result is the following central limit theorem.
CLT FOR THE EDWARDS MODEL 575
Theorem 2. For every β ∈0 ∞ there exists a σ
∗
β∈0 ∞ such that
lim
T→∞
P
β
T
B
T
− θ
∗
βT
σ
∗
β
√
T
≤ C
B
T
> 0
= ⺞ −∞C for all C ∈ R(0.7)
where ⺞ denotes the normal distribution with mean 0 and variance 1. The
scaled mean and variance are given by
θ
∗
β=b
∗
β
1/3
σ
∗
β=c
∗
(0.8)
Theorem 2 says that the fluctuations around the asymptotic mean have the
classical order
√
T, are symmetric, and even do not depend on the interaction
strength.
The numerical values of the constants in (0.6) are
a
∗
= 2189 ± 0001b
∗
= 111 ± 001c
∗
= 07 ± 01(0.9)
The values for a
∗
and b
∗
were obtained in van der Hofstad and den Hollander
(1995), Section 0.5, by estimating ρa for a range of a-values. This can be
done very accurately via a discretization procedure. (A rigorous upper bound
for a
∗
is given in Lemma 6 in Section 4.1.) The same data produce the value
for c
∗
. Note that c
∗
< 1. Apparently, as the path is pushed out to infinity, its
fluctuations are squeezed compared to those of the free motion with θ
∗
0=0,
σ
∗
0=1.
0.3. Scaling in β. It is noteworthy that the scaled mean depends on β
in such a simple manner and that the scaled variance does not depend on β
at all. These facts are direct consequences of the Brownian scaling property.
Namely, we shall deduce from (0.7) that for every β ∈0 ∞,
θ
∗
β=θ
∗
1β
1/3
σ
∗
β=σ
∗
1(0.10)
Indeed, for a, T>0,
B
T
LT x
x∈R
=
⺔
a
−1/2
B
aT
a
−1/2
LaT a
1/2
x
x∈R
(0.11)
where =
⺔
means equality in distribution [see Revuz and Yor (1991), Chap-
ter VI, Example (2.11), 1
◦
]. Apply this to a = β
2/3
to obtain, via (0.3), that
P
β
T
B
T
−1
= P
1
β
2/3
T
β
−1/3
B
β
2/3
T
−1
(0.12)
where we write µX
−1
for the distribution of a random variable X under a
measure µ. In particular, we have for all C ∈ R,
P
β
T
B
T
− θ
∗
1β
1/3
T
σ
∗
1
√
T
≤ C
B
T
> 0
= P
1
β
2/3
T
B
β
2/3
T
− θ
∗
1β
2/3
T
σ
∗
1
β
2/3
T
≤ C
B
β
2/3
T
> 0
(0.13)
576 R. VAN DER HOFSTAD, F. DEN HOLLANDER AND W. K
¨
ONIG
The r.h.s. tends to ⺞ −∞C as T →∞[in (0.7) pick β = 1 and replace
T by β
2/3
T]. Since the pair θ
∗
βσ
∗
β is uniquely determined by (0.7), we
arrive at (0.10).
0.4. Outline of the proof. Theorem 2 is the continuous analogue of the
central limit theorem for the Domb–Joyce model proved by K
¨
onig (1996). We
shall be able to use the skeleton of that paper, but the Brownian context will
require new ideas and methods. The remaining sections are devoted to the
proof of Theorem 2. We give a short outline.
In Section 1, we use the well-known Ray–Knight theorems for the local
times of Brownian motion to express the l.h.s. of (0.7) in terms of two- and zero-
dimensional squared Bessel processes. The former describes the local times in
the area 0B
T
; the latter describes the local times in −∞ 0 (respectively,
B
T
∞).
In Section 2, with the help of some analytical properties of the operator ⺛
a
proved in van der Hofstad and den Hollander (1995), we introduce a Girsanov
transformation of the two-dimensional squared Bessel process. The goal of this
transformation is to absorb the random variable exp−β
B
T
0
LT x
2
dx into
the transition probabilities. The transformed process turns out to have strong
recurrence properties. The Gaussian behavior of B
T
− θ
∗
βT/
√
T is traced
back to the asymptotic normality of the inverse of a certain additive functional
of this transformed process. Thus, the central limit behavior is determined by
those parts of the Brownian path that fall in the area 0B
T
.
In Section 3, we prove a central limit theorem for the inverse process. Fur-
thermore, as a second important ingredient in the proof, we derive a limit
law and a rate of convergence result for the composition of the transformed
process with the inverse process.
In Section 4, we finish the proof of Theorem 2 by showing that the contribu-
tion of the local times in −∞ 0∪B
T
∞ remains bounded as T →∞and is
therefore cancelled by the normalization in the definition of the transformed
path measure in (0.3).
1. Brownian local times. Since the dependence on β has already been
isolated [see (0.13)], we may and shall restrict to the case β = 1.
Throughout the sequel we shall frequently refer to Revuz and Yor (1991),
Karatzas and Shreve (1991), van der Hofstad and den Hollander (1995). We
shall therefore adopt the abbreviations RY, KS and HH for these references.
The remainder of this paper is devoted to the proof of the following key
proposition.
Proposition 1. There exists an S ∈0 ∞ such that for all C ∈
R,
lim
T→∞
expa
∗
TE
exp
−
R
LT x
2
dx
1
0<B
T
≤b
∗
T+C
√
T
= S⺞
c
∗2
−∞C
(1.1)
CLT FOR THE EDWARDS MODEL 577
where a
∗
, b
∗
and c
∗
are defined in (0.6), and ⺞
σ
2
denotes the normal distribu-
tion with mean 0 and variance σ
2
.
Theorem 2 follows from Proposition 1, since it implies that the conditional
distribution of B
T
−b
∗
T/
√
T given B
T
> 0 converges to ⺞
c
∗2
[divide the l.h.s.
of (1.1) by the same expression with C =∞and recall (0.3)].
Sections 1.1 and 1.2 contain preparatory material. Section 1.3 contains the
key representation in terms of squared Bessel processes on which the proof of
Proposition 1 will be based.
1.1. Ray–Knight theorems. This subsection contains a description of the
time-changed local time process in terms of squared Bessel processes. The
material being fairly standard, our main purpose is to introduce appropriate
notation and to prepare for Lemma 1 in Section 1.2 and Lemma 2 in Sec-
tion 1.3.
For u ∈ R and h ≥ 0, let τ
u
h
denote the time change associated with Lt u;
that is,
τ
u
h
= inft>0 Lt u >h(1.2)
Obviously, the map h → τ
u
h
is right-continuous and increasing, and therefore
makes at most countably many jumps for each u ∈ R. Moreover, PLτ
u
h
u=
h for all u ≥ 0=1 (see RY, Chapter VI). The following lemma contains the
well-known Ray–Knight theorems. It identifies the distribution of the local
times at the random time τ
u
h
as a process in the spatial variable running
forwards, respectively backwards, from u. We write C
2
c
R
+
to denote the set
of twice continuously differentiable functions on R
+
=0 ∞ with compact
support.
RK theorems. Fix u h ≥ 0. The random processes Lτ
u
h
u+ v
v≥0
and
Lτ
u
h
u− v
v≥0
are independent Markov processes, both starting at h.
(i) Lτ
u
h
u+ v
v≥0
is a zero-dimensional squared Bessel process BESQ
0
with generator
G
#
fv=2vf
vf∈ C
2
c
R
+
(1.3)
(ii) Lτ
u
h
u − v
v∈0u
is the restriction to the interval 0u of a two-
dimensional squared Bessel process BESQ
2
with generator
Gfv=2vf
v+2f
vf∈ C
2
c
R
+
(1.4)
(iii) Lτ
u
h
−v
v≥0
has the same transition probabilities as the process in (i).
For the proof, see RY, Sections XI.1-2 and KS, Sections 6.3 and 6.4.