Two-sided random walks conditioned to have no intersections

Let $S^{1},S^{2}$ be independent simple random walks in $\mathbb{Z}^{d}$ ($d=2,3$) started at the origin. We construct two-sided random walk paths conditioned that $S^{1}[0,\infty) \cap S^{2}[1, \infty) = \emptyset$.


Introduction
Let S = (S(n)) be a simple random walk in Z d (d = 2, 3) started at the origin. Take integers k < n. A time k is called cut time up to n if S[0, k] ∩ S[k + 1, n] = ∅, (1.1) where S[0, k] = {S(j) : 0 ≤ j ≤ k}. We call S(k) a cut point if k is a cut time. Lawler [4] has shown that there are constants 0 < c, c ′ < ∞ such that for all n, where ξ = ξ d is the intersection exponent (see Section 2.1 below). Lawler, Schramm and Werner [6] have proved that ξ 2 = 5 4 by using the SLE techniques. The value of ξ 3 is not still known. Let J k be the indicator function of the event that k is a cut time up to n and let R n = n k=0 J k . Lawler [4] also proved that there exists c > 0 such that P (R n ≥ cn 1− ξ 2 ) ≥ c for d = 2, R n ≈ n 1− ξ 2 with probability one for d = 3, where ≈ denotes that the logarithms of both sides are asymptotic. While the understanding of the number of cut times has been advanced, there is a few results about the geometrical structure of the path around cut points, which is the purpose of this paper. We consider the following problem. If we condition that S[0, n] ∩ S[n + 1, 2n] = ∅, then what kind of structure does the path have around S(n)? Let S 1 , S 2 be independent simple random walks started at the origin. Then, thanks to the translation invariance and the reversibility of the simple random walk, our problem may be deduced to clarify the structure of S 1 , S 2 around the origin when we condition that S 1 [0, n] ∩ S 2 [1, n] = ∅. Letting n → ∞, we will face the following problems: By ( 1.2), the probability that S 1 [0, ∞) ∩ S 2 [1, ∞) = ∅ is 0 for d = 2, 3, so question (i) is not trivial. For Brownian motions, Lawler [3], and Lawler, Vermesi [8] have constructed Brownian paths conditioned to have no intersections. More precisely, let B 1 , B 2 be Brownian motions in R d (d = 2, 3) starting distance one apart and In [3], it was proved that for d = 2, the limit lim n→∞ P (· | B 1 [0, T 1 (e n )] ∩ B 2 [0, T 2 (e n )] = ∅) (1.6) exists and the rate of convergence is bounded above by O(e −δ √ n ) for some δ > 0.
For d = 3, it was shown in [8] that the limit of ( 1.6) also exists and the rate of convergence is at most O(e −δn ) (see Proposition 2.4.1). In this paper we will answer the question (i) and (iii). We will construct the path in (1.3) by proving the existence of the limit as in ( 1.6) for simple random walk (Theorem 1.2.1). Furthermore, we will derive same rates of convergence as Brownian cases. Since the speed of convergence in Theorem 1.2.1 is relatively fast, it would give evidence that the gap considered in (1.5) is small.
Even though the conditioned Brownian paths were already constructed as in ( 1.6), it is not straightforward to construct it for the simple random walk. Both in [3] and [8], the scaling property of Brownian motion is crucial in the construction and hence the same arguments cannot be applied for the simple random walk case. To overcome this problem, we will use the strong approximation of Brownian motion by simple random walk derived from the Skorohod embedding. By this approximation, we can define simple random walks S 1 , S 2 and Brownian motions B 1 , B 2 on the same probability space so that with high probability, the paths of S i are very close to those of B i . However, if S 1 and S 2 start from a same point, then the difference between the path of S i and that of B i is too large to control the difference between P (B 1 [0, n] ∩ B 2 [1, n] = ∅) and P (S 1 [0, n] ∩ S 2 [1, n] = ∅). (See Proposition 2.2.1 for the difference between S i [0, n] and B i [0, n]. We must admit the fact that the difference may be of order n 1 4 .) This difficulty can be dealt with using the following ideas. Even if starting points of S 1 and S 2 are very close, they gradually have a good chance of being reasonably far apart because of the conditioning not to intersect. Once S 1 and S 2 are far apart, we can use the Skorohod embedding to control the non-intersection probability of simple random walks (see Proposition 3.3.16 for details).
The question (iii) will be discussed in a forthcoming paper [9]. Let S 1 , S 2 be the associated two-sided random walks whose probability law is P ♯ in Theorem 1.2.1. In order to show that paths of S i have different structures from those of usual simple random walk S i , we will consider a simple random walk on (Here we regard G as the subgraph consisting of all the vertices visited and edges traversed by either S 1 or S 2 .) In [9], it will be shown that the simple random walk on G, say X, has subdiffusive behavior for d = 2. This is due to that G has many so called bottleneck edges and it takes much longer for X to move away from its starting point compared to the simple random walk in Z 2 . Throughout this paper, we use c, c ′ , c 1 , c 2 , · · · to denote arbitrary constants that depend only on the dimension d. The values of them may change from place to place.
and P ♯ extends uniquely to a probability measure on Γ(∞).
The paper is organized as follows. Section 2 gives some preliminary propositions about Brownian motions and simple random walks. In particular, we state the Skorohod embedding which is crucial in this paper. Key estimates are given in Section 3 by using this approximation. We give the proof of Theorem 1.2.1 in Section 4.

Known Results
In this section, we give a list of definition of the objects and known results commonly used throughout this paper.

Intersection Exponent
In this subsection, we review the intersection exponent for Brownian motion and simple random walk. Let d = 2 or 3. Let B 1 , B 2 be independent Brownian motions in R d . We start by stating the estimate from [5]. Let T i (n) = inf{t ≥ 0 : |B i (t)| = n}, and write P x,y = P x,y 1,2 to denote probabilities assuming B 1 (0) = x, B 2 (0) = y. Then we have the following proposition. (2.1) Next we state the analogues for simple random walks. Let S 1 , S 2 be independent simple random walks in Z d . Again we write P x,y = P x,y 1,2 to denote probabilities assuming S 1 (0) = x, S 2 (0) = y. Let Then the following proposition was proved in [4].
for all m ≤ n.

Skorohod Embedding
In this subsection, we state the strong approximation of Brownian motion by simple random walk derived from the Skorohod embedding (see [4] for details).  We will be using the strong Markov property at time T (n). However, one slight complication that arises is the fact that {B(t), S(td) : t ≤ T (n)} might contain a little information about B(t) beyond time T (n). To overcome this problem, we need the following proposition.
such that on the event Ψ(n), are conditionally independent given B(T (2n)).

Beurling Estimate
We need some estimates that say intuitively two random walks that get close each other are very likely intersect. For d = 2, it is a case of the Beurling estimate. For d = 3, corresponding estimates were obtained in [4]. Here we state them. Let B be the Brownian motion in R 2 and S be the simple random walk in Z 2 . Then the following are well-known (see [7] for the continuous case and [2] for discrete case).
(ii) ([2], Theorem 2.5.2.) There exists a constant K < ∞ such that for any n ≥ 1, any x ∈ Z 2 with |x| ≤ n, any connected set A ⊂ Z 2 containing the origin and such that sup{|z| : z ∈ A} ≥ n, For d = 3, there is no useful analogue of Proposition 2.3.1. So we need some more work. Let B, B ′ be two independent Brownian motion in R 3 . For each ǫ > 0 and b < ∞, let where the supremum is over all z with |z| ≤ n such that dist(z, B ′ [0, T ′ (2n)]) ≤ bn 1−ǫ , and T (n) (resp. T ′ (n)) be the first hitting time of B (resp. B ′ ) to the boundary of disk centered at the origin with radius n. Note that P z denotes the probability with B(0) = z and Z n is a function of B ′ [0, T ′ (2n)]. The following proposition says that Brownian path is a 'hittable set' with high probability.
Finally, we state an analogue of this proposition for simple random walks. Let S, S ′ be two independent simple random walks in Z 3 . For each ǫ > 0 and b < ∞, let where the supremum is over all z with |z| ≤ n and and τ (n) (resp. τ ′ (n)) be the first hitting time of S (resp. S ′ ) to ∂B(n). Again note that P z denotes the probability with S(0) = z and Z ♯ n is a function of S ′ [0, τ ′ (2n)]. Then we have the following.

Nonintersecting Brownian motions
In this subsection, we state convergence theorems for Brownian motion in R 2 and R 3 obtained in [3] and [8], respectively. Let d = 2 or 3, and B 1 , Here ξ = ξ d is the intersection exponent defined as in Section 2.1. In [3] and [8], it was shown the following convergence theorems for d = 2 and d = 3, respectively. For each K 1 , K 2 ⊂ D and w = (w 1 , w 2 ) ∈ ∂D 2 with w j ∈ K j ∩ ∂D, the limit exists. Moreover there exist c < ∞ and β > 0 depending only on the dimension such that the following holds.

Approximation of non-intersection probabilities 3.1 Preliminary
Fix L ∈ N and γ = (γ 1 , γ 2 ) ∈ Γ L . We write w i = γ i (lenγ i ) for the end point of γ i . Assume 10L < m < n. Let S 1 , S 2 be two independent simple random walks in Z d starting at w 1 , w 2 respectively. Let A m (γ) denote the event The goal of this section is to prove the following proposition.

Several Lemmas
By the strong Markov property, Applying Proposition 2.3.3 with ǫ = 0.01, b = 1, S = S 1 and S ′ = S 2 , we see that there exist δ > 0 and c < ∞ such that . By the strong Markov property, For each i = 1, 2, define Lemma 3.2.2. There exist δ > 0 and c < ∞ such that for each N ≥ m, Proof. By the strong Markov property, It is easy to see that there exist δ > 0 and c < ∞ such that where the supremum is over all z with . By Proposition 2.3.3, we see that there exist δ > 0 and c < ∞ such that

Coupling
Using the strong Markov property, we see that Here S 1 and S 2 are independent simple random walks starting at S 1 (τ 1 m 3 ) and , respectively, and we use same notation τ i (R), τ i k for the hitting time of S i . More precisely, let . Throughout this section we will let (B 1 , S 1 ) and (B 2 , S 2 ) be two independent Brownian motion -random walk pairs coupled as in Section and . From now on, we assume the event A m 3 (γ) ∩ F m ∩ G m ∩ H m holds and compare the probability that two Brownian motions do not intersect each other with the probability that simple random walks do not intersect. For this purpose, let on the event F m , we see that P (J m,N , β = k) for m 3 < k ≤ N assuming that F m , G m and H m hold.
Then there exist c < ∞ and δ > 0 such that (3.21) Now we give an upper bound of By the strong Markov property, this probability is bounded above by Assume Q c holds. Then it is easy to see that For such s and t, we have Namely, the following event holds, where the supremum is over all z with z ∈ B(2 k + 2 31k 60 ) and We let H k be the event {Z k ≤ 2 −δk }. By Proposition 2.3.2, there exists δ > 0 such that Therefore, we have only to estimate which is a hittable set. By the strong Markov property, and this finishes the proof.

3.3.2
Bounds for 21m 60 < k ≤ N − 3 From now we assume that 21m 60 < k ≤ N − 3. The similar argument in the proof of Lemma 3.3.1 gives the following lemma, so we omit the proof.
for 21m 60 < k ≤ N − 3, it is enough to show the following lemma. Lemma 3.3.4. There exist δ > 0 and c < ∞ such that Proof. By the strong Markov property, the right hand side of ( 3.26) is bounded above by On the other hand, on the event Q c , we have Since 31k 60 ≤ m 3 , we have For k > 21m 60 , a standard estimate shows that Using the strong Markov property at T 1 (2 k−1 − 2 31k 60 ) first, and then estimating P (J m,k−2 ), we have Therefore, the proof for d = 3 and 21m 60 < k ≤ 20m 31 is finished. Next we assume d = 3 and 20m  Since this event occur with probability at most c2 − k 3 , the lemma is proved for d = 3.
Assume d = 2. In this case, the probability of the event ( 3.27) is bounded below by 1/k, so we need to change the proof. Assume 21m 60 < k ≤ 20m 31 . (For the otherwise, the proof is almost same in this case. So we only consider this case.) Let We already showed that if ] ∪ γ 2 ) = ∅ and Q c hold, then η ≤ T 1 (2 k + 2 31k 60 ). By the Proposition 2.3.1, we see that and the lemma is proved for all cases.

Bounds for
Finally, we give estimates for N − 2 ≤ k ≤ N . Since a proof is similar for each case, we only consider for k = N . By definition of β in ( 3.16), we see that We will only give bounds on the probability of the event for i = 1 in ( 3.28). First we show the following lemma.
For d = 3, the probability of the event ( 3.31) is bounded above by c2 − N 3 . Therefore, by the strong Markov property, Next we consider the two dimensional case. Assume Therefore, Using Proposition 2.3.1, the probability of the event ( 3.32) is bounded above by c2 − N 3 . Hence by the strong Markov property, ] ∪ γ 2 ) = ∅ and Q c holds. This implies that Again by using Proposition 2.3.1, and the lemma is proved.
To estimate the probability of ( 3.28), we have only to show the following lemma.
Before we start to prove this lemma, we need to prepare several lemmas.
Proof. Let Q be the event defined in the proof of Lemma 3.3.5. Let If Q c holds and σ < τ 2 N , then It is easy to see that the probability of ( 3.36) is bounded above by c2 −δN for some c < ∞ and δ > 0. Hence by the strong Markov property, If Q c and ( 3.37) hold, we see that For any x ∈ ∂B(2 N − 2

2N
3 ), we have By the strong Markov property, and hence prove the lemma.
Proof. Let Q be the event defined in the proof of Lemma 3.3.5. If 3 )] = ∅ and Q c holds, then we have Then by Proposition 2.3.2, for some δ > 0 and c < ∞. Therefore, Using the strong Markov property for B 1 , we see that this probability is bounded above by and hence the proof is finished.

Conclusion Lower Bound
Combining estimates obtained in subsections 3.3.1, 3.3.2 and 3.3.3 with ( 3.18), we have the following proposition.

Upper bound
From this subsection, we will give an upper bound of P Note that β ♯ ≤ N almost surely and β ♯ = m 3 implies J m,N holds. Therefore, We will give bounds for the second term in the right hand side of ( 3.44). For (3.45) We only consider for i = 1 in ( 3.45).

Bounds for 21m
60 ≤ k ≤ N − 3 Lemma 3.3.11. There exist δ > 0 and c < ∞ such that Proof. Since the idea is quite similar as in the proof of Lemma 3.3.1, we will just sketch the proof. By the strong Markov property, the probability in the left hand side of ( 3.46) can be bounded above by with probability at least 1 − c exp(−2 δk ), for some δ > 0 and c < ∞. By Proposition 2.3.3, once S 1 gets close to S 2 [0, τ 2 k ] during [τ 1 k−1 , τ 1 k ], then S 1 intersects S 2 [0, τ 2 k+1 ] until τ 1 k+1 with probability at least 1 − 2 −δk . Hence by using the strong Markov property, the lemma can be proved.
Proof. Similar ideas as in the proof of Lemma 3.3.4 works here. So we just state the idea of the proof.
First let d = 3. The probability that The probability that such an entrance occurs is at most c2 −(k− m 3 ) . Finally, using P 3 )ξ and the strong Markov property, the lemma is finished for d = 3.
Next let d = 2. In this case, if B 1 enters in B(2 Therefore, by using the strong Markov property, we finish the proof of the lemma for d = 2.

Bounds for
We can prove the following lemma by a same idea of Lemma 3.3.11. So we omit its proof.
Lemma 3.3.13. There exist δ > 0 and c < ∞ such that , we have only to show the following lemma. Lemma 3.3.14. There exist δ > 0 and c < ∞ such that Proof. We will give a full proof for this lemma. Recall the definition of PATH 2 f in ( 3.15). Let 11m 60 +1 } be the set obtained by letting PATH 2 f be fattened twice. Let for some δ > 0 and c < ∞. Recall that on the event Ψ, are conditionally independent given B 1 (T 1 k+2 ) and B 2 (T 2 k+2 ). Therefore, From now we will estimate for P If Q c holds, then it is easy to see that ]. Here the last inequality comes from that k ≤ 21m 60 . Hence, Then the right hand side in ( 3.51) is bounded above by ] ∪ γ 2 is a path from the origin to ∂B(2 k+1 ), by using Proposition 2.3.1, we see that for some δ > 0 and c < ∞. Hence ( 3.52) is bounded above by and the proof for d = 2 is finished.
Next we consider for d = 3. Recall the events F m , G m and H m in ( 3.12). By ( 3.51), we need to estimate on the event F m ∩ G m ∩ H m . For this end, we decompose 2PATH 2 f into three parts U 1 , U 2 and U 3 as follows.
Since γ 2 ∈ B(2 L ) and L ≤ m 10 , it is easy to see that for some δ > 0 and c < ∞. However, on the event F m . So the probability of ( 3.53) is bounded above by c2 − m 24 for some c < ∞. Using the strong Markov property, for some δ > 0 and c < ∞. Finally we consider for U 2 . Let Then by the strong Markov property, Hence on the event H m , the right hand side of ( 3.54) can be bounded above by c2 −δk 2 −(k− m 3 )ξ for some δ > 0 and c < ∞, and the lemma is proved.
3.3.8 Bounds for k = N − 2, N − 1, and k = N Again, we will only consider for k = N as in Section 3.3.3. Other cases can be estimated by a similar argument given below.
Proof. We will sketch the proof. First we consider the following probability, 3 ), it follows from Proposition 2.3.1 that the probability that S 1 does not intersect S 2 [0, τ 2 N ] until it reaches ∂B(2 N ) is bounded above by c2 − N 6 . Combining these estimate, we see that ( 3.56) can be bounded above by c2 − N 6 2 −(N − m 3 )ξ for d = 2. Therefore, in order to show ( 3.55), we need to estimate the following probability,   exists. If we write P ♯ (γ) for the limit, then Remark 4.1.4. In order to simplify the notations, all results above were stated for the first hitting time of ∂B(2 N ) instead of ∂B(N ). However there is no essential difference between them and similar arguments also work for the latter case. Since it is easy to extend above results to the hitting time of ∂B(N ), we leave the details to the reader.