Two observations on the capacity of the range of simple random walks on $\mathbb{Z}^3$ and $\mathbb{Z}^4$

We prove a weak law of large numbers for the capacity of the range of simple random walks on $\mathbb{Z}^{4}$. On $\mathbb{Z}^{3}$, we show that the capacity, properly scaled, converges in distribution towards the corresponding quantity for three dimensional Brownian motion.


Introduction
Let (X n ) n be a simple random walk on Z d with d = 3 or 4. We are interested in the scaling limit of the capacity of the random set where the capacity Cap(F ) of a set F of vertices on Z d is defined as the sum of escaping probabilities: The capacity of the range of random walks is closely related with the intersection probability of two independent random walks. In fact, many estimations on Cap(X[0, n]) were deduced from that. We refer the reader to the Lawler's classical book [7] and the reference there.
In the present paper, we show that when d = 4, the second moment of Cap(F ) is asymptotically equivalent to the square of the first moment, which implies a weak law of large numbers: Such a weak law of large numbers was conjectured by Asselah, Schapira and Sousi [2, Section 6]. Besides, they also expect a random scaling limit for Cap(X[0, n])/E[Cap(X[0, n])] for d = 3 and there is no such kind of weak law of large numbers on Z 3 . We affirm this as a corollary (see Remark 4.1) of our second main result, which states that as n → ∞, Cap(X[0, n])/ √ n has a random limit in distribution, which is the corresponding quantity for three dimensional Brownian motion. To be more precise, let (M t ) t≥0 be the standard Brownian motion on Z 3 . Recall the Green function for Brownian motions on R 3 , see e.g.
[9, Theorem 3.33]: The corresponding (Brownian motion) capacity of a Borel set F is given by The law of large numbers for (Cap(X[0, n])) n had already been obtained in dimension 5 and larger by Jain and Orey [5]. In [2], Asselah, Schapira and Sousi established a central limit theorem in dimension larger than or equal to 6. To understand the model of random interlacement invented by Sznitman [12], Ráth and Sapozhnikov [10,11]  by showing that it is comparable with the square of the first moment. Theorem 1.1 sharpens our result in [4], which implies a weak law of large numbers for Cap(X[0, n]) n on Z 4 . Soon after this and very recently, Asselah, Schapira and Sousi [1] greatly improved the result in Z 4 by proving the strong law of large numbers and the central limit theorem. In another paper [3] by the same authors, the strong law of law numbers was established for the Wiener sausage, which is the continuous counterpart of the discrete simple random walk. We refer the reader to [3] for more references and historical remarks on the Wiener sausages.
Finally, we briefly outline the proof. The argument for Theorem 1.1 is a refinement of that in [4]. We consider two independent simple random walks (X Organization of the paper We introduce necessary notation in Section 2. Then, we prove Theorem 1.1 and 1.2 in separate sections.

Notation
We collect several notation in the following.
• Green function for Brownian motions on R d (d ≥ 3): • SRW capacity of a set F : • Brownian motion capacity of a Borel set F : It is known that the capacity of a set is closely related to the hitting probability of that set, see Lemma 3.1.
We will prove Theorem 1.1 by refining the argument in [4, Lemma 2.4].
Proof of Theorem 1.1. Let (X 0 n ) n≥0 , (X 1 n ) n≥0 , (X 2 n ) n≥0 be three independent simple random walks. Denote by E x (i) the expectation corresponding to the random walk X i with initial point x. Similarly, we define (E x,y (i),(j) ) i =j . For simplicity of notation, we denote by E x,y,z (or P x,y,z ) the expectation (or probability) corresponding to X 0 , X 1 and X 2 with initial points x, y and z, respectively. Recall that X 0 [0, n] is the range of X 0 up to time n. Similarly, we define X 1 [0, ∞) and X 2 [0, ∞).
• (M 0 t ) t and (M 1 t ) t are both Brownian motions on Z 3 starting from 0.
As we mentioned above, by Skorokhod approximation, P[E c 1 ] ≤ C · e −n γ(ǫ) where C < ∞ does not depend on n. By union bounds and Hoeffding's inequality, P[E c 2 ] ≤ Cne −n ǫ/2 /C where C < ∞ does not depend on n. By [6, Lemmas 2.4, 2.6], for all N ≥ 1, ∃δ > 0 (in the definition of E 3 ) and C < ∞ such that for all n ≥ 1, P[E c 3 ] ≤ C · n −N . We define E = E 1 ∩ E 2 ∩ E 3 , take N = 2 and choose δ accordingly such that Take y n ∈ Z d such that ||y n || 2 = ⌊n 1 2 +ǫ ⌋. By the independence between (X 0 , M 0 ) and (X 1 , M 1 ), the last passage time decomposition and the Green function estimate, we have that (12) and similarly, by [ We will show that (12) = 1 3 (1 + o(1)) · (13) which is equivalent to We first find several quantities, which are asymptotically equivalent to the left hand side of (14). By the definition of E 3 and the strong Markov property of X 1 , we get that Next, we will show that X 1 could be replaced by M 1 in the following sense: ≤ 1 E · c · (n −6ǫ + n − 1 2 −2ǫ Cap(Nbd(X 0 [0, n], n