Discrete Approximations of Determinantal Point Processes on Continuous Spaces: Tree Representations and Tail Triviality

We prove tail triviality of determinantal point processes μ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mu $$\end{document} on continuous spaces. Tail triviality has 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 Sine2,Airy2,Bessel2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {Sine}_2 , \hbox {Airy}_2 , \hbox {Bessel}_2 $$\end{document}, and Ginibre point processes. Our main theorem proves all these point processes are tail trivial.


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. Here K : S × S → C is a measurable kernel and x = (x 1 , . . . , x m ). We refer to Sects. 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 K f (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 Δ = {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 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 to these examples we obtain that all have trivial tails. We shall present these examples in Sect. 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 (x, y) be the discrete kernel on I ( ) such that Then K can be regarded as a discrete kernel on I ( ). If K satisfies (A1), then K generates determinantal point field μ K . Indeed, Spec(K ) ⊂ [0, 1] follows from Spec(K) ⊂ [0, 1] and the Fubini theorem. One can expect the convergence of the kernel K to K, and as a result, the weak convergence of μ K to μ, at least for continuous K. Because μ K 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 ) below. Let G be the sub-σ -field of B(S) given by 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 Δ( ) = {A ,i } i∈I ( ) as a discrete, countable set with the interpretation that each element A ,i is a point. Thus, μ(·|G ) can be regarded as a point process on the discrete set Δ( ).
If μ(·|G ) were a determinantal point process for each , then Theorem 1 would follow from (1.7) immediately because determinantal point processes on discrete spaces always have trivial tails, and as discussed above, μ(·|G ) is naturally regarded as a determinantal point process on the discrete space Δ( ). 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( ) ( ∈ N) in Sect. 2 with base space Δ( ) with fiber consisting of a set of binary trees. We further expand I( ) to Ω( ) in (2.27), which has a fiber whose element is a product of a tree i and a component B ,i of partitions. See notation after Theorem 2.
Let μ| G denote the restriction of μ on G . By construction μ| G (A) = μ(A|G ) for all A ∈ G . In Theorems 2 and 3, we construct a lift ν F( ) m F( ) of μ| G on the fiber bundle Ω( ), and prove tail triviality of the lift ν F( ) m F( ) in Theorem 5, which establishes tail triviality of μ| G in Theorem 6. Combining Theorem 6 with the martingale convergence theorem in Lemma 6, we obtain Theorem 1.
The key point of the construction of the lift ν F( ) m F( ) is that we construct a consistent family of orthonormal bases F( ) = { f ,i } i∈I( ) in (2.15) and (2.16), and that we introduce the kernel K F( ) on I( ) in (2.21) such that (2.21) We shall prove in Lemma 2 that K F( ) is a determinantal kernel on I( ), and present ν F( ) as the associated determinantal point process on We shall indeed prove in Theorem 2 that their correlation functions ρ m G and ρ m F( ) satisfy the identity: which is a key to construct the lift ν F( ) m F( ) . 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 Sect. 2, we introduce definitions and concepts and state the main theorems (Theorems 2-6). We give tree representations of μ. In Sect. 3, we prove Theorem 2. In Sect. 4, we prove Theorems 3-6. In Sect. 5, we prove Theorem 1. In Sect. 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 (Theorems 2-6) other than Theorem 1.
A configuration space S over S is a set consisting of configurations on S such that where δ s i 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 Here A 1 , . . . , A j ∈ B(S) are disjoint and k 1 , . . . , k j ∈ N such that k 1 + · · · + k j = m. If 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 If we replace S r by any increasing sequence {O r } of relatively compact open sets such that In consequence of (1.4), we assume without loss of generality that each element i of the parameter set I ( ) 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.
Example 1 (Binary partitions of R) Typically we can take S = R, m(dx) = dx, and I = Z. For i = ( j 1 , . . . , j ) ∈ I ( ), we set J 1,i = j 1 and, for ≥ 2, For a given sequence of m-partitions satisfying (A2), such an orthonormal basis exists. We present here an example.
We next introduce the -shift of above objects such as I, B i , and F = { f i } i∈I . Let I(1) = I and, for ≥ 2, (2.14) We set rank(i) = r for i ∈ I ( )×{0, 1} r −1 . By construction rank(i) = r for i ∈ θ −1,r ( I). m). This follows from assumptions (2.15) and (2.16) and the fact that F = { f i } i∈I is an orthonormal basis.

2.25)
Assume that A n ∈ Δ( ) for all n = 1, . . . , m. Then Let Ω( ) be the single configuration space over Ω( ). Then by definition each element ω ∈ Ω( ) is of the form ω = i∈i δ (i,s i ) such that s i ∈ B ,i . Hence Let m f ,i be as in (2.22). We set By definition m f ,i in (2.29) is a product measure on the product space i∈i B ,i with (unordered) parameter i ∈ i, whereas m f ,i in (2.23) is a product measure on the product space B i 1 × · · · × B i m with (ordered) parameter i = (i 1 , . . . , i m ).

Remark 3 (1)
We can naturally regard the probability measures in (2.31) as a point process on i∈i B ,i supported on the set of configurations with exactly one particle configuration s = δ s on i∈i B ,i , that is, s = (s i ) i∈i is such that s i ∈ B ,i for each i ∈ i.
We can naturally regard I( ) as binary trees. Hence we call ν F( ) m F( ) a tree representation of μ of level .

Proof of Theorem 2
The purpose of this section is to prove Theorem 2. In Lemma 1, we present the identity of kernels K and K F( ) using the orthonormal basis F( ), where K F( ) is the kernel given by (2.21) and F( ) is as in (2.15) and (2.16). In Lemma 2, we prove (K F( ) , λ I( ) ) is a determinantal kernel and the associated determinantal point process ν F( ) exists. We will lift the the identity between K and K F( ) to that of correlation functions of μ| G and ν F( ) in Theorem 2.
Let λ I( ) be the counting measure on I( ) as before. We can regard K F( ) as an operator on L 2 (I( ), λ I( ) ) such that . We now prove that the (K F( ) , λ I( ) )-determinantal point ν F( ) process exists.

Proof of Theorems 3-6
In this section, we prove Theorem 3-Theorem 6.

Proof of Theorem 3
Let m be the m-point correlation function of (ν F( ) m f ) • u −1 | G . Then it suffices for (2.32) to prove By the definition of correlation functions, (2.30), and (2.31), we see that Combining (4.2) and (4.3), we deduce that From (4.4), we obtain (4.1). This completes the proof of Theorem 3.

Proof of Theorem 4
Theorem 4 follows from Theorem 3 immediately.

Proof of Theorem 5
It is known that determinantal point processes on discrete spaces are tail trivial [7,11]. Hence ν F( ) is tail trivial by Lemma 2.

Proof of Theorem 6
Let B ∈ Tail(S) ∩ G . Then we deduce that Hence from Theorems 3 and 5, we deduce that This completes the proof.

Proof of Theorem 1
In this section, we complete the proof of Theorem 1.

Lemma 4 Let X be a Tail(S)-measurable and integrable random variable. Then E μ [X |G ]
is Tail(S) ∩ G -measurable.
Proof Recall that Δ( ) = {A ,i } i∈I ( ) . Let π T r be the projection with T r such that Then X ∈ L 1 (S, μ) is σ [π T c r ]-measurable because X ∈ L 1 (S, μ) is Tail(S)-measurable and each A ,i is relatively compact. Hence for each r ∈ N X (s) = X • π T c r (s).

Examples Related to Random Matrices
In this section, we give typical examples of determinantal point processes related to random matrix theory [3,9]. All examples below are tail trivial because of Theorem 1. All the kernels K(x, y) below are continuous. In Examples 3-5, 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.
Example 4 (Airy point process) Let S = R and m(dx) = dx. Let 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 [3,9].
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.