The existence phase transition for two Poisson random fractal models

In this paper we study the existence phase transition of the random fractal ball model and the random fractal box model. We show that both of these are in the empty phase at the critical point of this phase transition.


Introduction
In order to better explain the rest of the paper, we shall start by a rather informal description of the general setup (see for example [1] for details). Let M be the set of bounded subsets of R d with non-empty interior, and let M be some (suitable) σ-algebra on M. We consider a measure µ on (M, M) which is scale invariant in the following sense. If A ∈ M is such that µ(A) < ∞, then µ(A s ) = µ(A) where 0 < s < ∞ and A s := {K : K/s ∈ A}.
We will also assume that µ is translation invariant in that µ(x + A) = µ(A) for every A ∈ M. Here of course, x + A = {L ⊂ R d : L = x + K for some K ∈ A}.
In order to define a model which will exhibit a non-trivial behaviour, it is often necessary to restrict µ to sets of diameter smaller than some cutoff. This is indeed what we do in this paper (see also the discussion in Section 2). For such measures, the property µ(A s ) = µ(A) will still hold, but only if neither A nor A s contains sets with diameter larger than the cutoff. We shall call such a measure semi scale invariant. In the rest of this introduction, any measure µ we refer to will be semi scale invariant.
Using λµ where 0 < λ < ∞ as the intensity measure, one can define a Poisson process Φ λ (µ) on M. Thus constructed, Φ λ (µ) is a semi scale and translation invariant random collection of bounded sets of R d . This setup contains many interesting examples such as the Brownian loop soup introduced in [4], and the semi scale invariant Poisson Boolean model studied for instance in [2] (see also the references therein). Throughout, this latter model will be referred to simply as the fractal ball model, and we shall give an exact definition of it in Section 2. In this fractal ball model, the measure µ above is supported on the set of open balls of R d . Of course, one could also consider a process of closed balls, or indeed a mix of open and closed balls. As we will see, the results of this paper are also valid for these cases, see further the remark after the statement of Theorem 1.1.
Throughout this paper, we will let and we will usually write C(λ) or simply C. Thus, with µ as above, C is a semi scale invariant random fractal and we will be concerned by various properties of C(λ) as λ varies. It is useful to observe that by using a standard coupling, C(λ) is decreasing in λ. Random fractal models exhibits several phase transitions (see for instance [3]). However, the perhaps two most natural are the existence and the connectivity phase transitions as we now explain. Define Therefore, for λ > λ e , C(λ) is almost surely empty, and we say that it is in the empty phase. If instead λ < λ e , then P(C(λ) = ∅) = 1. We say that λ e is the critical point of the existence phase transition. Analogously, we can define λ c := sup{λ > 0 : P(C(λ) contains connected components larger than one point) = 1}.
Of course, whenever such phase transitions occur, it is natural and interesting to ask what happens at the critical points. In [1] it was proven in full generality that P(C(λ c ) contains connected components larger than one point) = 1, so that at λ c the fractal is in the connected phase. Thus, this phase transition is very well understood.
The existence phase transition is much less understood. Hitherto, the only exact results appear to be in dimension 1. Indeed, in [5], exact conditions for when random intervals cover a line were established. However, there has been some progress (see [2]) on the case of the fractal ball model in d ≥ 2, see Section 2 for a precise statement of these results.
In analogy with how the fractal ball model is defined, we can also define the fractal box model (again see Section 2) for which the measure µ is supported on boxes of the form (a, b) d for a < b. In this case, Φ λ (µ) is then a random semi scale invariant collection of boxes in R d . Whenever we need to distinguish between the ball and the box model, we shall write C ball and C box etc.
Let v d be the volume of the unit ball in R d . The main result of this paper is the following.
Remarks: The fact that λ ball e = d/v d is easily deduced from results in [2], while we determine λ box e by a straightforward second moment argument. Thus, the main contribution of this paper is to determine what happens at the critical point of these phase transitions.
If we choose to consider closed balls (boxes) in place of open, then of course we would have that C closed ⊂ C open (using obvious notation). However, when determining λ e , one sees that the argument does not depend on whether we use open or closed sets so that λ e (C closed ) = λ e (C open ). It then follows trivially that Theorem 1.1 holds also for the case of closed balls (boxes).
The result does not depend on the specific value of the cutoff (as is clear from the proofs). However, it requires some cutoff.
The rest of the paper is organized as follows. In Section 2 we give precise definitions of our models and also provide some further background. In Section 3, we will prove Theorem 1.1.

Models
We start by defining the fractal ball model, although we will later reuse much of the notation for the box model.
Let ν be a locally finite measure on (0, 1], and let µ = dx × ν (where dx denotes d-dimensional Lebesgue measure) denote the resulting product measure on R d × (0, 1]. Then, we let Φ λ (µ) be a Poisson process on R d × (0, 1] using λµ as the intensity measure. This definition might seem to clash with Φ λ (µ) defined in the introduction (which was a Poisson process on sets). However, this is easily resolved by associating the point (x, r) ∈ R d × (0, 1] with the open ball B(x, r) centered at x and with radius r. Thus, we might write (1.1) as B(x, r).
. We observe that if ν(dr) = r −d−1 dr, then we have that (with I(·) being an indicator function) and an analogous calculation shows that also µ( We observe that µ cannot be fully scale invariant since we have that µ(A ǫ ) = 0. This follows since A ǫ only contains sets with r ≥ 1. However, if we in the above replace ν bỹ ν(dr) = r −d−1 dr supported on (0, ∞), we would obtain a fully scale invariant measureμ. Thus, our measure µ is the restriction ofμ to sets with r ≤ 1, which is then our cutoff. In particular we have that µ(A) = µ(A s ) as long as neither A nor A s contains sets with r > 1. We note that it would perhaps be more proper to write ν(dr) = I(0 < r ≤ 1)r −d−1 dr.
However, we will allow ourselves to slightly abuse notation by writing ν(dr) = r −d−1 dr, and remembering that ν is supported on (0, 1].
It is certainly possible to consider other choices of ν, but in this paper we shall focus on the semi scale invariant case. However, we want to mention the following result from [2] which deals with other choices of ν.
so that also this model is semi scale invariant. Whenever convenient, we will write K ∈ Φ to mean either a ball or a box, depending on the context.

Proofs
We start this section by introducing some useful notation. First, let IfX ∈X n , we shall refer toX as a level n box. Note that the membersX ofX n are deterministic, closed boxes. These should not be confused with the open boxes X(x, r) that belong to the Poisson process Φ box λ . The interpretation of the following definitions differ depending on whether we are considering the ball model or the box model. However, we believe that this should not lead to any confusion. For these models, we let then either a ball or a box). Thus, C n ↓ C. For m > n, let so that C n m ∩ C n = C m , and C n m , C n are independent. For any integer n, let M n := {X ∈X n : ∃K ∈ Φ n :X ⊂ K}.
Thus, M n is the set of level n boxes which are not covered by a single set in the Poisson process Φ n . Then, let m n := {X ∈X n : ∃K ∈ Φ n :X ∩ K = ∅}, which is the set of level n boxes untouched by the Poisson process Φ n . We see that if X ∈ m n , then in factX ⊂ C n . Obviously, |m n | ≤ |M n | since an untouched box cannot be covered.
The following proposition is a part of Theorem 1.1.
Proposition 3.1. For the box model we have that λ e ≥ d.

Proof.
We start by noting that if m n = ∅ for infinitely many n ≥ 1, then C n ∩ [0, 1] d = ∅ for every n ≥ 1. Since C n ⊃ C n+1 for every n, and the sets C n ∩ [0, 1] d are compact, we must then have that We will prove that for λ < d, there exists c = c(λ) > 0 such that We shall proceed by proving (3.1) using a second moment argument. To that end, observe that by translation invariance, for anyX ∈X n , we conclude that e −λ2 d n −λ ≤ P(X ∈ m n ) ≤ n −λ . Next, let k = (k 1 , . . . , k d ) be such thatX 2 =X 1 +k/n, and define k max := max{|k 1 |, . . . , |k d |}.
We get that for k max ≥ 2, where the last inequality follows by using the calculations in (3.2) combined with the upper bound of (3.3). Therefore, ifX 1 =X 2 and k max ≥ 2, we have that by using (3.6) and (3.7), P(X 1 ,X 2 ∈ m n ) = exp (−λµ(R n 1 ∪ R n 2 )) = exp(−2λµ(R n 1 ) + λµ(R n 1 ∩ R n 2 )) (3.8) If however k max ≤ 1, then we simply use that Thus, by (3.4) and (3.8), Here, the first inequality uses that there are n d possible choices ofX 1 , and given the choice ofX 1 there are at most 3 d choices ofX 2 that are either the same as, or immediate neighbours to,X 1 . The remaining boxesX 2 have k max ≥ 2. We see that if λ < d, then there exists a C = C(λ) > 0 such that E[m 2 n ] ≤ Cn 2(d−λ) . Using (3.5) we conclude that as desired.
Our next lemma gives a useful consequence of C(λ) surviving, but first we need some more notation. Let D n = D n (C n ) be a minimal collection of boxes inX n such that Note that D n is not necessarily unique, as a point x ∈ C n sitting on the boundary between two boxesX 1 andX 2 can be covered by either one of them. If there is more than one way of choosing such a set D n , we pick one according to some predetermined rule. Let L n = |{X ∈X n :X ∈ D n }|. Remarks: The reason for proving Lemma 3.2 along a subsequence (2 n ) n≥1 , is that this will avoid unnecessary technical details. It is also all that we need in order to prove Theorem 1.1.
Proof. Let E 2 n := X ∈D 2 nX , and observe that by definition of D 2 n , we have that We have that for some α = α(λ) > 0, By using the FKG inequality for Poisson processes together with the semi scale invariance of the models, we conclude that Therefore, if there exists L < ∞ such that L 2 n ≤ L for infinitely many n, we can use Lévy's Borel-Cantelli lemma, to conclude that almost surely C ∩ [0, 1] d = ∅.
We can now prove our main result. Proof of Theorem 1.1. The fact that λ ball e = d/v d is an immediate consequence of Theorem 2.1 as explained in Section 2. Furthermore, Proposition 3.1 shows that λ box e ≥ d. Therefore, it remains to prove that λ box e ≤ d and that both the ball and the box models are in the empty phase at their respective critical points.
Obviously, ifX ∈ D n , thenX cannot be covered by a single set in the Poisson process Φ n . Therefore, L n ≤ |M n |. We now turn to the case of C ball . First we observe that for any K ∈ Φ, [0, 1/n] d ⊂ K iff the closed ballB( 1 2n (1, . . . , 1), √ d/(2n)) ⊂ K, simply because of the fact that the sets K ∈ Φ are balls. We then get that for some constant C = C(λ) < ∞, and n > where the last inequality follows as above. As for the box model, we obtain that for λ e = d/v d P(C ball (λ e ) ∩ [0, 1] d = ∅) = 0.