数理解析研究所講究録 1275 巻 2002 年 112-118

112
Afamily of processes interpolating the Brownian
motion and the self-avoiding process on the Sierpi\’{n}ski
gasket and$\mathbb{R}$Ben HAMBLY

Kumiko HATTORI

Tetsuya HATTORI

University of Oxford

Shinshu University

Nagoya University

Abstract

We construct aone-parameter family of self-repelling processes on the Sierpinski
gasket, by taking scaling limits of self-repelling walks on the pre Sierpiiki gaskets.
We prove that our model interpolates between the Brownian motion and the selfavoiding process on the Sierpinski gasket. Namely, we prove that the process is
continuous in the parameter in the sense of convergence in law, and that the orderof Holder continuity of the sample paths is also continuous in the parameter. We
also establish alaw of the iterated logarithm for the self-repelling process. Finally
we show that this approach yields anew class of one-dimensional self-repelling
processes.

1. Our question
and arandom
To illustrate our questions, first let us consider the Euclidean lattice,
walk on it. The simple random walk (RW) is awalk that jumps to one of its nearest
neighbor points with equal probability. On the other hand, aself-avoiding walk (SAW)
is awalk that is not allowed to visit any point more than once.
If you take the scaling limit, that is, the limit as the lattice spacing (bond length)
tends to 0, the RW converges to the Brownian motion (BM) in .
The scaling limit of aSAW is far more difficult. It is because aSAW must remember
all the points it has once visited. In short, it lacks Markov property. For the l-dimensionallattice, that is, aline, it is trivial –the scaling limit is aconstant speed motion to theright or to the left. For 4or more dimensions, the scaling limit is the Brownian motion.
Since the space is large enough, the RW is not much different from the SAW. However,for the 2and 3-dimensional lattice, the scaling limit is not known.
iFrom this viewpoint, the Sierpinski gasket is arare example of alow dimensional
gasketspace, where the scaling limit of aSAW is known. The SAW on theconverges to anon-trivial self-avoiding process, which is not astraight motion along anedge, nor deterministic, and moreover, whose path Hausdorff dimension is greater than 1.
gasket, has infinitely fine creases and
It implies that the path spreads in theis self-avoiding. Let us emphasize here that in alow-dimensional space the existence of a
non-trivial self-avoiding process itself is “something.”
gasket has been constructed
On the other hand, the Brownian motion on theby Barlow, Perkins and Kusuoka as the scaling limit of the simple random walk on the
pre-Sierpinski gasket. (See [4], [5].
$\mathbb{Z}^{d}$$\mathbb{P}$$\mathrm{p}\mathrm{r}\mathrm{e}- \mathrm{S}\mathrm{i}\mathrm{e}\mathrm{r}\mathrm{p}\mathrm{i}\acute{\mathrm{n}}\mathrm{s}\mathrm{k}\mathrm{i}$$\mathrm{S}\mathrm{i}\mathrm{e}\mathrm{r}\mathrm{p}\mathrm{i}\acute{\mathrm{n}}\mathrm{s}\mathrm{k}\mathrm{i}$$\mathrm{S}\mathrm{i}\mathrm{e}\mathrm{r}\mathrm{p}\mathrm{i}\acute{\mathrm{n}}\mathrm{s}\mathrm{k}\mathrm{i}$113
Our question is :Now that we have two completely different processes on thegasket-the Brownian motion and the self-avoiding process, can we construct
afamily of processes that interpolates continuously these two?
We construct the interpolating process as the limit of aself-repelling walk. Aselfrepelling walk is something between the RW and aSAW. Visiting the same points morethan once is not prohibited, but suppressed compared with the. We want to constructaone-parameter family of self-repelling walks such that at one end of the parameter it
corresponds to the, at the other end the SAW. And we take the scaling limit.
$\mathrm{S}\mathrm{i}\mathrm{e}\mathrm{r}\mathrm{p}\mathrm{i}\acute{\mathrm{n}}\mathrm{s}\mathrm{k}\mathrm{i}$$\mathrm{R}\mathrm{W}$$\mathrm{R}\mathrm{W}$self-repelling walkRW
scalinglimit

SAW$\Leftrightarrow$$\Leftrightarrow\downarrow|$$\mathrm{B}\mathrm{M}\downarrow|$$\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{c}\mathrm{e}\mathrm{s}\mathrm{s}\downarrow|$

SA

Here we will further explain what is meant by interpolation. There is avery important
exponent that characterizes walks and their scaling limits. The most well-known scene
where it appears is the mean square displacement of the walk on an infinite lattice (graph).
For awalk starting at (the origin), let us assume$\mathrm{O}$$E[|X_{n}|^{2}]\sim n^{2\gamma}$,$narrow\infty$

,

where
is the walker’s location after steps, anddenotes the Euclidean distancefrom the starting point.
is our exponent. If you take the scaling limit, this exponent
governs the short-time behavior,$X_{n}$$|X_{n}|$$\mathrm{n}$$\gamma$$E[|X(t)|^{2}]\sim t^{2\gamma}$,$t\downarrow \mathrm{O}$.

The same determines also other path properties of the scaling limit such as H\"older
continuity and the law of the iterated logarithm.
For comparison, in the case of the one-dimensional integer lattice, , for the,$\gamma=1$
is known to be 1/2 (the well-known exponent for the, for the SAW,), andobviously, because it is astraight motion in one direction. In general, exponents are very
resistent to changes. Bolthausen proved for amodel of self-repelling walk on , that isalways 1regardless of the strength of self-repulsion. T\’oth constructed adifferent modelsuch that varies from 1/2 to 2/3. There are afew other models, but none of themconnects 1/2 to 1. (See [6, 7, 8, 9, 10, 11].)
It is interesting enough if we can connect the BM and the self-avoiding process on thegasket continuously in the sense of weak convergence of path measures. But
can we ask for more? So, our question is rephrased as:can we construct an interpolating
family of processes that connects the exponent for thecontinuously all theway to for theprocess? As we have seen above, it’s not easy even on the line-the simplest lattice.
However, on the Sierpinski gasket, we give an affirmative answer and the same method
works also on the line, R.
$\gamma$$\mathrm{R}\mathrm{W}$$\mathbb{Z}$$\gamma$$\mathrm{B}\mathrm{M}$$\mathbb{Z}$$\gamma$$\gamma$$\mathrm{S}\mathrm{i}\mathrm{e}\mathrm{r}\mathrm{p}\mathrm{i}\acute{\mathrm{n}}\mathrm{s}\mathrm{k}\mathrm{i}$$\gamma$$\gamma$$\mathrm{R}\mathrm{W}/\mathrm{B}\mathrm{M}$$\mathrm{S}\mathrm{A}\mathrm{W}/\mathrm{S}\mathrm{A}$

2. Our Model
The

gaskets and the Sierpinski gasket are defined as follows. Let $O=$
, $b=(1, 0)$ , and let
be the set of all the points on the vertices and$\mathrm{p}\mathrm{r}\mathrm{e}- \mathrm{S}\mathrm{i}\mathrm{e}\mathrm{r}\mathrm{p}\mathrm{i}\acute{\mathrm{n}}\mathrm{s}\mathrm{k}\mathrm{i}$$(0,0)$ ,$a=( \frac{1}{2}, \frac{\sqrt{3}}{2})$$F_{0}’$

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edges of AOab. We define asequence of sets

$F_{0}’$,$F_{1}’$,$F_{n+1}’= \frac{1}{2}F_{n}’\cup\frac{1}{2}(F_{n}’+a)\cup\frac{1}{2}(F_{n}’+b)$$F_{2}’$,, inductively by$\ldots$$n=0,1,2$ ,,$\ldots$,

where $A+a=\{x+a : x\in A\}$ axsd $kA$ $=\{kx : x\in A\}$ . Let$F_{n}=F_{n}’\cup(F_{n}’-b)$.

We call Fn’s the (finite) pre-Sierpinski gaskets. As increases, the lattice (graph) getsfiner. If we superpose all the Fn’s and take the closure, we get the (finite)gasket, $F$ .
$n$$\mathrm{S}\mathrm{i}\mathrm{e}\mathrm{r}\mathrm{p}\mathrm{i}\acute{\mathrm{n}}\mathrm{s}\mathrm{k}\mathrm{i}$$F=d(\cup F_{n})n=0\infty$.

.
by
We denote the set of vertices inthe set of continuous functions $w:[0, \infty)arrow F_{n}$ such that there
Let us denote by
for whichexists$G_{n}$$F_{n}$$W_{n}$$L(w)\in \mathrm{N}$$w(0)=O$ ,$w(t)=a$ ,,,$t<L(w)$ ,$i=0$ ,, $L(w)-1$ ,$|w(i)-w(i+1)|=1$ ,$\overline{w(i)w(i+1)}\subset F_{n}$ ,$i=0$ ,, $L(w)-1$ ,$w(t)=(i+1-t)w(i)+(t-i)w(i+1)$ , $i\leq t<i+1$ , $i=0,1,2$ ,$w(t)$$t\geq L(w)$$\not\in G_{0}\backslash \{O\}$$\cdots$$\cdots$$\cdots$.

that go from $O$ to without hitting or or . $L(w)$ denotes
is the set of paths on
the steps needed to get to . (Between integer times, we interpolate by constant speed
motion. W\’e ve made the path continuous just for later convenience.)
To define a“self-repelling walk,” we assign weight to each path. Our model is uniquein the way of realizing self-repulsion. In other models on , they count the numbers of
returns to the same points or bonds, and define arepulsion factor using these numbers.
But we count turns at asharp angle and -turns as shown below.
For $w\in W_{1}$ , let $M_{1}(w)$ be the number of returns to the starting point, $O$ . Let $N_{1}(w)$be the number of -turns and sharp turns that occur at points other than . Here U$\overline{w(i)w(i+1)}<0$ , where
denotes theturns and sharp turns occur when
inner product of and in .
Let $0\leq u\leq 1$ and $x>0$ be parameters. For each path in Wi, we assign the following
weight.
$W_{n}$$b$$a$$F_{n}$

$c$

$d$

$a$

$\mathbb{Z}$$\mathrm{U}$$O$$\mathrm{U}$$\frac{1}{w(i-1)w(i)}\cdot$$\vec{b}$$\tilde{a}\cdot\vec{b}$$\mathrm{R}^{2}$$\vec{a}$$P_{1}^{u}(x)[w]= \frac{x^{L(w)}u^{M_{1}(w)+N_{1}(w)}}{\Phi(x,u)}$

,

where
$\Phi(x, u)=\sum_{w\in W_{1}}x^{L(w)}u^{M_{1}(w)+N_{1}(w)}$.

The factor involving is the repulsion factor.
, we note that we. In defining aprobability on
on
Next, we go on to define. First consider apath ofby adding finer structures to apath in
get apath in. Let us add to the first step of afiner structure on that goes from $v(0)=O$ to$v(1)$ without hitting any
inside the equilateralvertices other than $v(0)$ . The part of$v(1)$$v(0)$
Thus,.
we see that thisas two of the vertices is similar to
triangle withand$u$

$P_{2}^{u}$$W_{2}$$W_{2}$$W_{1}$$W_{2}$$W_{1}$$v$$v$$F_{2}$$F_{2}$$F_{1}$$F_{1}$

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finer structure between the start and the first step of corresponds to some element of. We give finer structures to each step of in asimilar way. This way we get apathin, patching up small
paths,,,, on arough path . Actually, each pathincan be constructed in this way, adding finer structures. Thus, for finer structures
between each step,andare defined. We define the weight for $w\in W_{2}$ by$v$$W_{1}$$v$$W_{2}$$W_{1}$$\mathrm{w}\mathrm{i}$$\cdots$$v$$w_{L(v)}$$W_{2}$$M_{1}$$P_{2}^{u}(x)[w]$$N_{1}$$=$$\frac{1}{\Phi_{2}(x,u)}x^{L(w)}$.

$u^{M_{1}(v)+N_{1}(v)}$[base path on$=$$\frac{1}{\Phi_{2}(x,u)}x^{L(w)}$$F_{1}$][finer structures]$u^{M_{2}(v)+N_{2}(v)}$where $L(w)$ is the number of the steps on$F_{2}$, and $\Phi_{2}(x, u)$ is the normalization factor$\Phi_{2}(x, u)=\sum_{w\in W_{2}}x^{L(w)}u^{M_{2}(w)+N_{2}(w)}$on

iFrom the fact that we constructed apath on$F_{1}$$F_{2}$, it is easy to see$P_{n}^{u}$$W_{n}$$F_{n-1}$.

by adding finer structures to apath$\Phi_{2}(x, u)=\Phi(\Phi(x, u),$ $u)$

We go on to defineonFirst, we consider apath on
have the recursion relation$L \prod_{i}^{(v)}u^{M_{1}(w:)+N_{1}(w:)}$

recursively.
and patch up small.
paths on it. For general$W_{1}$$\Phi_{n}(x, u)=\Phi_{n-1}(\Phi(x, u),$ $u)$

$n$

, we

.

This is one of the key properties of our model. We can see the meaning of the recursion
in this way. Consider aself-repelling walk on. Pick up all thewith propability-points the walk visits. Then we get aself-repelling walk on
with renormalized
probability $P_{n-1}^{u}(\Phi(x, u))$ .
Let us choose $x=x_{u}$ to be the unique positive solution to the equation,$P_{n}^{u}(x)$$F_{n}$$F_{n-1}$$F_{n-1}$$x_{u}=\Phi(x_{u}, u)$.

This choice of makes the measure self-similar, $u=1$ corresponds to the simple random
walk with the first exit at . In this case, -factor is absent and $x_{u}=1/4$ . It shows thewalker chooses one of its four nearest neighbors with equal probability. For $u=0$ only
self-avoiding paths survive. (For more details of our model, see [1].)
$x$

$a$

$\mathrm{u}$3. Results
We study the function $\Phi(x, u)$ (this corresponds to the partition function, or the generating
function)closely and get the following results.
Let.
$\lambda_{u}^{\mathrm{d}}=^{\mathrm{e}\mathrm{f}}E^{P_{1}^{u}}[L]$$\lambda_{u}$is the average steps from$\lambda_{1}=5$$\mathrm{O}$

to on$(RW)$ ,$F_{1}$. It is continuous in , and$\lambda_{0}=\frac{7-\sqrt{5}}{9}$

$u$

(SAW)

$2<\lambda_{0}<3$.

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Now we are going to take the continuum limit. It corresponds to the limit as $narrow\infty$ .
. We can -consider it as aprobability
as aprobability measure on
We definedmeasure defined on aspace of continuous functions $C$ on the Sierpinski gasket supported. Thus the base space is common to all . Let us consider an accelerated processon. Recall that for our path, it takes time 1to go to anearest neighborby the factor ofvertex. As the lattice gets finer, our walk gets slower. So we need aproper acceleration to, and denote the destribution
be aprocess that obeys
get anon-trivial limit. Let.
byof time-scaled process,Our first theorem states the existence of the scaling limit.
$W_{n}$$P_{n}^{u}$$\mathrm{r}\mathrm{e}$$n’ \mathrm{s}$$W_{n}$$(\lambda_{u})^{n}$$X_{n}(\cdot)$$X_{n}((\lambda_{u})^{n}\cdot$

Theorem 1$P\sim nu$$P_{n}^{u}$$)$$\tilde{P}_{n}^{u}$converges weakly to a probability measure$P^{u}$

on

as

$C$

$narrow\infty$.

corresponds to the Brownian motion conditioned that it hits before , , , (andcorresponds to the non-trivial self-avoiding process mentioned inis stopped at ).
Section 1.
$P^{1}$

$b$

$a$

$a$

$c$

$d$

$P^{0}$

Remarkgasket has been
In $[2, 3]$ , adifferent model of self-avoiding walk on the
goes
path
from
to , apositivethat
self-avoiding
model,
eachforstudied. In thisis assigned, where $\beta>0$ is aparameter. It has been proved
weight propotional to$\beta_{c}>0$
for which the scaling limit is aself-avoiding processthat there exisits auniquethan 1almost surely. Our scaling limit process
greaterwith path Hausdorff dimension
coincides with this limit process. Our model at $u=0$ is more restricted than usual
SAW because sharp turns are prohibited as well as returning to the same points. But it
produces the same scaling limit as the ’standard SAW.’$\mathrm{S}\mathrm{i}\mathrm{e}\mathrm{r}\mathrm{p}\mathrm{i}\acute{\mathrm{n}}\mathrm{s}\mathrm{k}\mathrm{i}$

$O$

$w$

$a$

$e^{-\beta L(w)}$

Our second theorem shows that our limit process is continuous in u and does connect
the BM and the self-avoiding process continuously.
Theorem 2(Continuity in ) For all $u_{0}\in[0, 1]$ ,$u$$P^{u}arrow P^{u_{\mathrm{O}}}$

weakly as u

$arrow u_{0}$The following theorems concern path properties of the limit process.
Theorem 3For all $p>0$ , there exist$C_{\dot{1}}$$=C_{\dot{l}}(p, u)>0$, $i=1,2$ such that,

$C_{1} \leq\lim_{tarrow}\inf\frac{E^{u}[|X(t)|^{p}]}{t^{\gamma_{u}\mathrm{p}}}\leq\lim_{tarrow}\sup_{0}\frac{E^{u}[|X(t)|^{p}]}{t^{\gamma_{u}p}}\leq C_{2}$where$\gamma_{u}=\frac{1\mathrm{o}\mathrm{g}2}{1\mathrm{o}\mathrm{g}\lambda_{u}}$and is continuous in .
$u$

Theorem 4(H\"older contintity)For any $M>0$ and any$b=b(M, \gamma’, \omega)>0$ and $H=H(M, \gamma’, \omega)>0$ such that$|X(t+h)-X(t)|\leq b|h|^{\gamma’}$$\forall t\in[0, M]$, $|h|\leq H$,$0<\mathrm{Y}$$<\gamma_{u}$, there exist $a.s$ .

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Theorem 5(Law of the Iterated Logarithm) There exist $C_{i}=C_{i}(p, u)>0$ , $i=3,4$ suchthat
$C_{3} \leq\lim_{tarrow}\sup_{0}\frac{|X(t)|}{\psi(t)}\leq C_{4}$,$a.s.$,

where$\psi(t)=t^{\gamma_{u}}$ $( \log \log \frac{1}{t})^{1-\gamma_{u}}$Thus, in our model, the exponent.

in Section 1is given by$\gamma$$\gamma_{u}=\frac{1\mathrm{o}\mathrm{g}2}{1\mathrm{o}\mathrm{g}\lambda_{u}}$and is acontinuous function in$\gamma_{0}=\frac{1\mathrm{o}\mathrm{g}2}{\log\frac{7-\sqrt{5}}{2}}$$u$connecting$\gamma_{1}=\frac{1\mathrm{o}\mathrm{g}2}{1\mathrm{o}\mathrm{g}5}$

for the simple random walk andfor the self-avoiding walk.

4. Self-repelling processes on$\mathbb{R}$(instead of the pre-Sierpinski gasket).
We start with asequence of random walks on
The vertex set that we will use for our walks is $G_{n}=\{k2^{-n}$ : $k=-2^{n},$ $-2^{n}+$,, 0, 1, 2,, }.
is the set of continuous functions such that at integer times
it takes values inwith nearest neighbor jumps from 0to 1. $N_{k}(w)$ and $M_{k}(w)$ can be
defined similarly to the case of the Sierpinski gasket.
The generating function
is given by$\mathbb{Z}$$1$$\cdots$$\cdots$$2^{n}$$W_{n}$$G_{n}$$\Phi_{1}(x, u)$$\Phi_{1}(x, u)=\frac{x^{2}}{1-2u^{2}x^{2}}$.

In particular, we have $\Phi_{n}(x, 0)=x^{2^{n}}$ , which implies that when $u=0$ we have asingle
path which connects 0andby astraight line (i.e., the self-avoiding path on ), andfor $u=1$ we reproduce the generating function for the simple random walk.
.
We can give explicit formulas for $x_{u}>0$ and$2^{n}$$\mathbb{Z}$$\lambda_{u}>0$$x_{u}= \frac{1}{4u^{2}}(\sqrt{1+8u^{2}}-1)$,$\lambda_{u}=\frac{2}{x_{u}}=\sqrt{1+8u^{2}}+1$.

Once we have established these properties of the generating function the subsequent
analysis follows quite similar lines to the Sierpinski gasket case. For example, the probability measures on the paths are defined in asimilar way to the case of Sierpinski gasket ,and the existence of acontinuum limit (Theorem 1) and the weak continuity of the pathin $u\in[0,1]$ (Theorem 2) hold. The sample path properties such as Theorems
measure$P^{u}$

3through 5also hold with$\gamma_{u}=\frac{1\mathrm{o}\mathrm{g}2}{1\mathrm{o}\mathrm{g}\lambda_{u}}$.

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