PERCOLATION DIMENSION OF BROWNIAN MOTION IN RRR 3

Let B ( t ) be a Brownian motion in R 3 . A subpath of the Brownian path B [0 ; 1] is a continuous curve γ ( t ) , where γ [0 ; 1] (cid:18) B [0 ; 1] , γ (0) = B (0) , and γ (1) = B (1) . It is well-known that any subset S of a Browni an path must have Hausdor(cid:11) dimension dim h ( S ) (cid:20) 2 : This paper proves that with probability one there exist subpaths of B [0 ; 1] with Hausdor(cid:11) dimension strictly less than 2. Thus the percolation dimension of Brownian motion in R 3 is strictly less than 2.


Introduction
. In this paper we prove that for d = 3, with probability one, there exists a subpath of B[0, 1] with Hausdorff dimension strictly less than 2.
A double point of B[a, b] is any point x ∈ R d such that there exist a ≤ t 1 < t 2 ≤ b with B(t 1 ) = B(t 2 ) = x.
For such t 1 , t 2 it is natural to call B[t 1 , t 2 ] a "loop". For d ≥ 4, Brownian motion has no double po ints [7] and hence has no loops. However, for d = 3 Brownian motion has double points (but no triple points [8]), and thus in order to find a lower dimensional subpath one might proceed by erasing the loops. This arises naturally as an e xtension of loop-erased random walk (e.g. see Lawler [11]). The cut points are points x such that x = B(t), where t is a cut time. Clearly, any subpath of B[0, 1] must contain the set of cut points B(L[0, 1]). Lawler [12] has shown that for d = 3, the set B(L[0, 1]) has Hausdorff dimensi on dim h (B(L[0, 1])) > 1. The infimum of the dimensions of all subpaths of B[0, 1] is known as the percolation dimension of Brownian motion [3]. The 51 result of Lawler shows the percolation dimension of three dimensional Brownian motion is stri ctly greater than 1. The main result of this paper is the following theorem, This result tells us that the percolation dimension of three dimensional Brownian motion is distinct from the Hausdorff dimension, 2, and the topological dimension, 1. In four or more dimensions, the percolation dimension is 2, and in two dimensions the conjectured value for the percolation dimension is 1.
Another useful dimensional quantity is boundary dimension, which takes on a non-trivial value in d = 2. The other area of interest to which this paper relates is the self-avoiding walk. Erasing loops from a Brownian path may make it possible to create a self-avoiding Brownian motion. Although this would not be the limit of the usual self-avoiding walk, it may be the continuum limit of the loop-erased walk. It should be noted that loop-erased random walk is conjectured to be in a different universality class than self-avoiding walk. [11] The main idea of the paper is as follows. We define a random subset of the Brownian motion B(F ), where F ⊂ [0, 1]. The random set F is defined as the intersection of the nested sets F n , i.e. F n+1 ⊆ F n , so that standard techni ques may be used to give an upper bound for the Hausdorff dimension of B(F ). We define a particular subpath of the Brownian motion, γ(t) = γ(ω, t), as the subsequential limit of subpaths {γ n (t)} such that γ([0, 1]) ⊂ B(F ). One should note that the method of proof (using subsequences, etc.) does not lend itself to proving that this subpath γ([0, 1]) is a random curve. That is, there is no reason to believe that the mapping taking each ω in Wiene r space to γ(ω) is measurable. Nonetheless, we prove that there exists an a > 0 such that dim h (B(F )) ≤ 2(1 − a) and that γ( The crucial definitions of the random set F and the subpath γ(t) are given in the next section, and the main theorem is proven in section 3. The main technical tools needed to obtain this result are detailed estimates concerning the dimension of sets of cut points. These technical estimates are left to the end of the paper.
I thank Greg Lawler for his advice and assistance with this paper. He suggested the problem, and his advice in finding a solution has been invaluable. I also thank the referee for helpful comments and advice.

Notation and Definitions
To define the particular choice of Brownian subset B(F ) studied below, it will be useful to define the cut points of B The local cut points are points x such that x = B(t), where t is a local cut time. Let Definition 2.1. For 1 ≤ k ≤ 2 n let A(k, n) be the event Let F n and F be the random sets Proof. To prove F n+1 ⊆ F n , it is sufficient to prove that if I(k, n) is not in F n , then I(2k − 1, n + 1) and I(2k, n + 1) are both not in F n+1 . If I(k, n) ⊂ F n , then because there is now a larger set intersecting the same piece of the Brownian motion, so the intersection is non-empty. Likewise, This says that the relative cut points of a section of Brownian motion only becomes larger as the set is made larg er and the relative set stays the same. Thus F n+1 ⊆ F n .
Also, notice that since F n is compact for all n, standard topological arguments imply that Roughly, the construction of γ(t) is as follows (see Figure 1 below). Split the interval [0, 1] into 2 n pieces, labeled I(k, n). Let t 1 be the last time that B(t) intersects B(I(1, n)), the first piece of the Brownian path. Let s 1 ∈ I(1, n) be the unique time such that B(s 1 ) = B(t 1 ). Remove [s 1 , t 1 ] from [0, 1], thereby removing a loop (see Figure 1, first arrow). Let k 2 be such that t 1 ∈ I(k 2 , n), making B(I(k 2 , n)) the second piece of the loop-erased Brownian path. Let t 2 be the last time that B(t) intersects B(I(k 2 , n)), and choose s 2 ∈ I(k 2 , n) such that B(s 2 ) = B(t 2 ). Remove the loop [s 2 , t 2 ]. Continue this process until t i = 1. Parametrize the remaining pieces of [0, 1] to give a continuous curve, γ n (t), connecting B(0) to B(1). Define γ(t) as a subsequential limit of {γ n (t)}.
The precise definition of γ n (t) is as follows. Most of the elements in the definition of γ n (t) depend on n in an obvious way, but we often suppress the dependence on n. Let Then for i > 1 define Since, with probability one, ki 2 n is not a local cut point [12], then t i > ki 2 n . Hence for k i < 2 n , we have k i > k i−1 > . . . > k 1 . Let i * = inf{i : k i = 2 n } and notice that t i * = 1. Define s i as the unique time such that Since, with probability one, the Brownian path in d = 3 has only double po ints (no triple points, etc.), this is well-defined. Note that To parametrize the pieces I(k i , n), we define a time change φ n (t). Let t 0 = 0 and notice that, almost surely, Below we show that there exists a subsequence {n(i)} such that the subsequential limit of {γ n(i) (t)}, called γ(t), exists, is continuous, and is a subpath of the B[0, 1]. Also, we show that the sets B(F n ) contain the subpath γ(t). In Section 3 we prove the following theorem: and ther e exists a > 0 such that Using this theorem and the well-known result that Brownian motion in d ≥ 2 maps all Borel subsets of [0, 1] into sets of twice the dimension [10,15] proves Theorem ?? and Corollary 1.2.
A convention that is used throughout this paper is that c, c 1 , c 2 will be used to stand for constants that may change from line to line. The constants c 3 , c 4 , etc. will remain the same throughout the paper.
Start two independent Brownian motions B 1 , B 2 at the origin. Let the cut points of the second Brownian motion be denoted by . Also denote stopping times Then the main technical estimate nee ded is that there exists a c 3 < ∞, a > 0 such that To obtain these results, we need an estimate on the intersection exponent ζ 3 for Brownian motion in R 3 [4]. The intersection exponent is defined as follows. If B 1 , B 2 are independent Brownian motions in R d (d < 4) s tarting at distinct x, y respectively, with |x| = |y| = 1, then the intersection exponent is the number ζ d such that The symbol denotes that the two side s are bounded within a constant as n → ∞. The existence and rigorous bounds for this exponent are given in [4,11,12], In p articular, this paper uses the fact that ζ 3 < 1/2. To be precise, also notice that a ≤ ζ 3 < 1/2. This fact is used in the next section.

Proof of the Main Theorem
The proof of Theorem 1.1 proceeds as follows. First we show that the functions γ n (t) are uniformly equicontinuous. This implies that a subsequence of {γ n (t)} converges to a continuous function γ(t), which is a subpath of the Brownian path. The following lemma uses a path-by-path comparison of γ n (t) and B(t) to show that the path of γ n (t) has at most the same oscillation as the path of the Brownian motion from which it is generated. In particular, Lemma 3.1 proves that the functions γ n (t) are uniformly equiconti nuous. Proof. Fix δ > 0. For any s < t such that |t − s| < δ, let Then |r 1 − r 2 | ≤ |s − t|, and γ n (r i ) = B(r i ), which implies that This proves the lemma. Proof. Using the previous lemma and the Arzela-Ascoli theorem, we can construct a function γ(ω, t) which is a subsequential limit of the {γ n (ω, t)}. Proof. This proof uses the notation in the definition of γ n (t). Specifically, Since B(F n ) is compact, it is sufficient to prove that Γ i * ,n is contained in F n for all n. Given r ∈ Γ i * ,n , there is a j such that r ∈ [t j−1 , kj 2 n ]. Since r ∈ Γ i * ,n , then B(r, 1]∩B [0, This fact follows directly from the construction of Γ i * ,n . Specifically, if 0 ≤ t ≤ kj −1 2 n and t ∈ L[0, kj 2 n ], then either t i−1 ≤ t ≤ s i , or s i < t < t i , for some i ≤ j − 1. The second possibility violates the fact that t ∈ L[0, 2 n for some i ≤ j − 1, which implies that t ∈ Γ i * ,n . Thus Γ i * ,n ⊆ F n for all n, and so γ([0, 1]) ⊆ B(F n ) for every n. Therefore, γ([0, 1]) ⊆ ∞ n=1 B(F n ) = B(F ). Therefore, proving the set F has dim h (F ) ≤ 1 − a, with probability 1, proves Theorem 1.1 .
Define the random variable K n as K n = #{k : 1 ≤ k ≤ 2 n and A(k, n) holds}.
By (3), there exists a constant c > 0 such that Then for any > 0, Markov's Inequality implies P{K n ≥ (2 n ) 1−a+ } ≤ c(2 n ) − , and then using the Borel-Cantelli Lemma, Thus, for all sufficiently large n, K n ≤ (2 n ) 1−a+ , which says that the set F can be covered by (2 n ) 1−a+ intervals of length (2 n ) −1 . Now, using standard arguments, dim h (F ) ≤ 1 − a + . With probability one, this holds for every > 0, so This completes the proof of the upper bound on the dimension of the set F .

Cut Point Estimates
In this section we prove two technical results about the cut point properties of the Brownian path. These results, Propositions 4.1 and 4.2, are needed in the next section to prove (3). First, some definitions and notation are pr esented: In the following, B(t) is a Brownian motion starting at the origin. The following notation is used throughout the rest of the paper: S i = sphere of radius e i centered at the origin. B(a, r) = closed ball of radius r centered at a.
We will think of G i as the indicator of a "good" number: an i such that after traveling distance e i from the origin, the Brownian motion never returns to the ball of radius e i−1 and after hitting the ball of radius e i+2 never returns t o the ball of radius e i+1 . Proposition 4.1 proves that there exists a fraction > 0 such that, with high probability, at least this fraction of the G i equal 1. Specifically, This proposition says that a positive fraction of the sets B[0, τ n ] ∩ (B(0, e i+1 )\B(0, e i )) are in effect isolated from the rest of the Brownian path. This includes information from both the past and the f uture. This "isolation" of annuli leads us to suspect that the probability of having cut points in each annulus is bounded below by a positive constant. This is true and is proven as: There exists c 5 > 0 such that, for n ≥ 2, the following is true: where ζ = ζ 3 , the intersection exponent, and B(0, r) c is the complement of the ball of radius r.
This proposition could also be written as: There exists c 5 > 0 such that for all i ≥ 0, for all n ≥ i + 1, We start by proving Proposition 4.1. It is well-known (see for example [6, p. 29]) that in d = 3, for r < |x| < R, where P x is the probability given that B(0) = x. Using this fact, for any j. Therefore, since knowing that the Brownian motion never returns to S ρl−1+1 only increases this probability.
Also, the ρ l cannot be very far apart. In particular, we can prove similarly that Now, using standard techniques, there exists an s > 0 such that (9) implies that P{ m l=1 G ρl ≤ sm} is exponentially decreasing in m. Similarly, there exists an r > 0 such that (10) implies that p{ m l=1 (ρ l − ρ l−1 ) ≥ rm} is exponentially decreasing in m. Therefore, Choose n 2r ≤ m ≤ n r and then equation (11) becomes 2r finishes the proof of Proposition 4.1.
The proof of Proposition 4.2 uses two main ideas. First, if G i = 1 then the Brownian motion never returns to S i−1 after hitting S i and never returns to S i+1 after hitting S i+2 . Thus, assuming G i = 1 allows the possibility of local cut times t ∈ [τ i , τ i+1 ], because the Brownian motion is "pulled away" from the annulus B(0, e i+1 )\B(0, e i ). Second, we use the fact [12] t hat, with probability one, the dimension of the local cut points of a Brownian motion B[0, 1] is 2 − 2ζ. Using these two ideas, we prove that there is a positive probability of having local cut times t ∈ [τ i , τ i+1 ]. Moreover, the di mension of the cut points corresponding to these cut times is 2 − 2ζ. These heuristics are now made precise.
Specifically, in order to prove Proposition 4.2, we need to show: There exists a c 5 > 0 such that It suffices to prove that there exists c 5 > 0 such that the following is true: To prove this, let and let J n = #{k : 1 4 2 n < k ≤ 3 4 2 n and A(k, n) holds }.
The next lemma provides the technical estimate to prove that E(J n ) ≥ c 1 (2 n ) 1−ζ , for some c 1 > 0, where ζ = ζ 3 , the in tersection exponent. A result of Lawler [12,Lemma 2.3] says that there exists c 2 < ∞ such that E(J 2 n ) ≤ c 2 (2 n ) 2(1−ζ) . Using these two bounds on the first and second moments, standard second moment arguments yield the result (see for example [12,Proposition 2.2]).
So, in order to finish the proof of Proposition 4.2 it suffices to prove the following:  16 2 n . Proof. The events are chosen so that Q 1 is independent of the rest, so and by a result of Lawler [12,Proposi tion 3.16], there exists a c 7 > 0 such that This last probability is clearly a positive probability event, because it doesn't depend on n or k in any sub stantial way.

Proving Power Law Decay
The inequality (3) is the key estimate for this paper. This section is devoted to proving this estimate using Propositions 4.1 and 4.2. We use two standard facts and assume throughout this section that d = 3. The first is that in d = 3 any set of Hausdorff dimension strictly greater than 1 has positive capacity [9,Theorem 4.13]. The second is that Brownian motion hits sets of positive capacity with positive probability, i.e. there exists a c > 0 such that if A ⊂ B(0, r) with Cap(A) > 0, for every x ∈ B(0, r), where T A = inf{t : B(t) ∈ A}. Bass [1, Chap. II., sect. 5] discusse s this fact at length.
Combining the fact [10,15] that Brownian motion B[0, ∞) doubles the dimension of subsets of [0, ∞) with Proposition 4.2 proves that there exists a c > 0 (independent of n) such that Let A = C [0,τn] [τ 1 , τ n ]. Since ζ < 1/2, then A has Hausdorff dimension strictly greater than 1, and hence Cap(A) is positive. Proposition 4.1, Brownian scaling, and the scaling law for capacity combine to prove the following lemma, Lemma 5.2. There exists > 0, b > 0, and c 10 < ∞ such that Proof. If G 1 = 1, then (15) implies that there exists a c 1 > 0 such that Therefore, since sets of Hausdorff dimension greater than 1 have positive capacity in R 3 , there exists 1 > 0 such that Using a similar argument, if G k = 1 for k ≥ 1, Now, using the fact that capacity scales inversely with the diameter of the set in R 3 [11], we see that P{Cap(C [0,τn+k−1] [τ k , τ k+1 ]) > 1 e k−1 } ≥ c 1 2 . Since where τ i j = inf{t : B i (t) ∈ S j }. Then rewriting the inequality for non-exponential stopping times, (16) implies A few extra results are needed here about how this probability works for arbitrary times (not just distance stopping times). To this end, standard techniques yield the following: Claim: For every β > 0 there exists a c = c(β) such that P{T n ≤ n 2 (ln n) −2 } ≤ cn −β .
Proof. This need only be proven for 1-dimensional Brownian motion.