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.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
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.
Let \(\mu \) be a \(\sigma \)-finite Borel probability measure on \(\mathbb {R}\), and let \(\Pi (x,y)\) be the kernel of a locally trace-class operator of orthogonal projection acting in \(L_2(\mathbb {R}, \mu )\). Recall that the determinantal point process \(\mathbb {P}_{\Pi }\) is a Borel probability measure on \({\mathrm {Conf}}(\mathbb {R})\) defined by the condition that for any bounded measurable function g, for which \(g-1\) is supported in a bounded set B, we have
The Fredholm determinant in (1) is well-defined since \(\Pi \) is locally of tracel class. The equation (1) determines the measure \({\mathbb {P}}_{\Pi }\) uniquely. For any pairwise disjoint bounded Borel sets \(B_1,ldots,B_l\subset \mathbb {R}\) and any \(z_1,\cdots ,z_l\in {\mathbb C}\) from (1) we have
For further results and background on determinantal point processes, see e.g. [2, 10, 12, 13, 17–19].
We now formulate a sufficient condition for the rigidity of a determinantal point process on \(\mathbb {R}\).
Proposition 1.1
Let \(U\subset {\mathbb {R}}\) be an open subset, let \(\mu \) be the Lebesgue measure on 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
$$\begin{aligned} \int \limits _{x:|x|\le R}|\Pi (x,y)|^2{\hbox {d}}\mu (x)\le \frac{C(R)}{1+y^{1+\varepsilon }}. \end{aligned}$$
Then the point process \({\mathbb {P}}_{\Pi }\) is rigid.
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
Given a bounded measurable function f on M, we introduce the additive functional \(S_f\) on \(\mathrm {Conf}(M)\) by the formula
We recall the sufficient condition for rigidity of a point process given by Ghosh [6], Ghosh and Peres [7].
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
For the reader’s convenience, we recall the elegant short proof of Ghosh [6], Ghosh and Peres [7]. Let \(B^{(n)}\) be an increasing sequence of nested bounded Borel sets exhausting 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
Since, for any bounded B and sufficiently large n, we have
we thus obtain the equality
for \(\mathbb {P}\)-almost every 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
We next recall that if \(\mu \) is a \(\sigma \)-finite Borel measure on 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
It therefore suffices, in order to establish the rigidity of the point process \(\mathbb {P}_{\Pi }\), to find an increasing sequence of bounded Borel subsets \(B^{(n)}\) exhausting 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
Take \(R>0\), \(T>R\) and set
To establish Proposition 1.1, it suffices to prove
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
We estimate the double integral (2) for the additive statistic \(f=\varphi ^{(R,T)}\). Of course, if \(|x|, |y|<R\) or if \(|x|, |y|>T\), then the expression under the integral sign is equal to zero. We will now estimate our integral over the domain
and complete the proof by estimating the smaller contribution of the domains
We consider these three cases separately.\(\square \)
The First Case: \(x,y\in \mathbb {R}^2: R<|x|,|y|<T\).
It is clear that for any x, y satisfying \(|x|, |y|>R\) there exists a constant C(R) depending only on R such that we have
Using the first assumption of Proposition 1.1, we now estimate the integral (2) for the additive statistic \(f=\varphi ^{(R,T)}\) from above by the expression
where the implied constant depends only on 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: \(x,y\in \mathbb {R}^2: |x|>R,|y|>T\).
Next, we consider the integral
which (upon recalling that \(x\le y\) and making a scaling change of variable) can be estimated from above by the expression
The Third Case. \(\{x,y\in \mathbb {R}^2: 0<|x|<R<|y|<\infty \}\).
Finally, we consider the integral
in order to estimate which it suffices to estimate the integral
which, using the second assumption of Proposition 1.1, we estimate from above by the expression
where the implied constant, as always, depends only on R. The proposition is proved completely.
2.2 The case of integrable kernels
In applications, one often meets kernels admitting an integrable representation
with smooth functions A, B; the diagonal values of the kernel \(\Pi \) are given by the formula
In this case, Proposition 1.1 yields the following
Corollary 2.2
If the kernel \(\Pi \) admits an integrable representation (5) and, furthermore, there exist \(R>0\), \(C>0\) and \(\varepsilon >0\) such that
-
(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 \(|A(x)|\le C|x|^{1/2-\varepsilon }\); \(|B(x)|\le C|x|^{1/2-\varepsilon }\),
then the process \({\mathbb {P}}_{\Pi }\) is rigid.
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
Take \(s>-1\) and recall that the Bessel kernel is given by the formula
By the Macchi–Soshnikov theorem, the Bessel kernel induces a determinantal point process \({\mathbb {P}}_{\mathbf {J}_s}\) on \({\mathrm {Conf}}({\mathbb {R}}_+)\).
Proposition 3.1
The determinantal point process \({\mathbb {P}}_{\mathbf {J}_s}\) is rigid.
Proof
Indeed, this follows from Corollary 2.2, the estimate \(J_s(x)\sim x^{s/2}\), valid for small x (cf. e.g. 9.1.10 in in Abramowitz and Stegun [1]) and the standard asymptotic expansion
of the Bessel function of a large argument (cf. e.g. 9.2.1 in Abramowitz and Stegun [1]). Proposition 3.1 is proved. \(\square \)
3.2 The determinantal point process with the Airy kernel
Recall that the Airy kernel is given by the formula
where
is the standard Airy function.
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
Again, we take \(R>0\), \(T>R\) and set
Since \({\mathbb {P}}_{{\mathrm {Ai}}}\)-almost every trajectory admits only finitely many particles on the positive semi-axis, the additive functional \(S_{\varphi ^{(T)}}\) is \({\mathbb {P}}_{{\mathrm {Ai}}}\)-almost surely well-defined. It is immediate from (2) that its variance is finite. \(\square \)
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
Let \(\Pi (x,y)\) be a kernel yielding an operator of orthogonal projection acting in \(L_2({\mathbb Z})\). 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
$$\begin{aligned} \sum \limits _{x:|x|\le R}|\Pi (x,y)|^2\le \frac{C(R)}{1+y^{1+\varepsilon }}. \end{aligned}$$
Then the point process \({\mathbb {P}}_{\Pi }\) is rigid.
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
If the kernel \(\Pi \) admits an integrable representation (5) and, furthermore, there exist \(R>0\), \(C>0\) and \(\varepsilon >0\) such that for all \(|x|>R\) we have
then the process \({\mathbb {P}}_{\Pi }\) is rigid.
4.2 The determinantal point process with the Gamma-kernel
Let \({\mathbb {Z}}^{\prime }=1/2+{\mathbb Z}\) be the set of half-integers. The Gamma-kernel with parameters \(z, z^{\prime }\) is defined on \({\mathbb {Z}}^{\prime }\times {\mathbb {Z}}^{\prime }\) by the formula
Following Borodin and Olshanski (cf. Proposition 1.8 in [3]), we consider two cases: first, the case of the 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
(cf. e.g. 6.1.47 in Abramowitz and Stegun [1]) allows us directly to apply Corollary 2.2 and thus to complete the proof of
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.
References
Abramowitz, M., Stegun, I.: Handbook of Mathematical Functions, National Bureau of Standards, Department of Commerce of the United States of America, Tenth Printing (1972)
Borodin, A.M.: Determinantal Point Processes, in the Oxford Handbook of Random Matrix Theory. Oxford University Press, Oxford (2011)
Borodin, A., Olshanski, G.: Random partitions and the Gamma kernel. Adv. Math. 194, 141–202 (2005). arXiv:math-ph/0305043
Bufetov, A.I.: Quasi-symmetries of determinantal point processes. arXiv:1409.2068
Daley, D.J., Vere-Jones, D.: An Introduction to the Theory of Point Processes, vols. I–II. Springer, New York (2008)
Ghosh, S.: Determinantal processes and completeness of random exponentials: the critical case. arXiv:1211.2435
Ghosh, S., Peres, Y.: Rigidity and tolerance in point processes: Gaussian zeros and Ginibre eigenvalues. arXiv:1211.2381
Ghosh, S.: Rigidity and tolerance in Gaussian zeros and Ginibre eigenvalues: quantitative estimates. arXiv:1211.3506
Holroyd, A.E., Soo, T.: Insertion and deletion tolerance of point processes. arXiv:1007.3538
Hough, J.B., Krishnapur, M., Peres, Y., Virág, B.: Determinantal processes and independence. Prob. Surv. 3, 206–229 (2006)
Kolmogoroff, A.: Grundbegriffe der Wahrscheinlichkeitsrechnung. Springer, New York (1933)
Lyons, R.: Determinantal probability measures. Publ. Math. Inst. Hautes Études Sci. 98, 167–212 (2003)
Lytvynov, E.: Fermion and boson random point processes as particle distributions of infinite free Fermi and Bose gases of finite density. Rev. Math. Phys. 14(10), 1073–1098 (2002)
Macchi, O.: The coincidence approach to stochastic point processes. Adv. Appl. Prob. 7, 83–122 (1975)
Olshanski, G.: The quasi-invariance property for the Gamma kernel determinantal measure. Adv. Math. 226(3), 2305–2350 (2011)
Osada, H., Shirai, T.: Absolute continuity and singularity of Palm measures of the Ginibre point process. arXiv:1406.3913
Shirai, T., Takahashi, Y.: Random point fields associated with certain Fredholm determinants. I. Fermion, Poisson and Boson point processes. J. Funct. Anal. 205(2), 414–463 (2003)
Shirai, T., Takahashi, Y.: Random point fields associated with certain Fredholm determinants. II. Fermion shifts and their Ergodic and Gibbs properties. Ann. Prob. 31(3), 1533–1564 (2003)
Soshnikov, A.: Determinantal random point fields (Russian). Uspekhi Mat. Nauk 55 (2000), no. 5(335), 107–160; translation in Russian Math. Surveys 55 (2000), no. 5, 923–975
Tracy, C.A., Widom, H.: Level-spacing distributions and the Airy kernel. Commun. Math. Phys. 159(1), 151–174 (1994)
Tracy, C.A., Widom, H.: Level spacing distributions and the Bessel kernel. Commun. Math. Phys. 161(2), 289–309 (1994)
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Bufetov, A.I. Rigidity of determinantal point processes with the Airy, the Bessel and the Gamma kernel. Bull. Math. Sci. 6, 163–172 (2016). https://doi.org/10.1007/s13373-015-0080-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13373-015-0080-z