Discrete approximations of determinantal point processes on continuous spaces: tree representations and tail triviality

We prove tail triviality of determinantal point processes $ \mu $ on continuous spaces. Tail triviality had been proved for such processes only on discrete spaces, and hence we have generalized the result to continuous spaces. To do this, we construct tree representations, that is, discrete approximations of determinantal point processes enjoying a determinantal structure. There are many interesting examples of determinantal point processes on continuous spaces such as zero points of the hyperbolic Gaussian analytic function with Bergman kernel, and the thermodynamic limit of eigenvalues of Gaussian random matrices for Sine$_2 $, Airy$_2 $, Bessel$_2 $, and Ginibre point processes. Tail triviality of $ \mu $ plays a significant role in proving the uniqueness of solutions of infinite-dimensional stochastic differential equations (ISDEs) associated with $ \mu $. For particle systems in $ \R $ arising from random matrix theory, there are two completely different constructions of natural stochastic dynamics. One is given by stochastic analysis through ISDEs and Dirichlet form theory, and the other is an algebraic method based on space-time correlation functions. Tail triviality is used crucially to prove the equivalence of these two stochastic dynamics.


Introduction
Let S be a locally compact, complete, separable metric space with metric d(·, ·). We assume S is unbounded. We equip S with a Radon measure m such that m(O) > 0 for any non-empty open set O in S. Let S be the configuration space over S (see (2.1) for definition). S is a Polish space equipped with the vague topology.
A determinantal point process µ on S is a probability measure on (S, B(S)) for which the m-point correlation function ρ m with respect to m is given by the determinant (1.1) Here K : S × S → C is a measurable kernel and x = (x 1 , . . . , x m ). We refer to Section 2 and e.g. [1,3,10] for the definition of correlation functions and related notions. µ is said to be associated with (K, m) and also a (K, m)-determinantal point process.
We set Kf (x) = S K(x, y)f (y)m(dy). We regard K as an operator on L 2 (S, m) and denote it by the same symbol. We say K is of locally trace class if K A f (x) = 1 A (x)K(x, y)1 A (y)f (y)m(dy) is a trace class operator on L 2 (S, m) for any compact set A.
In the last two decades, determinantal point processes have been extensively studied. They contain many interesting examples; e.g., spanning trees and Schur measures on discrete spaces, zero points of the hyperbolic Gaussian analytic function with Bergman kernel, and thermodynamic limits of eigenvalues of Gaussian random matrices such as Sine 2 , Airy 2 , Bessel 2 , and Ginibre point processes on continuous spaces [1,5,10].
Determinantal point processes on discrete spaces have a well-behaved algebraic structure; as a result, some important facts are only known for discrete determinantal point processes [4,6,7,8,12]. One such example is tail triviality, which says that each event of a tail σ-field Tail(S) takes value 0 or 1. We refer to (2.3) for the definition of Tail(S).
The purpose of this paper is to prove that the tail σ-field Tail(S) of S is trivial with respect to µ. If the space S is discrete, then tail triviality has been proved by Shirai-Takahashi [11] for Spec(K) ⊂ (0, 1), and by Russell Lyons [7] for Spec(K) ⊂ [0, 1]. If the space S is continuous, the problem remained open [8].
To prove tail triviality we introduce a discrete approximation for determinantal point processes, called the tree representation. This representation has a determinantal structure, and so belongs to determinantal point processes on discrete spaces.
A m-partition ∆ = {A i } i∈I of S is a countable collection of disjoint relatively compact, measurable subsets of S such that ∪ i A i = S and that m(A i ) > 0 for all i ∈ I. For two partitions ∆ = {A i } i∈I and Γ = {B j } j∈J , we write ∆ ≺ Γ if for each j ∈ J there exists i ∈ I such that B j ⊂ A i . We assume: (A2) There exists a sequence of m-partitions {∆(ℓ)} ℓ∈N satisfying (1.2)-(1.4).
Condition (1.4) is just for simplicity. This condition implies that the sequence {∆(ℓ)} ℓ∈N has a binary tree-like structure. We remark that (A2) is a mild assumption and, indeed, satisfied if S is an open set in R d and m has positive density with respect to the Lebesgue measure. We now state one of our main theorems: Theorem 1.1. Assume (A1) and (A2). Let µ be the (K, m)-determinantal point process. Then µ has a trivial tail. That is, µ(A) ∈ {0, 1} for all A ∈ Tail(S).
Many interesting determinantal point processes arise from random matrices such as Sine 2 , Airy 2 , and Bessel 2 point processes in R and the Ginibre point process in R 2 . Applying Theorem 1.1 to these examples we obtain that all have trivial tails. We shall present these examples in Section 6.
We now explain the idea of the proof. We have two candidates for the discrete approximations of µ. One is the approximation of the kernel K. Let K ℓ (x, y) be the discrete kernel on I(ℓ) such that . Then K ℓ can be regarded as a discrete kernel on I(ℓ). If K ℓ satisfies (A1), then K ℓ generates deteminantal point field µ K ℓ . Indeed, Spec(K ℓ ) ⊂ [0, 1] follows from Spec(K) ⊂ [0, 1] and the Fubini theorem. One can expect the convergence of the kernel K ℓ to K, and as a result, the weak convergence of µ K ℓ to µ, at least for continuous K. Because µ K ℓ is a determinantal point process on the discrete space, its tail σ-field is trivial. Such weak convergence, however, does not suffice for the convergence of the values on the tail σ-field Tail(S). Taking the above into account, we consider the second approximation given by µ(·|G ℓ ) below. Let G ℓ be the sub-σ-field of B(S) given by (1.5) Combining (1.2) and (1.3) with (1.5), we obtain Let µ(·|G ℓ ) be the regular conditional probability of µ with respect to G ℓ . Using (1.6), we shall prove in Lemma 5.3 that for all U ∈ B(S) We see that the convergence in (1.7) is stronger than the weak convergence. In particular, the convergence in (1.7) is valid for all U ∈ Tail(S) because Tail(S) ⊂ B(S). We can naturally regard ∆(ℓ) = {A ℓ,i } i∈I(ℓ) as a discrete, countable set with the interpretation that each element A ℓ,i is a point. Thus, µ(·|G ℓ ) can be regarded as a point process on the discrete set ∆(ℓ).
If µ(·|G ℓ ) were a determinantal point process for each ℓ, then Theorem 1.1 would follow from (1.7) immediately because determinantal point processes on discrete spaces always have trivial tails, and as discussed above, µ(·|G ℓ ) is naturally regarded as a determinantal point process on the discrete space ∆(ℓ). This is clearly not the case because determinantal point processes are supported on single configurations and (1.8) Hence we introduce a sequence of fiber bundle-like sets I(ℓ) (ℓ ∈ N) in Section 2 with base space ∆(ℓ) with fiber consisting of a set of binary trees. We further expand I(ℓ) to Ω(ℓ) in (2.27), which has a fiber whose element is a product of a tree i and a component B ℓ,i of partitions. See notation after Theorem 2.1. Let µ| G ℓ denote the restriction of µ on G ℓ . By construction µ| G ℓ (A) = µ(A|G ℓ ) for all A ∈ G ℓ . In Theorem 2.1 and Theorem 2.2, we construct a lift ν F(ℓ) ⋄ m F(ℓ) of µ| G ℓ on the fiber bundle Ω(ℓ), and prove tail triviality of the lift ν F(ℓ) ⋄ m F(ℓ) in Theorem 2.4, which establishes tail triviality of µ| G ℓ in Theorem 2.5. Combining Theorem 2.5 with the martingale convergence theorem in Lemma 5.3, we obtain Theorem 1.1.
The key point of the construction of the lift ν F(ℓ) ⋄ m F(ℓ) is that we construct a consistent family of orthonormal bases F(ℓ) = {f ℓ,i } i∈I(ℓ) in (2.15) and (2.16), and that we introduce the kernel K F(ℓ) on I(ℓ) in (2.21) such that (2.21) We shall prove in Lemma 3.2 that K F(ℓ) is a determinantal kernel on I(ℓ), and present ν F(ℓ) as the associated determinantal point process on I(ℓ). To some extent, ν F(ℓ) is isometric to µ| G ℓ through the orthonormal basis F(ℓ) = {f ℓ,i } i∈I(ℓ) . We shall indeed prove in Theorem 2.1 that their correlation functions ρ m G ℓ and ρ m F(ℓ) satisfy the identity: which is a key to construct the lift ν F(ℓ) ⋄ m F(ℓ) . While preparing the manuscript, we have heard that Professor A. Bufetov has proved independently tail triviality of determinantal point processes on continuous spaces independently of us (a seminar talk at Kyushu University in October 2015). His method is completely different from ours and requires a restriction on an integrability condition of the determinantal kernel K(x, y). An improved version of the work is now available in [2].
The organization of the paper is as follows. In Section 2, we introduce definitions and concepts and state the main theorems (Theorems 2.1-2.5). We give tree representations of µ. In Section 3, we prove Theorem 2.1. In Section 4, we prove Theorem 2.2-Theorem 2.5. In Section 5, we prove Theorem 1.1. In Section 6, we present motivational examples such as Sine 2 , Airy 2 , and Bessel 2 , and Ginibre point processes.

Set up and main results
In this section, we recall various essentials and present the main theorems (Theorem 2.1-Theorem 2.5) other than Theorem 1.1 .
A configuration space S over S is a set consisting of configurations on S such that where δ si denotes the delta measure at s i . A probability measure µ on (S, B(S)) is called a point process, also called random point field. A symmetric function ρ m on S m is called the m-point correlation function of a point process µ with respect to a Radon measure m if it satisfies We fix a point o ∈ S as the origin, and set S r = {x ∈ S ; d(o, x) < r}. Each S r is assumed to be relatively compact, and thus s(S r ) < ∞ for all s ∈ S and r ∈ N. In this sense, each element s of S is a locally finite configuration. We note that this notion depends on the choice of metric d on S.
For a Borel set A we set π A : S → S by π A (s)(·) = s(· ∩ A). We set π S c r : S → S such that π S c r (s) = s(· ∩ S c r ). We denote by Tail(S) the tail σ-field such that In consequence of (1.4), we assume without loss of generality that each element i of the parameter set I(ℓ) is of the form We denote by I the set of all such parameters: We can regard I as a collection of binary trees and I is the set of their roots.
For a given sequence of m-partitions satisfying (A2), such an orthonormal basis exists. We present here an example.
(3) By construction, we see that where we set, for j = θ −1 ℓ−1,r (i) such that rank(i) = r, (2.20) Using the orthonormal basis F(ℓ) = {f ℓ,i } i∈I(ℓ) , we set K F(ℓ) on I(ℓ) by Let λ I(ℓ) be the counting measure on I(ℓ). We shall prove in Lemma 3.2 that (K F(ℓ) , λ I(ℓ) ) satisfies (A1). Hence we obtain the associated determinantal point process ν F(ℓ) on I(ℓ) from general theory [10,12]. For i ∈ I(ℓ), let m f ℓ,i (dx) be the probability measure on S such that Let G ℓ be the sub-σ-field as in (1.5). Let ν F(ℓ) be the (K F(ℓ) , λ I(ℓ) )-determinantal point process as before. Let ρ m G ℓ and ρ m F(ℓ) be the m-point correlation functions of µ| G ℓ and ν F(ℓ) with respect to m and λ I(ℓ) , respectively. We now state one of our main theorems: (2.25) Assume that A n ∈ ∆(ℓ) for all n = 1, . . . , m. Then Let Ω(ℓ) be the single configuration space over Ω(ℓ). Then by definition each element ω ∈ Ω(ℓ) is of the form ω = i∈i δ (i,si) such that s i ∈ B ℓ,i . Hence Let m f ℓ,i be as in (2.22). We set By definition m f ℓ,i in (2.29) is a product measure on the product space i∈i B ℓ,i with (unordered) parameter i ∈ i, whereas m f ℓ,i in (2.23) is a product measure on the product space B i1 × · · · × B im with (ordered) parameter i = (i 1 , . . . , i m ).
Let ν F(ℓ) ⋄ m F(ℓ) be the probability measure on Ω(ℓ) given by (1) We can naturally regard the probability measures in (2.31) as a point process on i∈i B ℓ,i supported on the set of configurations with exactly one particle configuration s = δ s on i∈i B ℓ,i , that is, s = (s i ) i∈i is such that s i ∈ B ℓ,i for each i ∈ i.
(2) We can regard ν F(ℓ) ⋄ m F(ℓ) as a marked point process as follows: The configuration i is distributed according to ν F(ℓ) , while the marks are independent and for each i the mark s is distributed according to m f ℓ,i . Thus the space of marks depends on i.
We remark that µ| G ℓ is not a determinantal point process. Hence we exploit ν F(ℓ) ⋄ m F(ℓ) instead of µ| G ℓ . As we have seen in Theorem 2.2, ν F(ℓ) ⋄ m F(ℓ) is a lift of µ| G ℓ in the sense of (2.32), from which we can deduce nice properties of µ| G ℓ . Indeed, an application of Theorem 2.2 combined with Theorem 2.4 is tail triviality of µ| G ℓ : (2.35) We shall apply Theorem 2.5 to prove Theorem 1.1 in Section 5.

Proof of Theorem 2.2-Theorem 2.5
In this section, we prove Theorem 2.2-Theorem 2.5.

Proof of Theorem 2.2
Let ̺ m be the m-point correlation function of (ν F(ℓ) ⋄ m f ℓ ) • u −1 ℓ | G ℓ . Then it suffices for (2.32) to prove . . , i k ) ∈ I(ℓ) m such that i n ∈ I(ℓ) mn . From Theorem 2.1, we see that By the definition of correlation functions, (2.30), and (2.31), we see that Combining (4.2) and (4.3), we deduce that Proof of Theorem 1.1 In this section, we complete the proof of Theorem 1.1.
Lemma 5.1. Let X be a Tail(S)-measurable and integrable random variable. Then Proof. Recall that ∆(ℓ) = {A ℓ,i } i∈I(ℓ) . Let π Tr be the projection with T r such that is Tail(S)-measurable and each A ℓ,i is relatively compact. Hence for each r ∈ N From this we deduce that By construction S r ⊂ T r . Then from this and (5. Combining these completes the proof of Lemma 5.1 for µ-a.s. s.. We have thus proved (5.6).

Examples related to random matrices
In this section, we give typical examples of determinantal point processes related to random matrix theory [9,3]. All examples below are tail trivial because of Theorem 1.1. All the kernels K(x, y) below are continuous. In Examples 6.1-6.3, we define the kernels only off diagonal. On diagonal, they are defined by continuity. be the sine kernel. The associated determinantal point process µ sin is called the sine 2 point process. be the Airy kernel. Here Ai is the Airy function, and Ai ′ is its derivative. The associated determinantal point process µ Ai is called the Airy point process [9,3]. Let K Be,α be the Bessel kernel such that 2(x − y) (x = y).
Let µ Be,α be the associated determinantal point process. µ Be,α is called the Bessel 2,α point process.
Here we identify R 2 as C by the obvious correspondence R 2 ∋ x = (x 1 , x 2 ) → x 1 + √ −1x 2 ∈ C, andȳ = y 1 − √ −1y 2 is the complex conjugate in this identification. The associated determinantal point process µ Gin is called the Ginibre point process.