# Rigidity of determinantal point processes with the Airy, the Bessel and the Gamma kernel

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## Abstract

A point process is said to be *rigid* if for any bounded domain in the phase space, the number of particles in the domain is almost surely determined by the restriction of the configuration to the complement of our bounded domain. The main result of this paper is that determinantal point processes with the Airy, the Bessel and the Gamma kernels are rigid. The proof follows the scheme used by Ghosh, Ghosh and Peres: the main step is the construction of a sequence of additive statistics with variance going to zero.

## Keywords

Determinantal point processes Airy kernel Bessel kernel Gamma kernel Rigidity## 1 Introduction

### 1.1 Rigid point processes

Let *M* be a complete separable metric space. Recall that a *configuration* on *M* is a purely atomic Radon measure on *M*; in other words, a collection of particles considered without regard to order and not admitting accumulation points in *M*. The space \(\text {Conf}(M)\) of configurations on *M* is itself a complete separable metric space with respect to the vague topology on the space of Radon measures. A point process on *M* is by definition a Borel probability measure on \(\text {Conf}(M)\).

Given a bounded subset \(B \subset M\) and a configuration \(X \in \text {Conf}(M),\) let \( \#_B(X)\) stand for the number of particles of *X* lying in *B*. Given a Borel subset \(C \subset M,\) we let \(\mathcal {F}_C\) be the \(\sigma \)-algebra generated by all random variables of the form \(\#_B, B\subset C.\) If \(\mathbb {P}\) is a point process on *M*, then we write \(\mathcal {F}_C^{\mathbb {P}}\) for the \(\mathbb {P}\)-completion of \(\mathcal {F}_C\).

The following definition of rigidity of a point process is due to Ghosh [6] (cf. also Ghosh and Peres [7]).

### **Definition**

A point process \(\mathbb {P}\) on M is called **rigid** if for any bounded Borel subset \(B \subset M\) the random variable \(\#_B\) is \(\mathcal {F}_{M\backslash B}^{\mathbb {P}}\)-measurable.

*g*, for which \(g-1\) is supported in a bounded set

*B*, we have

We now formulate a sufficient condition for the rigidity of a determinantal point process on \(\mathbb {R}\).

### **Proposition 1.1**

*U*, and let \(\Pi (x,y)\) be a kernel yielding an operator of orthogonal projection acting in \(L_2({\mathbb {R}}, \mu )\). Assume that there exists \(\alpha \in (0, 1/2)\), \(\varepsilon >0\), and, for any \(R>0\), a constant \(C(R)>0\) such that the following holds:

- (1)if \(|x|, |y|\ge R\), then$$\begin{aligned} |\Pi (x,y)|\le C(R)\cdot \frac{({x}/{y})^{\alpha }+ ({y}/{x})^{\alpha }}{|x-y|}; \end{aligned}$$
- (2)if \(|x|\le R\), then for all
*y*we have$$\begin{aligned} \int \limits _{x:|x|\le R}|\Pi (x,y)|^2{\hbox {d}}\mu (x)\le \frac{C(R)}{1+y^{1+\varepsilon }}. \end{aligned}$$

As we shall see below, Proposition 1.1 implies rigidity for determinantal point processes with the Airy and the Bessel kernels; in the last subsection of the paper, we shall obtain a counterpart of Proposition 1.1 for determinantal point processes with discrete phase space and, as its corollary, rigidity for the determinantal point process with the Gamma kernel.

### *Remark*

As far as I know, rigidity of point processes first appears (under a different name) in the work of Holroyd and Soo [9], who established, in particular, that the determinantal point process with the Bergman kernel is *not* rigid. For the sine-process, rigidity is due to Ghosh [6]. For the Ginibre ensemble, rigidity has been established by Ghosh and Peres [7]; see also Osada and Shirai [16].

### 1.2 Additive functionals and rigidity

*f*on

*M*, we introduce the additive functional \(S_f\) on \(\mathrm {Conf}(M)\) by the formula

### **Proposition 1.2**

(Ghosh [6], Ghosh and Peres [7]) Let \(\mathbb {P}\) be a Borel probability measure on \(\mathrm {Conf}(M)\). Assume that for any \(\varepsilon >0\) and any bounded subset \(B \subset M\) there exists a bounded measurable function *f* of bounded support such that \(f\equiv 1\) on *B* and \(\mathrm {Var}_{\mathbb {P}} S_f < \varepsilon \). Then the measure \(\mathbb {P}\) is rigid.

### *Proof*

*M*. Our assumptions and the Borel–Cantelli Lemma imply the existence of a sequence of bounded measurable function \(f^{(n)}\) of bounded support, such that \(f^{(n)}|_{B^{(n)}} \equiv 1\) and that for \(\mathbb {P}\)-almost every \(X \in \mathrm {Conf}(M)\) we have

*B*and sufficiently large

*n*, we have

*X*, and the rigidity of \(\mathbb {P}\) is proved. \(\square \)

### *Remark*

In fact, to prove rigidity, it suffices that the function *f* only satisfy the inequality \(|f-1|<\varepsilon \) on *B*; the proof of the proposition becomes slightly more involved, but the result is still valid.

### 1.3 Variance of additive functionals

*M*and \(\mathbb {P}\) is a determinantal point process induced by a locally trace class operator \(\Pi \) of orthogonal projection acting in the space \(L_2(M,\mu )\), then the variance of an additive functional \(S_f\), corresponding to a bounded measurable function

*f*of bounded support, is given by the formula

*M*and a sequence \(f^{(n)}\) of bounded Borel functions of bounded support such that \( f^{(n)}|_{B^{(n)}} \equiv 1\) and

## 2 Rigidity in the continuous case

### 2.1 Proof of Proposition 1.1

### **Lemma 2.1**

If \(\Pi \) satisfies the assumptions of Proposition 1.1, then, for any sufficiently large \(R>0\), as \(T\rightarrow \infty \) we have \( {\mathrm {Var}}_{{\mathbb {P}}_{\Pi }} S_{\varphi ^{(R,T)}}\rightarrow 0. \)

### *Proof*

*The First Case*: \(x,y\in \mathbb {R}^2: R<|x|,|y|<T\).

*x*,

*y*satisfying \(|x|, |y|>R\) there exists a constant

*C*(

*R*) depending only on

*R*such that we have

*R*. Introducing the variable \(\lambda =y/x\) and recalling that \(\alpha <1/2\), we estimate the integral (3) from above by the expression

*The Second Case*: \(x,y\in \mathbb {R}^2: |x|>R,|y|>T\).

*The Third Case.*\(\{x,y\in \mathbb {R}^2: 0<|x|<R<|y|<\infty \}\).

*R*. The proposition is proved completely.

### 2.2 The case of integrable kernels

*integrable*representation

*A*,

*B*; the diagonal values of the kernel \(\Pi \) are given by the formula

### **Corollary 2.2**

- (1)
for all \(|x|<R\) we have \(|A(x)|\le C|x|^{-1/2+\varepsilon }\); \(|B(x)|\le C|x|^{-1/2+\varepsilon }\);

- (2)
for all \(|x|>R\) we have \(|A(x)|\le C|x|^{1/2-\varepsilon }\); \(|B(x)|\le C|x|^{1/2-\varepsilon }\),

### *Proof*

Indeed, it is clear that both assumptions of Proposition 1.1 are verified in this case.\(\square \)

## 3 Examples: the Bessel and the Airy kernel

### 3.1 The determinantal point process with the Bessel kernel

### **Proposition 3.1**

The determinantal point process \({\mathbb {P}}_{\mathbf {J}_s}\) is rigid.

### *Proof*

*x*(cf. e.g. 9.1.10 in in Abramowitz and Stegun [1]) and the standard asymptotic expansion

### 3.2 The determinantal point process with the Airy kernel

By the Macchi-Soshnikov theorem, the Airy kernel infuces a determinantal point process \({\mathbb {P}}_{{\mathrm {Ai}}}\) on \({\mathrm {Conf}}({\mathbb {R}})\). In this case, we establish rigidity in the following slightly stronger form.

### **Proposition 3.2**

For any \(D\in {\mathbb {R}}\), the random variable \(\#_{(D, +\infty )}\) is measurable with respect to the \({\mathbb {P}}_{{\mathrm {Ai}}}\)-completion of the sigma-algebra \({\mathcal F}_{(-\infty , D)}\).

### *Proof*

### **Lemma 3.3**

For any fixed \(R>0\), as \(T\rightarrow \infty \), we have \( {\mathrm {Var}}S_{\varphi ^{(R,T)}}\rightarrow 0. \)

The proof of Lemma 3.3 is done in exactly the same way as that of Proposition 1.1 and Corollary 2.2 , using standard power estimates for the Airy function and its derivative for negative values of the argument (cf. e.g. 10.4.60, 10.4.62 in Abramowitz and Stegun [1]) as well as the standard superexponential estimates for the Airy function and its derivative for positive values of the argument (cf. e.g. 10.4.59, 10.4.61 in Abramowitz and Stegun [1]). Proposition 3.2 follows immediately.

## 4 Rigidity of determinantal point processes with discrete phase space

### 4.1 A general sufficient condition

Proposition 1.1 admits a direct analogue in the case of a discrete phase space.

### **Proposition 4.1**

- (1)if \(|x|, |y|\ge R\), then$$\begin{aligned} |\Pi (x,y)|\le C(R)\cdot \frac{({x}/{y})^{\alpha }+ ({y}/{x})^{\alpha }}{|x-y|}; \end{aligned}$$
- (2)if \(|x|\le R\), then for all
*y*we have$$\begin{aligned} \sum \limits _{x:|x|\le R}|\Pi (x,y)|^2\le \frac{C(R)}{1+y^{1+\varepsilon }}. \end{aligned}$$

The proof is exactly the same as that of Proposition 1.1. The Corollary for integrable kernels assumes an even simpler form in the discrete case.

### **Corollary 4.2**

### 4.2 The determinantal point process with the Gamma-kernel

*half-integers*. The Gamma-kernel with parameters \(z, z^{\prime }\) is defined on \({\mathbb {Z}}^{\prime }\times {\mathbb {Z}}^{\prime }\) by the formula

*principal*series, where \(z^{\prime }={\overline{z}}\notin {\mathbb {R}}\) and the case of the complementary series, in which \(z, {z}^{\prime }\) are real and, moreover, there exists an integer

*m*such that \(z, z^{\prime }\in (m, m+1)\). In both these cases, the Gamma-kernel induces an operator of orthogonal projection acting in \(L_2({\mathbb {Z}}^{\prime })\). We now establish the rigidity of the corresponding determinantal measure \({\mathbb {P}}_{\mathbf {\Gamma }_{z, {z}^{\prime }}}\) on \({\mathrm {Conf}}({\mathbb {Z}}^{\prime })\). We use Corollary 4.2. In the case of the principal series, the functions

*A*,

*B*giving the integrable representation for the Gamma-kernel, are bounded above, so there is nothing to prove. In the case of the complementary series, the standard asymptotics

### **Proposition 4.3**

The determinantal point process with the Gamma-kernel is rigid for all values of the parameters *z*, \({z}^{\prime }\) belonging to the principal and the complementary series.

## Notes

### Acknowledgments

I am deeply grateful to Grigori Olshanski and Yanqi Qiu for useful discussions. This work is supported by A*MIDEX project (No. ANR-11-IDEX-0001-02), financed by Programme “Investissements d’Avenir” of the Government of the French Republic managed by the French National Research Agency (ANR). It is also supported in part by the Grant MD-2859.2014.1 of the President of the Russian Federation, by the Programme “Dynamical systems and mathematical control theory” of the Presidium of the Russian Academy of Sciences, by a subsidy granted to the HSE by the Government of the Russian Federation for the implementation of the Global Competitiveness Program and by the RFBR grant 13-01-12449.

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