Translation invariant realizability problem on the $d-$dimensional lattice: an explicit construction

We consider a particular instance of the truncated realizability problem on the $d-$dimensional lattice. Namely, given two functions $\rho_1({\bf i})$ and $\rho_2({\bf i},{\bf j})$ non-negative and symmetric on $\mathbb{Z}^d$, we ask whether they are the first two correlation functions of a translation invariant point process. We provide an explicit construction of such a realizing process for any $d\geq 2$ when the radial distribution has a specific form. We also derive from this construction a lower bound for the maximal realizable density and compare it with the already known lower bounds.


Introduction
Let d be a positive integer.Given a point process P = {P i } i∈Z d on the d−dimensional lattice, P i ∈ {0, 1}, whose distribution is described by the probability measure µ, we define the first and second order correlation function as follows ρ 1 (i) := P i ρ 2 (i, j) := P i P j − ρ 1 (i)δ(i − j) , where i, j ∈ Z d , δ is the Dirac delta function and • denotes the expectation w.r.t.µ.
The truncated realizability problem addresses the inverse question.Namely, given two functions ρ 1 (i) and ρ 2 (i, j) non-negative and symmetric for all i, j ∈ Z d , does there exist a point process P for which these are the correspondent first and second order correlation functions?Clearly, the truncated realizability problem can be posed for any finite sequence of non-negative and symmetric functions (ρ k (i 1 , i 2 , . . ., i k )) n k=1 .When the question is asked for a given infinite sequence (ρ k ) k∈N , then the problem is addressed as full realizability problem (see e.g.[15,16] for a systematic study of the full realizability problem for point processes and [7,8] for recent developments).
In the following we will consider the important special case of translation invariant point processes, which actually contains all the essential difficulties of the problem.In this case the realizability problem asks if, for given ρ ∈ R + and g : Z d → R + symmetric, there exists a translation invariant point process such that its first two correlation functions are given by (1) ρ If such a process exists, then it is said to be realizing and the pair (ρ, g) is called realizable on Z d .Note that writing the second order correlation in this form is not an additional restriction beyond the assumption of translation invariance.The function g is known in classic fluid theory as radial distribution, [6].
The truncated realizability problem is in fact a longstanding problem in the classical theory of fluids (see e.g.[5,19,20]), but it has been investigated in many other contexts such as stochastic geometry [17], spatial statistics [3,23], spatial ecology [18] and neural spike trains [1,9], just to name a few.In particular, Stillinger, Torquato et al. developed fascinating applications in the study of heterogeneous materials and mesoscopic structures based on the solvability of the truncated realizability problem (see e.g.[4,21,22,24,25,26]).A structural investigation of this problem was recently started in [13], where the authors identify the realizability problem as a particular instance of the infinite-dimensional truncated moment problem (see [2,10,11,12,14] for further recent developments about the truncated realizability problem for point processes).As far as we know, the only earlier reference about the truncated infinite-dimensional moment problem is [28].
In this paper, we will show how to explicitly construct a point process on the d−dimensional lattice with d ≥ 2 such that, for given α ≥ 0, (1) holds for certain values of ρ and for g = g (α) defined as follows: (2) Explicit constructions of point processes realizing this lattice problem in the case d = 1 were provided in [12, Appendix 1].Such a problem has been extensively studied for the case α = 0 by Stillinger and Torquato in [21] (see also [4,25]).The function g (0) describes a model with on-site and nearest neighbour exclusion and with no correlation for pairs of sites separated by two or more lattice spacings.
From [12, Section 1], we know that for a fixed α the set of realizable densities ρ is an interval [0, ρα (d)] with 0 < ρα (d) ≤ 1.Moreover, in [12] the authors discuss: (i) certain general methods which, when applied to (2), yield lower bounds for ρα (d) in any dimension d. (ii) concrete upper and lower bounds for ρα (1).In particular, the lower bounds improve those obtained from the general methods (i).
Our d−dimensional construction combined with the one-dimensional lower bounds (ii) provides a lower bound for ρα (d) for any d ≥ 2 and any α ≥ 0. We will briefly compare this with the lower bound obtained from the general methods (i).We also follow techniques from [12] to get an upper bound for ρα (d).

An explicit realizing translation invariant point process on Z d
In the following, we explicitly construct a point process P = {P i } i∈Z d on the d−dimensional lattice with d ≥ 2 such that, for given α ≥ 0, the following hold for certain values of ρ (depending on α and on d): (3) P i = ρ and (4) that is, the radial distribution is given by (2).

Construction in dimension 2.
In order to build such a process on Z 2 we start from a realizing one-dimensional process achieving density γ.Namely, given α ≥ 0, we consider a point process {A i } i∈Z , A i ∈ {0, 1}, defined on the one-dimensional lattice and such that for some γ > 0 we have and We denote a process of this kind by BP γ that stays for basic process with density γ.
As pointed out in the introduction, there exists a good number of constructions of realizing processes in the one-dimensional case, see e.g.[12,Appendix 1].The results in the one-dimensional case relevant to our investigation (in particular the range where γ can vary) are recalled in Section 3, which is devoted to the discussion of the maximal realizable density in any dimension.
Let us define two processes on Z 2 as follows.For a fixed i 1 ∈ Z, the process and B (1) j1,j2 j2∈Z are independent.In particular, we have ( 5) B (1) In other words, the process B (1) can be seen as a sequence of vertical BP γ's independent one from each other (see Figure 1 for an example).

Bounds for the maximal realizable density
In this section, we will discuss the problem of estimating the maximal realizable density ρα (d).In particular, we will show a general upper bound for any d ≥ 1 using the technique introduced in [12] for d = 1.As for the lower bound, we will recall the results in [12,Appendix 1] for the one-dimensional case and combine them with the explicit construction proposed in Section 2 to produce a lower bound for ρα (d) for any d ≥ 2. We will compare this with the lower bound obtained by applying the general methods of [12] to the case when the radial distribution is given by (2).

Upper bounds for ρα (d).
For d ≥ 1 and α ≥ 0, the covariance matrix associated to a given pair (ρ, g (α) ) realizable on Z d must be positive semidefinite.This is equivalent to the nonnegativity of the corresponding infinite volume structure function Ŝ on R d (for more details see e.g.[12, Section 2]): This leads to an explicit upper bound for the maximal realizable density ρα (d).In fact, it is easy to see that for any k : Then, using the non-negativity of Ŝ on R d , we get that where The best upper bound is then obtained for the points of R d where f α attains the maximum.Hence, we have that ρα (d) ≤ 1 max By computing the maximum of f α over R d , we get our upper bound As mentioned above, this technique was employed in [12,Appendix 1] to get R F (α, 1).Furthermore, the authors provided another upper bound R Y (α, 1) in the one-dimensional case by using the Yamada condition (see [27]).Note that
Let us summarize in one formula the lower bounds coming from the two main constructions considered in [12, Appendix 1]: In [2] a further explicit construction is provided for the case α = 0, which slightly improves this lower bound to ρ0 (1) > 0.265.In the same work also the upper bound is improved to ρ0 (1) < (326 − √ 3115)/822 ≈ 0.3287.However, it remains an open problem to reduce the gap between lower and upper bounds for ρα (1) for any α ≥ 0.
Exactly as in the one-dimensional case, also for d ≥ 2, one can try to obtain better lower bounds than (9) by using explicit constructions.In the following, we will use the construction we proposed in Section 2 combined with the onedimensional lower bound (10) to compute a new lower bound for ρα (d), which we will briefly compare with (9).
If we apply the construction given for d ≥ 2 in Section 2 starting with a basic process with density ρα (1), then we get a point process on Z d which realizes the pair (ρ α (1)) d , g (α) for any α ≥ 0. This explicit construction guarantees that for any α ≥ 0, ρα (d) ≥ (ρ α (1)) d .