On Covering paths with 3 Dimensional Random Walk

In this paper we find an upper bound for the probability that a $3$ dimensional simple random walk covers each point in a nearest neighbor path connecting 0 and the boundary of an $L_1$ ball of radius $N$. For $d\ge 4$, it has been shown in [5] that such probability decays exponentially with respect to $N$. For $d=3$, however, the same technique does not apply, and in this paper we obtain a slightly weaker upper bound: $\forall \varepsilon>0,\exists c_\varepsilon>0,$ $$P\left({\rm Trace}(\mathcal{P})\subseteq {\rm Trace}\big(\{X_n\}_{n=0}^\infty\big) \right)\le \exp\left(-c_\varepsilon N\log^{-(1+\varepsilon)}(N)\right).$$


Introduction
In this paper, we study the probability that the trace of a nearest neighbor path in Z 3 connecting 0 and the boundary of a L 1 ball of radius N is completely covered by the trace of a 3 dimensional simple random walk.
First, we review some results we proved in a recent paper for general d's. For any integer N ≥ 1, let ∂B 1 (0, N) be the boundary of the L 1 ball in Z d with radius N. We say that a nearest neighbor path P = P 0 , P 1 , · · · , P K is connecting 0 and ∂B 1 (0, N) if P 0 = 0 and inf{n : P n 1 = N} = K. And we say that a path P is covered by a d dimensional random walk {X d,n } ∞ n=0 if Trace(P) ⊆ Trace(X d,0 , X d,1 , · · · ) := {x ∈ Z d , ∃n X d,n = x}.
In [5], we have shown that for any d ≥ 2 such covering probability is maximized over all nearest neighbor paths connecting 0 and ∂B 1 (0, N) by the monotonic path that stays within distance one above/below the diagonal x 1 = x 2 = · · · = x d . Theorem 1.1. (Theorem 1.4 in [5]) For each integers L ≥ N ≥ 1, let P be any nearest neighbor path in Z d connecting 0 and ∂B 1 (0, N). Then P Trace(P) ∈ Trace(X d,0 , · · · , X d,L ) ≤ P ր P ∈ Trace(X d,0 , · · · , X d,L ) Then noting that the probability of covering ր P is bounded above by the probability a simple random walk returns to the exact diagonal line for [N/d] times, one can introduce the Markov procesŝ where X i d,n is the ith coordinate of X d,n and see that {X d−1,n } ∞ n=0 is another d − 1 dimensional non simple random walk, which is transient when d ≥ 4. Thus, we immediately have the following upper bound: There is a P d ∈ (0, 1) such that for any nearest neighbor path P = (P 0 , P 1 , · · · , P K ) connecting 0 and ∂B 1 (0, N) and {X d,n } ∞ n=0 which is a d−dimensional simple random walk starting at 0 with d ≥ 4, we always have Here P d equals to the probability that {X d,n } ∞ n=0 ever returns to the d dimensional diagonal line. Theorem 1.2 implies that for each fixed d ≥ 4, the covering probability decays exponentially with respect to N.
For d = 3, the same technique we had may not hold since now {X 2,n } ∞ n=0 is a recurrent 2 dimensional random walk, which means that P d = 1 and that the original 3 dimensional random walk will return to the diagonal line infinitely often. To overcome this issue, we note that although the diagonal line D ∞ = {(0, 0, 0), (1, 1, 1), · · · } is recurrent, it is possible to find an infinite subsetD ∞ that is transient. And if we can further show for this specific transient subset we found, the returning probability is uniformly bounded away from 1 (which is not generally true for all transient subsets, as is shown in Counterexample 1 in Section 3), then we are able to show With this approach, we have the following theorem Theorem 1.3. For each ε > 0, there is a c ε ∈ (0, ∞) such that for any N ≥ 2 and any nearest neighbor path P = (P 0 , P 1 , · · · , P K ) ⊂ Z 3 connecting 0 and ∂B 1 (0, N), we have Note that the upper bound in Theorem 1.3 seems to be non-sharp. The reason is that we did not fully use the geometric property of path ր P to minimize the covering probability. I.e., although we require our simple random walk to visit the transient subset for O(N log −1−ε (N)) times, those returns may be not enough to cover every point inD ∞ ∩ ր P. We conjecture that the actual decay rate is also exponential for d = 3. Numerical simulations seem to support this as is shown in Section 5.
There is a c ∈ (0, ∞) such that for any N ≥ 2 and any nearest neighbor path P = (P 0 , P 1 , · · · , P K ) ⊂ Z 3 connecting 0 and ∂B 1 (0, N), we always have The structure of this paper is as follows: In Section 2, we construct the infinite subsetD ∞ of the diagonal line, calculate its density, and show it is transient. In Section 3, we show the returning probability ofD ∞ is uniformly bounded away from 1, no matter where onD ∞ the random walk starts from. With these results, in Section 4, we finish the proof of Theorem 1.3. Numerical simulations are given in Section 5 showing possible non tightness of our result.

Infinite Transient Subset of the Diagonal
Without loss of generality we can concentrate on the proof of Theorem 1.3 for sufficiently large N. Recall that is the path connection 0 and B 1 (0, N) that maximizes the covering probability. When d = 3, let be the points in ր P that lie exactly on the diagonal. Although it is clear that for simple random walk {X 3,n } ∞ n=0 starting at 0, D ∞ is a recurrent set, following a similar construction to Spitzer [6, Chapter 6.26], we find a transient infinite subset of D ∞ as follows: for n 1 = 0, n 2 = ⌈log 1+ε (2)⌉ = 1, and for all k ≥ 3 Since log 1+ε (k) > 1 for all k ≥ 3, it is easy to see that {n k } ∞ k=1 is a monotonically increasing sequence. Moreover, for each 1 ≤ k 1 < k 2 < ∞, This implies that for all k 2 ≥ 8 and 1 ≤ k 1 < k 2 , Recalling the definition of n k in (2.1), we also equivalently have Proof. Noting that for any k such that we must have that k > C N , and that Noting that K N → ∞ as N → ∞ and that which implies C N < K N and finishes the proof of the upper bound. On the other hand, note that Thus the proof of Lemma 2.1 is complete.
With Lemma 2.1, we next show thatD ∞ is transient for 3 dimensional simple random walk: Proof. According to Wiener's test (see Corollary 6.5.9 of [3]), it is sufficient to show that . Then according to the definition of capacity (see Section 6.5 of [3]), we have for all k ≥ 1 By Lemma 2.1,

Uniform Upper Bound on Returning Probability
Now we haveD ∞ is transient, i.e., P X n ∈D ∞ i.o. = 0, which immediately implies that there must be somex ∈ Z 3 \D ∞ such that ∞ is the first time a simple random walk visitsD ∞ , and P x (·) is the distribution of the simple random walk condition on it starting at x. Then note thatD ∞ is a subset of the diagonal line, which impliesD ∞ has no interior point while Z 3 \D ∞ is connected. Thus for any x k ∈D ∞ , there exists a nearest neighbor path Y = {y 0 , y 1 , · · · y m } with y 0 = x k , y m =x while y i ∈ Z 3 \D ∞ , for all i = 1, 2, · · · , m − 1. Combining this with the fact that for all x ∈ Z 3 \D ∞ , we have P y i (TD ∞ < ∞) < 1, for all i ≥ 1, which in turns implies that ∞ is the first returning time, i.e. the stopping time a simple random walk first visitD ∞ after its first step.
However, in order to use the transient setD ∞ as if it is just like one point in a transient random walk, (3.2) is not enough. We need to show that starting from each point x k = (n k , n k , n k ) ∈D ∞ , the probability P x k (TD ∞ < ∞) is uniformly bounded away from 1. And this is not generally true for all transient subsets A. First of all, when A has interior points, the returning probability of those points are certainly one. And even if A has no interior point and Z 3 \ A is connected, we have the following counter example:

Counterexample 1: Consider subsets
where the 2 dimensional projection of A is illustrated in Figure 1 (the distances between A k 's are not exact in the figure):

Figure 1. A counter example to uniform upper bound on returning probability
Using Wiener's test, it is easy to see A is a transient subset. However, for points a k = (2 k , 0, 0) ∈ A, k ≥ 1, in order to have a simple random walk starting at a k never returns to A, we must have the first k steps of the random walk be along the z−coordinate. Thus Remark 3.1. It would be interesting to characterize uniformly transient sets i.e. sets with uniformly bounded return probabilities.
Fortunately, for the specific transient subsetD ∞ , since it becomes more and more sparse as x → ∞, we can still have: Proof. With (3.2) showing all returning probabilities are strictly less than 1, it is sufficient for us to show that Actually, here we prove a stronger statement It suffices for us to show that To show (3.6) and (3.7), we first note the well known result that there is a C < ∞ such that for any x = y ∈ Z 3 , First, to show (3.6) we have according to (2.2), for any k ≥ 8 Thus it is again sufficient to show that Note that (3.10) For each k ≥ 8 and i ≤ [k 1/2 ], we have

Thus (3.11)
Then for each k ≥ 8 and i ∈ k 1/2 , k − 1 , Thus (3.12) Combining (3.9), (3.11) and (3.13), we obtain (3.6). Then, to show (3.7) we have according to (2.2), for any k ≥ 8 Now for each k we separate the infinite summation in (3.15) as For its first term we use similar calculation as in (3.12) and have (3.17) And since At last for the second term in (3.16), we have for each k ≥ 8 and i ≥ k 2 + 1, .

Finally, noting that
we have the tail term and thus By Lemma 2.1 we have for all N ≥ 4. Thus combining (3.21) and (3.22) log(1 − c ε,1 ). And the proof of Theorem 1.3 is complete.

Discussions
In Conjecture 1.4, we conjecture that the cover probability should have exponential decay just as the d ≥ 4 case. This conjecture seems to be supported by the following preliminary simulation which shows the log-plot of probabilities that the first 5000 steps of a 3 dimensional simple random walk starting at 0 cover D i = {(0, 0, 0), (1, 1, 1), · · · , (i, i, i)} for i = 1, 2, · · · , 9. The simulation result above seems to indicate that after taking logarithm, the covering probability decays almost exactly as a linear function, which implies the exponential decay we predicted, indicating that the upper bound we found in Theorem 1.3 is not sharp. For N = 9, if Theorem 1.3 were sharp and there were a correction greater than log(N) in the exact decaying rate, then in the log-plot, it would cause the point to be log log(9) ≥ 0.787 above the line. This is not seen in the simulation. However, the simulation above does not rule out the possibility that there is correction of a smaller order than log(N), since it could be so small for the initial 9 i's and thus has not be seen significantly yet in the current simulation.
Another possible approach towards a sharp asymptotic is noting that although {X 2,n } ∞ n=0 is recurrent and will return to 0 with probability 1, the expected time between each two successive returns is ∞. Moreover, in order to cover ր P, only those returns to diagonal before that {X 3,n } ∞ n=0 has left B 2 (0, N) ⊃ B 1 (0, N) forever could possibly help. This observation, together with the tail probability asymptotic estimations using local central limit theorem and techniques in [1] and [2] applied on the non simple random walk {X 2,n } ∞ n=0 , and some large deviation argument, enable us to find a proper value of T such that • with high probability {X 3,n } ∞ n=T ∩ B 2 (0, N) = Ø, • with high probability {X 2,n } T n=0 will not return to 0 for [N/3] times or more.
Right now this approach can only give us the following weaker upper bound (a detailed proof can be found in technical report [4]): Proposition 4.1. There are c, C ∈ (0, ∞) such that for any nearest neighbor path P = (P 0 , P 1 , · · · , P K ) ⊂ Z 3 connecting 0 and ∂B 1 (0, N), P Trace(P) ⊆ Trace {X 3,n } ∞ n=0 ≤ C exp −cN 1/3 . However, this seemingly worse approach might have the potential to fully use the geometric property of path ր P to minimize the covering probability. Note that in order to cover D [N/3] we not only need {X 2,n } ∞ n=0 return to 0 for at least [N/3] times before {X 3,n } ∞ n=0 forever leaving B 2 (0, N), but also must have the locations of X 3,n at such visits cover each point on the diagonal (let alone the request of covering the off diagonal points as well). I.e., define the stopping times τ l 3 ,0 = 0 τ l 3 ,1 = inf{n ≥ 1 :X 2,n = 0} and for all i ≥ 2 τ l 3 ,i = inf{n > τ l 3 ,i−1 :X 2,n = 0}.