Isomorphisms between determinantal point processes with translation invariant kernels and Poisson point processes

We prove the Bernoulli property for determinantal point processes on $ \mathbb{R}^d $ with translation-invariant kernels. For the determinantal point processes on $ \mathbb{Z}^d $ with translation-invariant kernels, the Bernoulli property was proved by Lyons and Steif and Shirai and Takahashi. As its continuum version, we prove an isomorphism between the translation-invariant determinantal point processes on $ \mathbb{R}^d $ with translation-invariant kernels and homogeneous Poisson point processes. For this purpose, we also prove the Bernoulli property for the tree representations of the determinantal point processes.


Introduction and the main result
We consider an isomorphism problem of measure-preserving dynamical systems among translation-invariant point processes on R d such as the homogeneous Poisson point processes and the determinantal point processes with translation-invariant kernel functions.
The homogeneous Poisson point process is a point process in which numbers of particles on disjoint subsets obey independently Poisson distributions. It is parameterized using intensity r > 0. From the general theory of Ornstein and Weiss [9], homogeneous Poisson point processes are isomorphic to each other regardless of the value of r.
The determinantal point process is a point process for which the determinants of its kernel function give its correlation functions. It describes a repulsive particle system and appears in various mathematical systems such as uniform spanning trees, the zeros S. Osada of a hyperbolic Gaussian analytic function with a Bergman kernel, and the eigenvalue distribution of random matrices.
These two classes of point processes have different properties in correlations among particles. For example, determinantal point processes have negative associations [4]. The sine point process is a typical example of a translation-invariant determinantal point process that has number rigidity [1]. In contrast, Poisson point processes do not have this property because the particles are regionally independent. Nevertheless, we prove that they are isomorphic to each other.
We start by recalling the isomorphism theory. An automorphism S of a probability space ( , F, P) is a bi-measurable bijection such that P • S −1 = P. Let S G = {S g : g ∈ G} be a group of automorphisms of ( , F, P) parametrized by a group G. A measure-preserving dynamical system of G-action is the quadruple ( , F, P, S G ). We call ( , F, P, S G ) the G-action system for short.
Let ( , F, P, S G ) and ( , F , P , S G ) be G-action systems. A factor map is a measurable map φ : → such that for each g ∈ G and a.e. x ∈ .
In this case, we call ( , F , P , S G ) the φ-factor of ( , F, P, S G ) or simply a factor of ( , F, P, S G ). An isomorphism is a bi-measurable bijection φ : → such that both φ and φ −1 are factor maps. If there exists an isomorphism φ : → , then ( , F, P, S G ) and ( , F , P , S G ) are said to be isomorphic.
Let ( , F, P, S G ) be a G-action system with a measurable map φ from ( , F) to ( , F ).
We also call the G-action system ( , F φ , P φ , S φ G ) the φ-factor of ( , F, P, S G ). A typical system with a discrete group action is a Bernoulli shift. A G-action Bernoulli shift is a system formed from the direct product of a probability space over G and the canonical shift. Ornstein [6,7] proved that the Z-action Bernoulli shifts with equal entropy are isomorphic to each other. We call a system ( , F, P, S G ) Bernoulli if ( , F, P, S G ) is isomorphic to a Bernoulli shift. Ornstein and Weiss [9] extended the isomorphism theory to amenable group actions. As a consequence of the general theory, all the homogeneous Poisson point processes on R d are isomorphic to each other regardless of their intensity. We is isomorphic to a homogeneous Poisson point process. We also refer to Kalikow and Weiss [2] for d = 1, who constructed an explicit isomorphism between the time-one map of a homogeneous Poisson point process and a Bernoulli shift with infinite entropy. In this paper, we do not consider to construct explicit isomorphisms.
Let X be a locally compact Hausdorff space with countable basis. We denote by Conf(X) the set of all non-negative integer-valued Radon measures on X. We equip Conf(X) with the vague topology, under which Conf(X) is a Polish space. We call a Borel probability measure μ on Conf(X) a point process on X. We say that μ is simple when ξ({x}) ∈ {0, 1} for each x ∈ X for a.e. ξ ∈ Conf(X).
Let μ be a point process on X. Throughout this paper, we write the completion of μ by the same symbol. We also write (Conf(X), μ, T G ) as the G-action system made of the completion of (Conf(X), B(Conf(X)), μ) and a G-action group of automorphisms T G .
A homogeneous Poisson point process with intensity r > 0 is the point process on R d satisfying: (1) ξ(A) has a Poisson distribution with mean r|A| for each A ∈ B(R d ); Here, ξ(A) is the number of particles on A for ξ ∈ Conf(X) and |A| is the Lebesgue measure of A. A determinantal point process μ on X is a point process associated with a kernel function K : X × X → C and a Radon measure λ on X, for which the n-point correlation function with respect to λ is given by (1.1) for each n ∈ N. See Definition 4.1 for the definition of the n-point correlation function. We call μ a (K, λ)-determinantal point process. If the context is clear, we omit λ, calling μ a K-determinantal point process. Throughout this paper, we assume that λ is the Lebesgue measure if X = R d . Now, we state the main theorem.
We remark that the assumption for K in Theorem 1.1 implies the following conditions (1)-(4) with X = R d and the Lebesgue measure λ.
The K-determinantal point process μ satisfying (1)-(4) above is translation invariant because its n-correlation functions are translation invariant.
For determinantal point processes on Z d with translation-invariant kernel and the counting measure, Lyons and Steif [5] and Shirai and Takahashi [12] independently proved the Bernoulli property, the latter giving a sufficient condition for the weak Bernoulli property under the assumption K : Z d × Z d → C satisfying (1), (2), Spec(K) ⊂ (0, 1), and (4). We recall that the weak Bernoulli property is stronger than the Bernoulli property.
One of the ideas in [5] is using the dbar distance, which is a metric on the set of Z d -action systems; the Bernoulli property is closed under this metric [8,9,14]. However, the dbar distance does not work for systems with infinite entropy because entropy is continuous with respect to the dbar distance. In general, a translation-invariant point process on R d has infinite entropy. Therefore, we cannot apply the dbar distance to our case. Therefore, we construct point processes on a discrete set that approximate the determinantal point process on R d . We prove the Bernoulli property of the discrete point processes. In turn, we can prove the Bernoulli property of the determinantal point process on R d via its tree representations [10].
To prove Theorem 1.1, we apply the general theory given by Ornstein and Weiss [9]. We quote them in the form applicable to the R d -and Z d -actions. We also refer to [8] for the Zand R-actions, and [14] for the Z d -action.
The outline of this paper is as follows. In §2, we recall notions related to the Bernoulli property. In §3, we introduce the kernel functions that approximate the determinantal kernel K in Theorem 1.1 uniformly on any compact set on R d . In §4, we introduce the tree representations of the determinantal point processes on R d . We combine these representations with the kernels introduced in §3. The tree representations are determinantal point processes on Z d × N and are translation invariant with respect to the first coordinate. In §5, we prove the Bernoulli property of the tree representations using the properties of the dbar distance introduced in §2. In §6, we prove Theorem 1.1 using the Bernoulli property of the tree representations.

Notions related to the Bernoulli property
In this section, we collect properties of point processes without determinantal structure and notions related to the Bernoulli property.
We first recall the notion of monotone coupling. For Let ν 1 and ν 2 be probability measures on {0, 1} Z d . We say that a probability measure γ on {0, 1} Z d × {0, 1} Z d is a monotone coupling of ν 1 and ν 2 if the following hold. (1) μ ≤ ν.
(2) There exists a monotone coupling of μ and ν.
We naturally regard a simple point process μ on Z d × N as a probability measure on {0, 1} Z d ×N , denoted by the same symbol μ. We write μ ≤ ν for simple point processes μ and ν if the corresponding probability measures on {0, 1} Z d ×N satisfy μ ≤ ν. We introduce the notion of monotone coupling for simple point processes on Z d × N from that of the corresponding probability measures on {0, 1} Z d ×N in an obvious fashion. Fix Proof. By assumption and Lemma 2.1, there exists a monotone coupling γ of μ and ν.
Then γ N is a monotone coupling of μ N and ν N . From this and Lemma 2.1, we obtain the claim.
We recall the notion of being finitely dependent, which is a sufficient condition for the Bernoulli property. See, e.g., [5].
Here, ν A denotes the conditional probability measure under A, is the collection of the couplings of ν| R and ν A | R , and The very weak Bernoulli property is equivalent to the Bernoulli property for elements of P inv (M).
Let μ and ν ∈ P inv (M). Defined : Thend gives a metric on P inv (M). The Bernoulli property is closed underd.

Approximations of the determinantal kernel
In this section, we introduce three approximations of the kernel K introduced in (1.2). For r > 0, let w r : R d → R be the product of the tent function such that We denote byŵ r its Fourier transform sin πrt j πt j 2 .

S. Osada
Proof. By construction, we see that

Tree representations of determinantal point processes
In this section, we introduce the tree representations of determinantal point processes on R d . Then we apply them to the determinantal point processes associated with the kernels introduced in §3. Before doing so, we recall the definition and well-known facts about determinantal point processes.
Let μ be a point process on X. A locally integrable symmetric function ρ n : X n → [0, ∞) is called the n-point correlation function of μ (with respect to a Radon measure λ for any disjoint Borel subsets A 1 , . . . , A k and for any n i ∈ N, i = 1, . . . , k, such that k i=1 n i = n. Let K : X × X → C. We call μ a determinantal point process with kernel K and Radon measure λ if the n-point correlation function ρ n of μ with respect to λ satisfies (1.1) for each n.
Assume that K : X × X → C satisfies Here, K in (4.3) is an integral operator on L 2 (X, λ) such that Kf (x) = X K(x, y)λ(dy) and K A in (4.4) is its restriction on L 2 (A, λ). Then there exists a unique determinantal point process on X with kernel function K. Next, we introduce the tree representations of the determinantal point processes. Let μ K be the determinantal point process on R d with kernel function K satisfying (4.2)-(4.4). First, we introduce a partition of R d and the associated orthonormal basis on L 2 (R d ). Let P = {P z : z ∈ Z d } be a partition of R d such that each P z is relatively compact and Here, A + x = {a + x; a ∈ A} for A ⊂ R d and x ∈ R d . Let = P = {φ z,l } (z,l)∈Z d ×N be an orthonormal basis on L 2 (R d ) such that suppφ z,l ⊂ P z and φ z+w,l (x) = φ z,l (x − w). (4.5) For the kernel function K above, let K : Proof. By assumption and (4.6), K satisfies (4.2) and (4.4). Equation (4.3) follows from [10].
From Lemma 4.1 and the general theory in [11,13], there exists a determinantal point process ν K, on Z d × N associated with K . We call ν K, the tree representation of μ K with respect to .
We apply the tree representations for the translation-invariant kernels on R d introduced in §3.
Assume that K is given by (1.2). Then K is translation invariant. Hence, by construction, K is translation invariant with respect to the first coordinate Z d . From this, we see that ν K, is translation invariant with respect to the first coordinate.
Define K r , K r , and K r similarly as (4.6) with replacement of K with K r , K r , and K r in (3.1)-(3.3), respectively. By construction, K r , K r , and K r satisfy (4.2)-(4.4). Hence, K r , K r , and K r satisfy (4.2)-(4.4) with respect to the counting measure on Z d × N by Lemma 4.2. Furthermore, K r , K r , and K r are translation invariant with respect to the first coordinate Z d .
Let ν K, r , ν K, r , and ν K, r be K r -, K r -and K r -determinantal point processes, respectively. We remark that a determinantal point process ν on Z d has no multiple points with probability 1. Hence, we can regard ν as a probability measure on {0, 1} Z d . We quote the following result. 1, 2). Assume that K 1 ≤ K 2 . Let ν K 1 and ν K 2 be the determinantal point processes with K 1 and K 2 , respectively. Then there exists a monotone coupling of ν K 1 and ν K 2 .
Applying Lemma 4.3, we obtain the following result. Proof. Recall that is the orthonormal basis of L 2 (R d ) given in (4.5). Let U :L 2 (R d )→ L 2 (Z d × N) be the unitary operator such that U(φ z,n ) = e z,n , where {e z,n } (z,n)∈Z d ×N is the canonical orthonormal basis of L 2 (Z d × N). Then, by Lemma 1 in §3 of [10], we see that K = UKU −1 . From this and Lemma 3.1, we obtain From (4.9) and (4.10) combined with Lemma 4.3, we conclude (4.7) and (4.8).
Recall that K r , K r , and K r are translation invariant with respect to the first coordinate. Hence, ν K, r , ν K, r , and ν K, r are also translation invariant with respect to the first coordinate. We regard We continue the setting of §4. Let K be the kernel defined by (4.6). Let ν K, be the K -determinantal point process as before. The purpose of this section is to prove the Bernoulli property for (Conf(Z d × N), ν K, , T Z d ).
Let N be the map defined by (2.2).
We also denote N -factors of (Conf (Z d Let d be the graph distance as before. Let r 0 > 0 be such that for each z, w ∈ Z d , Let P , Q ⊂ Z d × N be finite sets such that d(P , Q) ≥ r 0 . Then K r (z, l; w, m) = 0 for (z, l) ∈ P , (w, m) ∈ Q . (5.1) For P ⊂ Z d × N, we define a cylinder set by By construction, 1 P ∩ 1 Q = 1 P ∪Q . Therefore, The third equality follows from (5.1). Let R, S ⊂ Z d be such that d(R × N, S × N) ≥ r 0 . From (5.2) and the π -λ theorem, The last equation follows from the fact that γ N is a monotone coupling of ν K, r,N and ν K, N . Because of Lemma 5.1, (5.6) is true for (ν K, r,N , ν K, r,N ), (ν K, N , ν K, r,N ), and (ν K, r,N , ν K, r,N ). From this combined with (5.4) and (5.5), we obtain The purpose of this section is to complete the proof of Theorem 1.1.
We quote a general fact of isomorphism theory.
For n ∈ N, let P n = {P n,z : z ∈ Z d } be a partition of R d such that Then P n • T z (ξ) = T z • P n (ξ) for each z ∈ Z d and ξ ∈ Conf(R d ). Let μ K P n = μ K • −1 P n . Then (Conf(Z d ), μ K P n , T Z d ) is the P n -factor of (Conf(R d ), μ K , T Z d ).
Proof. Let n = {φ n z,l } (z,l)∈Z d ×N be an orthonormal basis on L 2 (R d ) such that φ n z+w,l (x) = φ n z,l (x − w) and supp φ n z,l = P n,z . Let ν K, be the tree representation of μ K with respect to n . Let π : Conf(Z d × N) → Conf(Z d ) be such that By construction, π • T z (η) = T z • π(η) for each z ∈ Z d and η ∈ Conf(Z d × N). From Lemma 4.2, ν K, • π −1 = μ K P n .
Hence, (Conf(Z d ), μ K P n , T Z d ) is the π -factor of (Conf(Z d × N), ν K, , T Z d ). From Theorem 5.4, (Conf(Z d × N), ν K, , T Z d ) is Bernoulli. From this and Lemma 6.1, the claim holds.

S. Osada
Proof. By construction, the sequence of partitions {P n : n ∈ N} is increasingly finer and separates the points of R d . From this, we obtain that {σ [ P n ]} n∈N is increasing and n∈N σ [ P n ] separates the points of Conf(R d ). Putting this result together with Lemma 6.2 and Lemma 2.9 implies the claim.
We quote Theorem 10 of III.6 in [9]. LEMMA 6.4. [9] For an R d -action system ( , F, P, S R d ), let S Z d = {S g : g ∈ Z d } be the limitation on Z d -action of S R d . If ( , F, P, S Z d ) is Bernoulli with infinite entropy, then ( , F, P, S R d ) is Bernoulli.
We are now ready to complete the proof of Theorem 1.1.
Proof of Theorem 1.1. From Lemma 6.3, (Conf(R d ), μ K , T Z d ) is Bernoulli. Because the restriction of μ K on [0, 1) d is a non-atomic probability measure, the entropy of (Conf(R d ), μ K , T Z d ) is infinite. Putting this and Lemma 6.4 together implies the claim.