The dimension of the range of a transient random walk

We find formulas for the macroscopic Minkowski and Hausdorff dimensions of the range of an arbitrary transient walk in Z^d. This endeavor solves a problem of Barlow and Taylor (1991).


Introduction
Throughout, we choose and fix an integer d 1, and let Z d denote the corresponding d-dimensional integer lattice. Our main object of study is the dimension of the range of arbitrary random walks with state space a subset of Z d . With this aim in mind, let X := {X n } ∞ n=0 denote a random walk on Z d , started at some deterministic point X 0 := a ∈ Z d , and denote its range by R X := ∪ ∞ n=0 {X n }. Our goal is to give an answer to the following question of Barlow and Taylor [2,Problem,p. 145]: What is Dim H (R X )? (1.1) Here, Dim H (G) denotes the macroscopic Hausdorff dimension of a set G ⊂ R d , as was defined in [1,2]. We will recall the formal definition and first properties of Dim H in §2.1 below. In informal terms, Dim H (G) measures the "local dimension of G at infinity," and describes the large-scale geometry of G in a manner that is similar to the way that the ordinary, microscopic, Hausdorff dimension of G describes the small-scale geometry of G. Barlow and Taylor [2] pose (1.1) only for transient random walks. We briefly explain why answering (1.1) in the case of a recurrent walk is straight-forward. Here, "recurrent" means "recurrent on its range"; the walk can still avoid significant portions of Z d . Recall that a point x ∈ Z d is possible for X if P 0 {X n = x} > 0 for some n 1, where P a denotes the conditional law of X given that X 0 = a, as usual. The collection of all possible points of X is an additive subgroup of Z d , whence homomorphic to Z k for some integer 0 k 2. [This is valid by the structure theory of finitely-generated abelian groups and the fact that X is transient provided that the collection of possible points of X is homomorphic to Z for 3.] Therefore, the strong Markov property of X implies that R X is homomorphic to Z k a.s., and hence we have Dim H (R X ) = Dim H (Z k ) = k a.s. [2].
As far as we know, the only existing positive result about (1.1) is due to Barlow and Taylor themselves [2,Cor. 7.9]. In order to describe their result, let g denote the Green function of X. That is, g(a , x) := ∞ n=0 P a {X n = x}, x, a ∈ Z d . (1.2) Of course, g(a , x) = g(0 , x − a) as well, and the transience of X is equivalent to the finiteness of the function g on all of Z d × Z d .
Question (1.1) was in part motivated by the following positive result. The principal goal of this article is to answer question (1.1) in general. We do this by following a suggestion of Barlow and Taylor and introducing a random-walk "index" that is equal to Dim H (R X ). Moreover, as is tacitly implied in [2], this index is defined solely in terms of the statistical properties of X.
It turns out that our index is related to the notion of a "recurrent set" for X. Recall that a set F ⊆ Z d is said to be recurrent for X, under P a , if the random set X −1 (F ) := {n 0 : X n ∈ F } is unbounded P a -a.s. This definition makes sense regardless of whether F is random or not. Because X is transient, a necessary condition for the recurrence of F is that F is unbounded. The following example shows that the converse implication is false: Let X denote the simple symmetric walk on Z 3 , and define F := ∪ ∞ k=1 {x k }, where x k := (0 , 0 , k 3 ). By the classical local central limit theorem, P 0 {x k ∈ R X } = O(k −3 ) as k → ∞. Therefore, Tonelli's theorem implies that The Dimension of the Range of a Random Walk A necessary-and-sufficient condition for the recurrence of a non-random set F was found first by Itô and McKean [7] in the case that X is the simple random walk on Z d and d 3. Lamperti [15] discovered a necessary-and-sufficient condition in the case that X belongs to a large family of transient random walks on Z d . Lamperti's theorem in fact holds for a large family of transient Markov chains X. When X is a general transient random walk on Z d , or more generally a Markov chain on a countable state space, there is also an exact condition for set recurrence, but that condition is more involved; see Bucy [4] and, more recently, Benjamini et al [3].
All latter works involve various notions of abstract capacity that are borrowed from probabilistic potential theory. Our answer to (1.1) is also stated in terms of a sort of abstract capacity condition, and appears later on as Corollary 5.2 to a master theorem [Theorem 5.1] on the large-scale potential theory of random walks. We have not included our answer to (1.1) in this Introduction since that answer is complicated and its description hinges on first introducing a certain amount of machinery. Still, our answer has a simpler form when the random walk X is sufficiently regular; see Corollary 5.4 for instance.
We conclude the Introduction with an outline of the paper. In §2 we include some of the technical prerequisites to reading this paper. Then, in §3 we develop a macroscopic theory of "fractal percolation" that is the large-scale analogue of the microscopic theory of fractal percolation [17,20]. Our macroscopic extension of the microscopic theory is not entirely trivial, but will ring familiar to many experts.
In §4 we introduce a forest representation of Z d and use it together with the theory of two-parameter processes [9] in order to characterize exactly when R X intersects a piece of a macroscopic fractal percolation set. This is the truly-novel part of the present article, and is likely to have other uses particularly in computing the ordinary and/or large-scale dimension of complex random sets.
Finally, in §5 we establish a master theorem on "hitting probabilities"; see Theorem 5.1. Subsequently, we use that master theorem, together with an adaptation of an elegant replica method of Peres [20] to the present setting, in order to compute Dim H (R X ∩ F ) for every recurrent non-random set F ; see Corollary 5.4. Our answer to the original question (1.1) of Barlow and Taylor is obtained by specializing the preceding to F := Z d .
In the section that follows the above discussion [ §6] we derive the following simpler and more elegant almost-sure representation for the macroscopic Minkowski dimension of an arbitrary random walk on Z d : In the last section §7 we state a few interesting remaining open problems and related conjectures.

Background material
This section introduces the prerequisite material, necessary for later use.

Macroscopic Hausdorff dimension
Throughout we follow the original notation of Barlow and Taylor [1,2] by setting for all k 0. For every integer n-positive as well as non positive-define D n to be the collection of all dyadic cubes Q (n) of the form where j 1 , . . . , j d ∈ Z are integers. In the sequel we tacitly will use the fact that the cubes in D n are actually disjoint. If a cube Q (n) has the form (2.2), then we say that j := (j 1 , . . . , j d ) is the southwest corner of Q (n) , and 2 n is the sidelength of Q (n) .
By D we mean the collection of all dyadic hypercubes of Z d ; that is, A special role is played by the collection of all dyadic cubes of sidelength not smaller than 1, which we denoted as where "side" temporarily denotes "sidelength," and the minima are taken over all possible coverings Q 1 , . . . , Q m of A ∩ S k such that every Q i ⊂ S k is a cube of sidelength 2 i 1 whose southwest corner is in 2 i Z d . Note in particular that Q 1 , . . . , Q m are all elements of D 1 .
Let us pause momentarily in order to make two small observations. Remark 2.1. The minimum in (2.3) is taken over a finite set. Therefore, the said minimum is attained for a covering Q 1 , . . . , Q m of A with respective sidelengths 2 1 , . . . , 2 m 1. We will appeal to the conclusion of this remark tacitly in the sequel.
where now the coverings use dyadic cubes in V k rather than dyadic cubes in S k . Assume for the moment that A ⊆ S k . Then it follows readily that there exists a real number c d > 0, that depends only on d, such that The first inequality above is immediate since every cover that uses S k -cubes is also cover that uses V k -cubes. In order to understand the second inequality observe that for every dyadic cube Q ⊆ V k there exist at most 2 2d dyadic cubes Q 1 , Q 2 , . . . in S k such that side(Q j ) side(Q) for all 1 j 2 2d and Q ∩ S k ⊂ ∪ 2 2d j=1 Q j . This is merely an assertion about the geometry of R d . In any case, it follows from this assertion and Jensen's inequality that EJP 23 (2018), paper 83. Therefore, if Q 1 , Q 2 , . . . form a minimizing V k -cover of A ∈ S k , then there exist dyadic S k -cubes {Q i,j } 1 j 2 2d , for each i 1, such that This proves the remaining portion of (2.4) because the cubes in the preceding sum are all in S k and they cover A.
We can identitfy N α (A , S k ) as the large-scale analogue of the α-dimensional Hausdorff measure of A, restricted to V k , and scaled so that ∞ k=1 N α (A , S k ) can serve as a proxy for "macroscopic α-dimensional Hausdorff measure." The latter is not a measure. Still, these kinds of remarks undoubtedly led M. T. Barlow and S. J. Taylor [1,2] to define the macroscopic Hausdorff dimension of A via 1 We adopt this definition and notation here and throughout.
One can glean many of the properties of Dim H from first principles. For instance, it follows readily from the definition of N α that Dim H (A) = 0 if A is a finite set. It is also easy to see that Dim H (A) Dim H (B) whenever A ⊆ B. This fact is a consequence of the observation that every covering of B is also a covering of A when A ⊆ B. Finally, let us mention that that Dim H (Z d ) = d [1, pp. 2622-2623], and therefore, (2.6) whenever A ⊆ B. The second, seemingly-natural, inequality in (2.6) is one of the novel features of the theory of Barlow and Taylor [1,2], and does not hold for some of the previously-defined candidates of large-scale dimension in the literature [18,19]. Let X denote the simple symmetric random walk on Z d where d 3. According to the local central limit theorem, g(x , y) ∼ const · x − y 2−d as x − y → ∞. Therefore, Proposition 1.1 applies, and implies the very appealing fact that the macroscopic Hausdorff dimension of the range of X is a.s. 2. Barlow and Taylor [2] have proved that macroscopic Hausdorff dimension of the range of transient Brownian motion is also a.s. 2, thus giving further credance to their assertion that Dim H is a natural large-scale variation of the classical notion of [microscopic] Hausdorff dimension, nowadays usually denoted by dim H .

Recurrent sets for Markov chains
We follow the existing related works on probabilistic potential theory [3,4,6,15], and consider a somewhat more general setting in which our random walk X is replaced by a transient Markov chain, still denoted by X. However, in contrast with prior works we continue to assume that our Markov chain X takes values in the special state space Z d , for some d 1, and not in a general countable state space. This assumption is needed for some, but not all, of the ensuing analysis. We make the assumption once and for all in order to avoid studying various special cases, and hence to simplify the exposition. We continue to write R X := ∪ ∞ n=0 {X n } for the range of the Markov chain X.
As was done for random walks, let P a denote the conditional law of X, given X 0 = a. A random or non-random set F ⊆ Z d is said to be recurrent for X under P a when R X ∩ F is unbounded P a -a.s. 2 We are aware of at least two characterizations of non-random recurrent sets for general chains, due to Bucy [4] and Benjamini et al [3]. In order to describe the second characterization, which turns out to be more relevant for our needs, let M 1 (F ) denote the collection of all probability measures on F , and c 1 (F ; a) the Martin capacity of F for the walk started at a ∈ Z d [3]. That is, Recall that a set G ⊂ Z d is said to be cofinite when Z d \ G is a bounded set.
The preceding capacity condition for cofinite sets is not so easy to verify in concrete settings. There is an older result, due to Lamperti [15], which contains a more easilyapplicable characterization of recurrent sets for "nice" Markov chains.

Example 2.5.
Suppose that X is a random walk that satisfies Condition (1.3) of Proposition 1.1. It readily follows that g(0 , y) > 0 for all y = 0, and simultaneously for all integers k 0 and µ ∈ M 1 (F ). As usual, we write "f (z) g(z) for all z ∈ Z" to mean that there exists a positive and finite constant C such that 2 One can imagine other variations on this definition. For instance, one could consider F to be recurrent if instead P a {R X ∩ F is unbounded} > 0. It should be possible to adjust our methods to study the latter notion of recurrence; see Theorem 5.1 for instance. We are interested mainly in the case where X is a random walk.
In that case, the two notions agree. Therefore, we will not puruse these matters further. describes the usual random-walk capacity of G ⊂ Z d . 3 It is easy to see from (1.3) that the Lamperti-type condition (2.8) also holds in this case. Therefore, Corollary 2.4 tells us

Macroscopic fractal percolation
We temporarily leave the topic of Markov chains and random walks in order to present some basic facts about macroscopic fractal percolation.
Let k 0 denote a fixed integer, and suppose {U (Q) : Q ∈ D, Q ⊂ V k } is a collection of independent random variables, defined on a rich enough probability space (Ω , F, P), such that each U (Q) is distributed uniformly between 0 and 1. We may define, for all p ∈ (0 , 1], I p (Q) := 1 (0,p) (U (Q)). Then: For all integers k 0 define: and then define iteratively for all integers n 0, Π p,n (V k ) := {Q ∈ D k−n : Q ⊆ Π p,n−1 (V k ) and I p (Q) = 1} .
In this way, we see for example that Π p,0 (V k ) is a random set of cubes with sidelength 2 k which result from the first step of a certain branching process; see  Fractal percolation on V k [with parameter 0 < p 1] is the random set 3 Standard last-exit arguments, and/or maximum principle arguments, show that our "cap g " is the same capacity form as Lamperti's "C" [15] and Barlow and Taylor's "Cap G " [2]. This fact can be found implicitly in Bucy [4], and might even be older.
One can see that this is the usual construction of Mandelbrot's fractal percolation [17], scaled to take place in the cube V k . Namely, we may write V k as a disjoint union of 2 d elements of D k ; each of those elements is selected independently with probability p and rejected with probability 1 − p. We then write every one of the selected cubes as a disjoint union of 2 d elements of D k−1 ; each resulting sub-cube is kept/selected or discarded/deselected independently with respective probabilities p and 1 − p; and we continue.
Elementary branching-process theory implies that P{Π p,∞ (V k ) = ∅} > 0 if and only if p > 2 −d ; see also Figures 1,2,and 4. Presently, we are interested in performing fractal percolation in V k but we will stop the subdivisions after k + 1 steps. In other words, we are interested in Π p,k (V k ), which is a random, possibly empty, collection of side-one cubes in D 0 .
Now let us construct all of these fractal percolations on the same probability space so that: In words, we appeal to the preceding procedure in order to construct the Π p,k (V k )'s simultaneously for all k, using an independent collection of weights I p (Q)'s for each V k .
By macroscopic fractal percolation we mean the random set Of course, Π 0 = ∅ and Π 1 = R d . Starting from here, we often assume tacitly that p ∈ (0 , 1) in order to avoid the trivial cases p = 0 and p = 1. In any case, it is easy to deduce our next result.
Lemma 3.1. The following are valid: . . are independent random variables; and 3. Π p ∩ S k is distributed as Π p,k (V k ) ∩ S k for every integer k 0. In fact, we have the equality of events, We will not include the elementary proof.
If A, B ⊆ R d are both unbounded, then we say that A is a recurrent set for B provided that there exist infinitely many shells S k1 , S k2 , . . . such that A ∩ B ∩ S kn = ∅ for all n 1. We often write "A ∼ (R) B " in place of "A is a recurrent set for B." The relation "∼ (R) " is reflexive and symmetric for all pairs of unbounded sets.  The enumeration scheme. Every annulus is enumerated by its index; this indexing enduces a partial ordering of cubes in each shell. Thus, cubes in S 0 are enumerated before those in S 1 and so on. Percolation in the shell S k is the result of percolation in V k according to the description that follows eq. (3.2). Each V k is subdivided into four cubes [the four quadrants] which we enumerate lexicographically; each quadrant is canonically associated with a vector in {−1 , 1} 2 , depending on the sign of the various coordinates, and by lexicographic order of the quadrants we mean the order of these vectors. When d = 2, as is the case in the above schematic, . Each of those cubes is then divided into four cubes; those are again enumerated lexicographically, etc. This enumeration procedure is continued until we are left with cubes of size 1 which are now-for the purposes of the figures-enumerated lexicographically. The resulting scheme for enumerating size-one cubes yields an isomorphism between the percolation set and a certain random forest. That scheme is illustrated further in Figure 5.
A word of notational caution at this point: We will use F ∼ (R) Π p as shorthand for P{F ∼ (R) Π p } = 1. However, when we condition on a given configuration from this a.s. event, F ∼ (R) Π p reverts to its original meaning.
As we have noted already, the [microscopic] fractal percolation set Π p,∞ (V k ) is nonvoid if and only if p > 2 −d . The large-scale analogue of becoming nonvoid is to become unbounded. The following shows that the large-scale result takes a different form than its small-scale counterpart in the critical case p = 2 −d .
We will be interested only in Π p when it is unbounded; the preceding tells us that we want to consider only values of p ∈ [2 −d , 1]. The said condition on p will appear several times in the sequel for this very reason.
Proof. By the Borel-Cantelli lemma for independent events, the random set Π p is un- the probability that Π p ∩ S k is nonempty is at most the probability that a Galton-Watson branching process with mean branch rate 2 d p survives after k + 1 generations. We shall denote by Z k the number of descendants in the k-th generation. When p < 2 −d , the said Galton-Watson process is strictly subcritical. It is Thus, we conclude that Π p is a.s. bounded when p < 2 −d .
By the monotonicity of p → Π p , it remains to prove that Π 2 −d is a.s. unbounded. One can define fractal percolation on any dyadic cube Q ∈ D in much the same way as we defined it on V k . Next we note that if k 2, then we obtain the following by first selecting a cube Q ∈ D k in V k and then another cube in D k−1 in S k : where E k denotes the event that fractal percolation with p = 2 −d on a dyadic cube of side 2 k−1 does not become void in k − 1 steps. In other words, up to a constant multiplicative factor which does not depend on k, the probability P{Π 2 −d ∩ S k = ∅} is bounded below by the probability that a certain critical Galton-Watson does not become extinct in its first k generations. A well-known theorem of Kolmogorov [14] asserts that lim k→∞ kP{Z k > 0} = 2σ −2 for that critical Galton-Watson branching process, where σ 2 denotes the variance of the offspring distribution; see also Kesten et al [8] and Lyons et al [16]. This fact implies that for every > 0 there exists a positive integer k 0 such that which concludes the proof.
The following is the result of an elementary computation.
According to (3.4), 0 λ(x , y) k + 1 for all x, y ∈ S k . Both bounds can be achieved: λ(x , y) = 0 when the most recent common ancestor of x and y in the branching process is the root V k ; and λ(x , y) = k + 1 when y = x.
We will use the preceding in order to prove the following.
where log 2 denotes the usual base-2 logarithm. Remark 3.6. Theorem 3.5 can be recast as follows: That case can be decided by a more delicate capacity criterion. We will not pursue those refinements since we will not need them.

Proof. For every set
; see Lemma 6.1 of Barlow and Taylor [2]. Therefore, we may assume without loss of generality that F is a subset of Z d ; otherwise, we can replace F by F z everywhere throughout the remainder of the proof. From now on we consider only F ⊆ Z d .
Let k 0 be an arbitrary integer. We can find dyadic cubes Q 1 , . . . , Q m ⊂ S k such that: 1. For every 1 i m, the sidelength of Q i is 2 i 1 for some integer 0 Thus, we obtain For the inequality we have used the fact that Π p, The preceding holds for all p ∈ (0 , 1]. Suppose for the moment that log 2 (1/p) > Dim H (F ).
The non recurrence of F for Π p follows by the Borel-Cantelli lemma.
We have proved that if p ∈ (0 , 2 − Dim H (F ) ), then F is not recurrent for Π p . It now remains to show that From now on we choose and fix an arbitrary δ ∈ (0 , Dim H (F )). Note that by the definition of Dim H .
Next we briefly state Theorem 4.2 of Barlow and Taylor [2], in our notation, and adapt it the statements to our settings. This is the version we use in the sequence.
We use the above theorem in the following way: Pick an arbitrary F ⊂ Z d and define Apply the theorem to every F k in order to construct a (discrete) measure µ k , for every k, that is fully supported on F k and satisfies property (3.8). Since the µ k 's are supported on disjoint sets, they can be pasted together to collectively define a sigma-finite measureμ on F ⊂ Z d . It follows readily that there exists a real number c > 0, that depends only on the ambient dimension d, such that for all integers k 0 and all dyadic cubes Q ⊂ V k with sidelength 2 1. The first inequality follows from (3.8) and Remark 2.2 (the measureμ = c δ d µ where µ is the measure in Theorem 4.2. [2]).
Define a measure µ by normalizingμ on S k as follows: in order to conclude the following for all integers k 0: , uniformly for all dyadic cubes Q ⊂ S k with sidelength 2 1. [This is valid even when N δ (F , S k ) = 0, provided that we define 1/0 := ∞.]  For a given x, define C j (x) := {y ∈ S k : λ(x , y) j}. Because µ(S k ) = 1, Therefore, there exists a dyadic cube Q ∈ D k+1−j such that C j (x) ⊂ Q, whence by property (ii) of the measure µ, This, in turn, implies that For the last inequality we have used the hypothesis of (3.6). The Paley-Zygmund inequal- . Therefore, our bounds for the two first moments of ζ k and equation (3.9) together lead us to The independence half of the Borel-Cantelli lemma implies (3.6).
Let us close this section with a quick application of Theorem 3.5. Proof. The proof uses the replica argument of Peres [20] without need for essential changes. More specifically, let Π q denote a fractal percolation set with parameter q ∈ (0 , 1] such that Π p and Π q are independent. Because Π p ∩Π q has the same distribution as Π pq , it follows that P{Π p ∼ (R) Π q } = P{Π pq is unbounded}.
Thus, by Lemma 3.3, we have We may first condition on Π p and then appeal to Theorem 3.5 in order to see that Dim H (Π p ) = − log 2 (q c ), where q c is the critical constant q ∈ (0 , 1] such that pq2 d 1. Because q c = p −1 2 −d the corollary follows. Remark 3.9. By Theorem 3.5-see also Remark 3.6-and thanks to Corollary 3.8, we We can now deduce Barlow The preceding is an immediate consequence of Proposition 8.2 of [2], which is a variation of Lamperti's test [15]; the general form of this sort of Lamperti's test is in fact Corollary 2.4, which we will prove in due time. We apply the preceding observation conditionally, with F := Π p , where p ∈ (2 −d , 1] is a fixed parameter. By Corollary 3.8, Dim H (Π p ) = d + log 2 p a.s.. Therefore, By the Hewitt-Savage 0-1 law, Dim H (R X ) is P a -a.s. a constant. Choose p > 2 −α and assume to the contrary that Restrict the probability space of the random walk to the full-P a probability event {Dim H (R X ) < − log 2 p} and fix a realization F = R X . Theorem 3.5 ensures that P{F ∼ (R) Π p } = 1 for almost all realizations of the random walk. This gives the desired contradiction since P a {R X ∼ (R) Π p } = 1 for P-a.e. realization of Π p , while − log 2 p a.s. as long as p > 2 −α . An analogous argument shows that Dim H (R X ) − log 2 p whenever p < 2 −α . Therefore, we conclude that Dim H (R X ) = − log 2 (p c ) P a -a.s. where p c = 2 −α .
The preceding describes a bijection between the points in Z d ∩ S k and a certain collection of (k + 2)-chains of dyadic cubes. We can now use this bijection in order to build a directed-tree representation of Z d ∩ S k : The vertices of the tree are comprised of all dyadic cubes Q ∈ D whose sidelength is 1. For the [directed] edges of our tree, we draw an arrow from a vertex Q to a vertex Q if and only if there exists an integer i = 0, . . . , k and a point x ∈ Z d such that Q = Q i (x) and Q = Q i+1 (x). The resulting graph is denoted by T k .
It is easy to observe the following properties of T k : 1. T k is a finite rooted tree, the root of T k being V k ; 2. Every ray in T k has depth k + 1; 3. There is a canonical bijection from the rays of T k to Z d ∩ S k . The question marks signify that we do not have information about that branch of the process if we are allowed to look only at the percolation cluster; they correspond to lattice squares outside the shell S k .
Since the directed tree T k is finite for every k 0, we can isometrically embed it in R 2 so that the vertices of T k that have the maximal depth lie on the real axis.
Of course, there are infinitely-many such possible isometric embeddings; we will choose and fix one [it will not matter which one]. In this way, we can think of every T k as a finite rooted tree, drawn in R 2 , whose vertices of maximal depth lie on the real axis and whose root lies k + 1 units above the real axis. Because every x ∈ Z d ∩ S k has EJP 23 (2018), paper 83. been coded by the rays of T k , and since those rays can in turn be coded by their last vertex [these are vertices of maximal depth], thus we obtain a bijection π k that maps each point x ∈ Z d ∩ S k to a certain point π k (x) on the real axis of R 2 . Note that, in this way, {π k (x)} x∈Z d ∩S k can be identified with a finite collection of points on the real line.
The collection {T k } ∞ k=0 is a forest representation of Z d . We use this representation in order to impose an order relation ≺ on Z d as follows: 1. If x ∈ Z d ∩ S k and y ∈ Z d ∩ S for two integers k, l 0 such that k < , then we declare x ≺ y; 2. If x, y ∈ Z d ∩ S k for the same integer k 0, and π k (x) π k (y), then we declare x ≺ y. It can be checked, using only first principles, that ≺ is in fact a bona fide total order on Z d . We might sometimes also write y x in place of x ≺ y.
If we identify x, y ∈ Z d ∩ S k with 2 rays of the tree T k , viewed as a tree drawn in R d as was described earlier, then x ≺ y iff the ray for x lies on, or to the left of, the ray for y. And if x ∈ S k and y ∈ S for k < , then our definition of x ≺ y stems from the fact that we would like to draw the tree T k to the left of the tree T , as we embed the forest T 0 , T 1 , . . ., tree by tree, isometrically in R 2 .

Martin capacity of fractal percolation
Now we return to Markov chains. Throughout this section, let X := {X n } ∞ n=0 denote a transient Markov chain on Z d . This chain is constructed in the usual way: We have a probability space (A , A , P ) together with a family {P a } a∈Z d of probability measures such that under P a , the Markov chain begins at X 0 = a for every a ∈ Z d . By E a we mean the corresponding expectation operator for P a for all a ∈ Z d [E a (f ) := f dP a ].
We assume that the Markov chain is independent of the fractal percolations. The two processes are jointly constructed on (A × Ω , A × F, P × P). We write P a := P a × P and E a the corresponding expectation operator [E a (f ) := f dP a ].
As is well known, the transience of X is equivalent to seemingly-simpler condition that g(x , x) < ∞ for all x ∈ Z d . This is because g(a , x) g(x , x) for all x ∈ Z d ; in fact, one can apply the strong Markov property to the first hitting time of x ∈ Z d in order to see that g(a , x) = g(x , x) · P a {X n = x for some n 0} for every x ∈ Z d .
We define an equivalence relation on Z d as follows: For all x, y ∈ Z d we write "x ↔ y" when there exists an integer k 0 such that x and y are both in the same shell S k . Symbol x ↔ y denotes that x and y are in different shells.
If µ is a probability measure on Z d , then we define two "energy forms" for µ. The first form is defined, for every fixed a ∈ Z d , as The set function c 1 is the same Martin capacity that appeared earlier in (2.7). It might help to observe that the Martin p-capacity satisfies the following monotonicity property: The main result of this section can be stated as follows.
where c p is defined by (5.2).
Theorem 5.1 implies the following.

Corollary 5.2.
If F ⊆ Z d is recurrent for X P a -a.s. and a := X 0 ∈ Z d and F are non-random, then According to Theorem 5.1, It follows from this and elementary properties of probabilities that This in turn shows that We may apply Lemma 3.2, conditionally on R X , in order to deduce from the preceding that R X ∩ F ∼ (R) Π p a.s. [P a ]. In particular, we apply Theorem 3.5, once again conditionally on R X , in order to see that Dim H (R X ∩ F ) − log 2 p c (F ; a) P a -a.s.

(5.5)
For the other bound, it is enough to consider the case that p c ∈ (2 −d , 1] since p c = 2 −d yields a trivial bound. Thus, we can consider instead a number p ∈ [2 −d , 1) such that inf c p (F ∩ G ; a) = 0, where once again the infimum is over all cofinite sets G ⊂ Z d . It is easy to deduce from this choice of p and Theorem 5.1 that Since the random walk X is transient, and because R X ∩ F is a.s. recurrent, elementary properties of probabilities imply that P a {R X ∩ F ∼ (R) Π p } = 0. Therefore, we may apply Theorem 3.5, one more time conditionally on R X , in order to see that Dim H (R X ∩ F ) − log 2 p. Optimize over our choice of p to see that The corollary follows.
Proof of Corollary 5.3. Let Π q denote an independent copy of Π q and denote the corresponding (independent of P a ) measure by P with corresponding expectation operator E . Theorem 5.1 ensures that Therefore we integrate [P ] in order to see that For the other bound, we recall that Π p ∩ Π q has the same law [P × P ] as Π pq does [P].
Therefore, Theorem 5.1 implies that   (5.9) as N → ∞. This is what we aim to prove.
By the strong Markov property, Let us observe that, thanks to (2.8), there exists a finite constant K > 1 such that g(x , y) Kg(a , y) whenever x ∈ S n and y ∈ S m for integers m > n K such that m n + K. Thus, it follows readily from the definition (5.2) of c 1 that uniformly for all integers k, l K. In accord with (5.8), Similarly, we can appeal to (2.8) in order to find a finite constant K > 1 such that P a {T k+l < T k < ∞} 256K P a {T k+l < ∞}P a {T k < ∞}, (5.11) as long as k, l K . Let K 0 := 256 max(K , K ). Because Eq. (5.10) and eq. (5.11) together imply that The corollary follows immediately.
Let us mention a final corollary of Theorem 5.1. That corollary presents a more tractable formula for Dim H (R X ∩ F ), valid under the Lamperti-type condition (2.8).
If, in addition, (2.8) holds, then Remark 5.5. If X is a random walk that satisfies the conditions of Proposition 1.1 and starts at 0, then our previous comments in Example 2.5 imply that Proof of Corollary 5.4. First we prove the upper bound on Dim H (R X ∩ F ).
If Dim H (R X ∩ F ) > − log 2 p for some p ∈ (2 −d , 1], then Theorem 3.5 ensures that X ∼ (R) Π p ; see especially Remark 3.6. This, the easy half of the Borel-Cantelli lemma, and Theorem 5.1 together imply that ∞ k=0 c p (F ∩ S k ; a) = ∞. Optimize over p ∈ (2 −d , 1] in order to deduce that Dim H (R X ∩ F ) − log 2 p c (F ; a) P a -a.s. In the reverse direction we assume that (2.8) holds, and strive to show that P a -a.s.

(5.13)
There is nothing to prove if p c (F ; a) = 1. Therefore, we assume without loss of generality that 2 −d p c (F ; a) < 1.
According to Theorem 5.1, and thanks to the definition of the critical probability (5.14) Next we verify that there exists a uniform positive constant A so that By the Markov property of X and the particular construction of Π p , where C := 32768. We now apply an argument very similar to the one used to produce (5.9) in order to see that there exists an integer K * > 1 such that as long as k j + K * . In this way we find that Therefore, Theorem 5.1 shows that E a (τ 2 N ) 4CK * [E a τ N ] 2 + 2K * E a τ N . Because of the 0-1 law [see Lemma 3.2], this and (5.14) together imply that τ N → ∞ a.s. [P a ] as N → ∞. This is another way to state that R X ∩F ∼ (R) Π p a.s. [P a ]. Theorem 3.5-see, in particular, Remark 3.6-then implies that Dim H (R X ∩ F ) − log 2 p P a -a.s. Since p ∈ (p c (F ; a) , 1] were arbitrary, the lower bound (5.13) follows.
We conclude this section with a proof of Theorem 5.1.
Proof of Theorem 5.1. Because P a {X n ∈ Π p ∩ F for some n 0} is equal to we can assume without loss of generality that F is a finite set.
The first inequality of the proposition follows readily by adapting the second-moment argument of Benjamini et al [3]. The few details follow.
For every x = (x 1 , . . . , x d ) ∈ Z d , there exists a unique positive integer k such that x ∈ S k . Let ∆(x) := k + 1 for this pairing of x ∈ Z d and k 1. Then we define, for all probability measures µ ∈ M 1 (F ), a nonnegative random variable where µ(w) := µ({w}) for every w ∈ Z d . The preceding display contains an almost surely well-defined sum because the summands are non negative and µ(X n )/g(a , X n ) 1 a.s.
[P a ] for all n 0. We can therefore rearrange the sum and write Similarly, we compute E a (J 2 µ ) 2 x,y∈Z d m n 0 P a {X n = x , X m = y} g(a , x)g(a , y) P {x, y ∈ Π p } p ∆(x)+∆(y) µ(x)µ(y) = 2I(µ ; a) + 2I p (µ ; a). If J µ > 0 for some µ ∈ M 1 (F ), then certainly X n ∈ Π p ∩ F for some n 0. Therefore, the Paley-Zygmund inequality implies that for every µ ∈ M 1 (F ), The left-most quantity does not depend on µ ∈ M 1 (F ); therefore, we may optimize the right-most quantity in (5.18) over all probability measures µ ∈ M 1 (F ) in order to see that P a {X n ∈ Π p ∩ F for some n 0} 1 2 c p (F ; a). This is the desired lower bound on the hitting probability of the theorem. Next we verify the complementary probability, still assuming without loss of generality that F is finite; that is, F ⊆ V k for a nonnegative integer k that is still held fixed throughout. Without loss of generality, we may also assume that P a {X m ∈ Π p ∩ F for some m 0} > 0. (5.19) Otherwise, there is nothing to prove. In order to establish the more interesting second inequality of the theorem we will need to introduce some notation. Let X n denote the sigma-algebra generated by X 0 , . . . , X n for all n 0.
Recall that, because of our forest representation of Z d , we identify every point ρ ∈ S k ∩ Z d with a ray in a finite tree T k , which was described in §4. Recall also that T k has been embedded in R 2 so that its deepest vertices lie on the real axis of R 2 . In this way, we can identify every point ρ ∈ S k ∩ Z d with a point, which we continue to write as ρ, on the real axis.
For every ρ ∈ V k ∩ Z d , let P ρ denote the sigma-algebra generated by the fractalpercolation weights I p (Q 0 (y)), . . . , I p (Q ∆(y)+1 (y)) for all y ∈ Z d ∩ V k such that y ≺ ρ. Similarly, let F ρ the sigma-algebra generated by all of the fractal-percolation weights I p (Q 0 (y)), . . . , I p (Q ∆(y)+1 (y)), where y ∈ Z d ∩ V k satisfies y ρ. If we think of ρ as a maximum-depth vertex of T k and the latter is embedded in R 2 , as was mentioned earlier, then we can think of P ρ as the information, on the fractal percolation, on T k that lies to the left of ρ [including ρ]; this is the "Past" of ρ. Similarly, we may think of F ρ as the information to the right of ρ; this is the "F uture" of ρ.
Next we define two "2-parameter martingales," Λ and V as follows: for all m 0 and ρ ∈ V k ∩ Z d . Because our random walk is independent of the fractal percolation, we may write the following after we appeal to independence: The Markov property implies the a.s.-inequality, We stress, once again, that the ratio of the Green's functions are well defined P a -a.s. Similarly, Therefore, with probability one, There are only a countable number of such pairs (m , ρ). Therefore, the previous lower bound on Λ m,ρ holds, off a single null set, simultaneously for all integers m 0 and integral points ρ ∈ V k ∩ Z d .
In order to simplify the typesetting, let us write We might note that, with probability one, (5.20) simultaneously for all integers m 0.
Now we apply an idea that is, in a different form due to Fitzsimmons and Salisbury EJP 23 (2018), paper 83.
The preceding is valid P a -a.s. for any probability measure µ on F . We apply it using the following particular choice: . (5.22) For this particular choice of µ ∈ M 1 (F ) we obtain the following: P a -a.s., thanks to the Cauchy-Schwarz inequality and out special choice of µ in (5.22).
[The conditional expectation is well defined since P a (G) > 0.] Recall that the forest representation of Z d in §4 identifies ρ ∈ V k ∩ Z d with a certain finite subset of the real line. With this in mind, we see that {Λ m,ρ } m 0,ρ∈V k ∩Z d is a 2-parameter martingale under the probability measure P a , in the sense of Cairoli [5], with respect to the 2-parameter filtration Because X and F are independent, the 2-parameter filtration G satisfies the commutation hypothesis (F4) of Cairoli [5]; see Khoshnevisan where log 2 is the usual logarithm in base two.
In analogy with the usual [small-scale] theory of the dimensions, Dim H (A) Dim M (A) for all sets A ⊆ Z d ; see Barlow and Taylor [2]. The Minkowski dimension is perhaps the most commonly used notion of large-scale dimension, in some form or another, in part because it is easy to understand and in many cases compute.
In the context of random walks, we have the following elegant formula for the Minkowski dimension of the range of a transient random walk on Z d .
The proof hinges on the analysis of the 0-potential measure, thanks to a combination of Tonelli's theorem and (5.1).
The following simple argument implies the first half of Theorem 6.1.
Proof of Theorem 6.1: Upper bound. We first prove that, with probability one, The more involved converse bound will be proved later.
Chebyshev's inequality and (6.3) together show that for all real numbers γ > 0 and integers k 1, Because g(x , y) g(0 , 0) < ∞ for all x, y ∈ Z d -see (5.1)-there exists a finite constant b such that U (S k ) b2 kd for all k 1. Therefore, the sum over k of the right-hand side of (6.5) is always finite when γ > d. If γ ∈ (0 , d) is such that the right-hand side of (6.5) forms a summable series [indexed by k], then so does the left-hand side. The Borel-Cantelli lemma shows that for any such value of γ the random variable L γ defined by is a.s. finite. In particular, This and (6.7) together imply (6.4).
For the remaining, more challenging, direction of Theorem 6.1 we need to know that the measure U is volume-doubling. That is the essence of the following result.
Next we rewrite γ c -see (6.1)-in a slightly different form. We will be ready to complete the proof of Theorem 6.1 once that task is done.
Proof. If x ∈ S k , then x d 1/2 2 k . Therefore, Therefore, we can infer from (6.8) that We apply (6.10) to rewrite γ c once more time: If γ > γ c , then U (S k ) = o(2 kγ ) as k → ∞. If on the other hand γ ∈ (0 , γ c ), then we can argue by contraposition to see that, for every fixed > 0, U (S k ) > 2 k(γ− ) for infinitely-many integers k. This means that γ c = lim sup n→∞ n −1 log 2 U (S n ). On one hand, (6.11) implies that γ c γ c . If, on the other hand, ϑ > γ c is an arbitrary finite number, then there exists a finite constant L ϑ such that U (S k ) L ϑ 2 kϑ for all integers k 1. In particular, for all n 1, whence follows that U (V n ) = O(2 nϑ ) as n → ∞. Since this is true for all ϑ > γ c , it follows that γ c γ c , as was promised.
Proof of Theorem 6.1: Lower bound. It remains to prove that Dim M (R X ) γ c a.s., (6.12) where γ c was defined in (6.1). If γ c = 0, then we are done. Therefore, from now on we assume without loss of generality that γ c > 0. thanks to the strong Markov property. If x ∈ V n and n 1 are held fixed, then y∈Vn\{x} P x {τ (y) < ∞} = U (V n − x) g(0 , 0) U (V n+1 ) g(0 , 0) , since V n − x ⊂ V n − V n = V n+1 . Therefore, (6.14) implies that The last part holds since (6.13) implies that E[card(R X ∩ V n )] 1 for infinitely-many integers n > 1. The preceding displayed inequality and Proposition 6.3 together imply that Dim M (R X ) γ c , with probability at least (d) > 0. Since Dim M (R X ) = Dim M (R X ∩ V c N ) for all N 1, an application of the Hewitt-Savage 0-1 law shows the desired result that Dim M (R X ) γ c almost surely.

Concluding remarks and open problems
Corollary 5.2 succeeds in yielding a formula for Dim H (R X ∩ F ) for every recurrent set F ⊂ Z d , though it is difficult to work with that formula. We do not expect a simple formula for Dim H (R X ∩ F ) when F is a general recurrent set in Z d . In fact, it is not even easy to decide whether or not a general set F is recurrent, as we have seen already. However, one can hope for simpler descriptions of Dim H (R X ∩ F ) when F = Z d . In this section we conclude with a series of remarks, problems, and conjectures that have these comments in mind.
Page 28/31 http://www.imstat.org/ejp/ whether or not Conjecture 7.1 is true, Dim H (R X ) always lies in the non-random interval [lim inf n→∞ n −1 log U (V n ) , lim sup n→∞ n −1 log U (V n )]. The extrema of this interval are typically not hard to compute; therefore, we at least will have easy-to-compute bounds for Dim H (R X ).
Let us state a third conjecture that is motivated also by Conjecture 7.1.
Choose and fix an arbitrary integer N 1, and define X (1) , . . . , X (N ) to be N independent copies of a symmetric, transient random walk X on Z d whose Green's function satisfies the Barlow-Taylor condition (1.3) for some α ∈ (0 , 2]. We can define an N -parameter additive random walk X := {X(n)} n∈Z N + as follows [9,Ch. 4]: For every n := (n 1 , . . . , n N ) ∈ Z N + , X(n) := X (1) n1 + · · · + X (N ) n N . Let R X := ∪ n∈Z N + {X(n)} denote the range of the random field X. Proposition 1.1 implies that Conjecture 7.3 is correct if N were replaced by 1; the case N > 1 has eluded our many attempts at solving this problem.
It is possible to adapt the arguments of [13] in order to derive Conjecture 7.1 from Conjecture 7.3. We skip the details of that argument. Instead, we conclude with two problems about the "continuous version" of Corollary 5.2, which we recall, contained our Hausdorff dimension formula for the range of a walk. Problem 7.4. Let {y(t)} t 0 be a transient, but otherwise general, Lévy process on R d whose characteristic exponent is Ψ, normalized as exp{iz · y(t)} = exp{−tΨ(z)} for all z ∈ R d and t 0, to be concrete. Is there a formula for the a.s.-constant quantity Dim H (y(R + )) that is solely in terms of Ψ?
Before we state our last question let us define the upper Minkowski dimension of a set A ⊆ R d as follows: Define A to be the union of all dyadic cubes Q ∈ D 0 of sidelength one that intersect A.