Point Processes, Hole Events, and Large Deviations: Random Complex Zeros and Coulomb Gases
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Abstract
We consider particle systems (also known as point processes) on the line and in the plane and are particularly interested in “hole” events, when there are no particles in a large disk (or some other domain). We survey the extensive work on hole probabilities and the related large deviation principles (LDP), which has been undertaken mostly in the last two decades. We mainly focus on the recent applications of LDPinspired techniques to the study of hole probabilities and the determination of the most likely configurations of particles that have large holes. As an application of this approach, we illustrate how one can confirm some of the predictions of Jancovici, Lebowitz, and Manificat for large fluctuation in the number of points for the (twodimensional) \(\beta \)Ginibre ensembles. We also discuss some possible directions for future investigations.
Keywords
Point processes Particle systems Coulomb gases Random matrices Random polynomials Hole probabilities Large deviations Empirical measuresMathematics Subject Classification
Primary 60G55 Secondary 60F101 Introduction
Random point configurations, also known as point processes, have been an object of key interest in the last few decades  both in probability theory and in the statistical physics literature. The most extensive results have been obtained in Euclidean spaces of dimensions 1 and 2, although higher dimensions and other geometries have also been studied.
A point process \(\Pi \), usually defined to live on a Polish space \(\Sigma \) equipped with a regular Borel measure \(\mu \), is a probability distribution over the space of locally finite point configurations on \(\Sigma \). We recall here that a Polish space is a separable and completely metrizable topological space. It is well known ([20, Chap. 9]) that, under mild conditions, the statistical behavior of a point process is described by its various kpoint intensities (\(k=1,2,\dots \)), which are roughly the joint probability densities of having particles at k specified locations in \(\Sigma \). For almost all interesting point processes, these k point intensities are absolutely continuous with respect to \(\mu ^{ \otimes k}\) (referred to as the background measure), and the resulting Radon Nikodym derivatives are known as the k point intensity functions. Often, in Euclidean spaces or other homogeneous spaces, key point processes exhibit invariance, which is to say that the law of the process is invariant under the isometries of \(\Sigma \).
The most fundamental example of a point process is the Poisson point process. The Poisson point process, defined on the space \(\Sigma \) with respect to the background measure \(\mu \), is the unique point process on \(\Sigma \) that exhibits statistical independence of its point configurations in disjoint domains, with the particle count in a domain \(D \subset \Sigma \) obeying a Poisson distribution with mean \(\mu (D)\). This characterizing property of spatial independence makes many important statistical properties easy to compute, which is the reason behind the popularity of the Poisson process as a probabilistic model for many realworld systems ([20, Chap. 2]). At the same time, it renders the Poisson process ineffective in modeling many natural phenomena, particularly those involving local repulsion, like electron systems.
In 1 and 2 dimensions, at inverse temperature \(\beta =2\), the Coulomb system with logarithmic interactions (a.k.a. Dyson log gas in 1D) is known to be a determinantal point process, meaning that its correlation functions are given by certain determinants. When \(V(x) = x^2\), these ensembles can also be described as the set of eigenvalues of certain random matrices  in 1 dimension it is the Gaussian Wigner matrix (GUE), and in 2 dimensions the Ginibre ensemble (having independent standard complex Gaussian entries). Both of these ensembles have welldefined weak limits that are determinantal point processes with infinitely many particles. In 1 dimension, the fundamental solution to the Laplacian is \(f(x)=x\), and it is natural to consider a Coulomb system with this interaction potential. This system has been extensively studied by Aizenmann, Lebowitz, Martin, Yalcin, and others (see, e.g., [2, 3, 42, 47] for some of the delicate results on this model).
An object of key interest in the study of point processes is the “hole” event \(\mathcal {H}_R\), entailing that a large disk (or interval, according to the dimension) of radius R around the origin does not contain any particles. Of course, this is a rare event, and \(\mathbb {P}[\mathcal {H}_R] \rightarrow 0\) as \(R \rightarrow \infty \). The quantitative asymptotics of how this decay takes place throws light on the statistical structure of the point process, and has been studied in fine detail for many key processes. A closely related but much less understood question pertains to what causes such a large “hole” to appear. This involves understanding the typical configuration of particles outside the “hole”, and until recently such results were available only for \(\beta =2\) Coulomb systems in 1 and 2 dimensions ([40]). Very recently, progress has been made on this front for the GEF zeros process, as well as for holes of general shapes for the Ginibre ensemble. This is based on large deviation techniques, which brings us to the third key object in this paper, namely large deviation principles (abbreviated henceforth as LDPs).
In [1, 33] the main ingredient of the approach to the “hole configuration” is to consider the “hole” event as a “rare” event in the setting of the LDP for the relevant matrix or polynomial ensemble. This intuitively leads to the conclusion that the (limiting) intensity measure of the particles outside the hole must be the minimizer of the large deviation rate functional, under the constraint of the existence of the hole. This approach seems to be rather promising in investigating related problems for point processes. We provide more details on this approach in Sect. 6, where we study the twodimensional \(\beta \)Ginibre ensembles (also known as jellium or the onecomponent plasma).
The main thrust of this work is on a certain set of ideas that tie together point processes, large deviations, and the study of the hole event. Such focus naturally leaves out several important strands of work related to various combinations of these concepts. For instance, we mention the recent series of works studying various fine properties of the large deviation principle for Coulomb systems. In particular, these works establish rigorous connections of the LDP to the concept of renormalized energy ([6, 14, 43, 45, 57, 61, 62]). Another direction of recent investigations involves the study of spatial rigidity structures that arise in several of these natural models ([10, 11, 12, 28, 29, 30, 34, 55]). Beyond that, there is the extensive research on universality in random matrix ensembles (see, e.g., [24, 65]). We will not pursue these matters here.
2 Large Deviations for Empirical Measures
Definition 2.1
A rate function \(I{:}\, \Sigma \rightarrow \mathbb {R}_+\) is good if all its level sets \(\{x {:}\, I(x) \le \alpha \}\) are compact subsets of \(\Sigma \).
2.1 Eigenvalues of Random Matrices
Large deviations for empirical measures of random matrices have been studied by multiple authors. In this section, we will only focus on LDPs for the empirical measure of some specific families of random matrices, including Gaussian (and other unitarily invariant) Hermitian ensembles in 1D, and the (real and complex) Ginibre ensemble in 2D. We direct readers interested in a more extensive survey, including dynamical aspects related to evolution under the Dyson Brownian motion, to [35].
2.1.1 The Ginibre Ensemble
We begin with the LDP for the Ginibre ensemble. For the original paper, we refer the reader to [9]. The (real or complex) Ginibre ensemble (of order n) is the ensemble of eigenvalues of \(n \times n\) random matrices with i.i.d. Gaussian entries (resp., real or complex) with mean zero and variance \(n^{1}\). The (infinite) Ginibre ensemble is the limit, in distribution, of the finite Ginibre ensembles.
Theorem 2.1
([9, 37]) The sequence of empirical measures \(\mathcal {E}(\mathcal {G}_n^{[S]})\) obey a large deviation principle in the space \(\mathcal {M}_1(\mathbb {C})\) with rate \(n^2\) and good rate function \(I^{[S]}\).
The rate function \(I^{[S]}\) is minimized by the uniform measure on the unit disk. Consequently, the (random) empirical measures \(\mathcal {E}(\mathcal {G}_n^{[S]})\) converge a.s. to the uniform measure on the unit disk.
2.1.2 OneDimensional \(\log \)Gas
Theorem 2.2
([8]) The sequence of empirical measures \(\mathcal {E}(\mathbb {P}^n_{V,\beta })\) obey a large deviation principle in the space \(\mathcal {M}_1(\mathbb {R})\) with rate \(n^2\) and good rate function \(I^V_\beta \).
We mention in passing that large deviation principles are also known for \(\beta \)Ginibre ensembles (defined analogously to the \(\beta \) ensembles in 1D by using a general \(\beta \) exponent on the Vandermonde in (2)); these correspond to the 2D Coulomb gas (for general inverse temperature \(\beta \)). For details, we refer to [38].
2.2 Zeros of Random Polynomials
The theory of large deviations for empirical measures of zeros of random polynomials is of more recent origin. One of the earliest articles in this direction, namely [67], deals with the crucial case of (complex) Gaussian random polynomials, i.e., random polynomials with independent Gaussian coefficients (with mean zero and possibly decaying variances). Depending on the mode of decay of the variances, we obtain several distinguished “standard ensembles”  Kac (constant variance of coefficients), Elliptic (coefficient of \(z^k\) has variance \({n \atopwithdelims ()k} k!\)) and Weyl (coefficient of \(z^k\) has variance 1 / k!). [67] covers all these cases, as well as more general scalings of coefficients. In fact, [67] works in the more general setting of Gaussian measures on polynomial spaces of degree n that live on the Riemann surface \(\mathbb {C}\mathbb {P}^1\). These Gaussian measures are determined by inner products naturally induced from a metric h and measure \(\nu \) on \(\mathbb {C}\mathbb {P}^1\), the only condition being that the pair \((h,\nu )\) satisfy the socalled Bernstein–Markov property. The results obtained on the Riemann sphere \(\mathbb {C}\mathbb {P}^1\) can be transferred (via the stereographic projection) to the complex plane. For a detailed exposition of this, we refer the reader to [15] (which also deals with the case of real Gaussian coefficients).
2.2.1 Weyl Polynomials
In what follows, we will denote by \(\mathcal {Z}_n = \{z_1, \dots , z_n \}\) the zero set of the Weyl polynomial of degree n, scaled down by \(\sqrt{n}\). Also recall that for any measure \(\mu \in \mathcal {M}_1(\mathbb {C})\), we denote by \(U_\mu \) and \(\Sigma (\mu )\) the logarithmic potential and the logarithmic energy of \(\mu \), respectively. We can now state the following LDP for zeros of Weyl polynomials:
Theorem 2.3
The minimizer (and, consequently, the a.s. limit of the \(\mathcal {Z}_n\)s ) of the above rate function is the uniform measure on the unit disk. The constant C is such that I evaluated at this measure is 0.
2.2.2 Other Polynomials
LDPs are known for empirical measures of zeros of many other random polynomial and polynomiallike ensembles, in addition to the models described above. Some key examples are [25, 32, 68] and [16].
Some of these ensembles pose specific technical challenges of their own in establishing the LDP. As an example, we can consider the LDP for the Kac polynomial ensemble with exponential coefficients ([32]). The major new difficulty is that all the coefficients are now positive a.s. This restricts the possible zero sets of such polynomials, the precise nature of which was not fully understood until recently. E.g., Obrechkoff’s theorem ([54]) provides a necessary (but not sufficient) condition that the number of zeros of such a polynomial in a conical sector (around \(\mathbb {R}_+\)) can grow at most linearly with the angle at the apex of the cone. This issue makes an impact even on the form of the LDP rate functional:
Theorem 2.4
The approach of [32] exploits certain aspects of a potential theoretic description of the set \(\mathcal {P}\) obtained by [7]. The universality results of [16] employ comparison techniques with appropriate ensembles already known to have an LDP.
3 Hole Events and Hole Probabilities
Hole events and hole probabilities have classically been a key object of interest in the study of point processes (a.k.a. particle systems). An important example of this is the wellknown result that hole probabilities for a determinantal point process are given by certain Fredholm determinants related to its kernel ([48, Chap. 6], [60]).
To fix ideas, let \(D_r\) denote the (open) disk (in dimension one an interval) of radius r, centered at the origin. The hole event, denoted \(\mathcal {H}_r\), is the event where there are no points of the point process in \(D_r\). The hole probability at radius r is \(\mathbb {P}[\mathcal {H}_r]\), which clearly decays to 0 as \(r \rightarrow \infty \). A very wellstudied question in point process theory is the manner of decay of \(\mathbb {P}[\mathcal {H}_r]\) (more precisely, its logarithmic asymptotics). Typically, the logarithm of the hole probability decays like a power law, whose exponent depends upon the point process under consideration, and is thought to shed light on its ‘rigidity’. By rigidity in this setting, we envisage latticelike behavior. In particular, the heuristic is that the faster the decay rate of the hole probability (that is, the higher the exponent discussed above), the stronger is the latticelike behavior. E.g., as we shall see below, the exponent for the Poisson process (in 2D) is 2. On the other hand, for compactly supported i.i.d. perturbations of the lattice \(\mathbb {Z}^2\), the hole probability is 0 for large enough hole sizes, and hence, heuristically speaking, the above exponent is \(\infty \).
In 2D, the simplest example of a homogeneous point process, namely the Poisson process (with unit intensity) gives a decay of \(\mathbb {P}[\mathcal {H}_r]=\exp (\mathrm {Area}(D_r))=\exp (\pi r^2)\), so the decay exponent is 2. For the 2D Coulomb gas (inverse temperature \(\beta =2\), a.k.a. the Ginibre ensemble), it has been shown that this exponent is 4 ([58, 39, Chap. 7]); i.e., the hole probability exhibits the decay \(r^{4}\log \mathbb {P}[\mathcal {H}_r] \rightarrow \frac{1}{4}\) as \(r \rightarrow \infty \). The larger exponent of the Ginibre process already attests to a stronger global spatial correlation compared to the Poisson process (the latter being characterized by the spatial independence of its points). For the application of LDP techniques to study hole probabilities for the Ginibre ensemble, see, for example, Sect. 4.3.
A key ingredient in the proof is the fact that the number of particles in \(D_r\), for any determinantal point process, is given by a sum of independent Bernoulli random variables (see, e.g., [39, Chap. 4]). The parameters (success probabilities) of these Bernoullis are essentially the eigenvalues of the integral operator given by the kernel of the determinantal process restricted to \(D_r\). An alternative approach for the Ginibre ensemble is to use the fact (first proved by Kostlan for the finite Ginibre ensemble) that the set of the squares of the moduli of the eigenvalues is distributed like a set of independent Gamma random variables (see [39, Theorem 4.7.3] also for the infinite ensemble).
Thus, the exponents of the Ginibre and the GEF zeros process match. This leads to the interesting question regarding the comparison of these two processes visavis their strength of correlations (or latticelike behavior). This has spawned an interesting collection of results. On the one hand, there are comparison theorems for finite order correlation functions of the Ginibre and GEF zero ensembles ([53]). On the other hand, there are recent results showing significant differences in the properties of their (spatially) conditional distributions ([33, 34]). A very interesting problem is to determine whether there is a natural invariant point process in the plane whose hole probability decays qualitatively faster than the decay rate of the Ginibre ensemble and the GEF zeros process.
We conclude this section by mentioning several other works related to hole probabilities, mostly of recent origin. An important instance is the study of hole probability asymptotics for zeros of a wide class of Gaussian analytic functions having a finite radius of convergence. This includes the wellstudied hyperbolic GAFs (with general intensity \(L>0\)), whose domain of convergence is the unit disk. For the case \(L=1\), the zero set has been shown to be a determinantal point process in [56]. In the same paper ([56]), the asymptotics of the hole probability (as \(r \uparrow \infty \)) has been worked out. In [17], the hole probability asymptotics have been worked out for general L. In the process, a surprising discovery is made to the effect that the form of the asymptotics (including its dependence on L) depend crucially on whether L is subcritical (\(0<L<1\)), critical (\(L=1\)), or supercritical (\(L>1\)). Another interesting family of results involves gap probabilities (essentially, hole probabilities in 1D) for important families of 1D Gaussian processes, in particular connecting these asymptotics with simple properties of their spectral measures and socalled “persistence probabilities” ([4, 21, 22, 26, 27]). In [58] and [59], the author obtains fine quantitative estimates on various aspects of the hole probability and the hole event for the Ginibre ensemble and related determinantal processes associated with higher Landau levels.
4 Conditional Distribution on the Hole Event
In this section, we consider the following problem: What is the principal cause of a (rare) event of a hole of large radius? Having understood hole probabilities, the next natural question, therefore, is to try and understand the point process conditioned to have a hole of a large radius. This question, however, turns out to be a surprisingly difficult one  even in expectation.
4.1 The Ginibre Ensemble
Until recently, the only 2D point process for which this was understood was the Ginibre ensemble ([40]; see [58] for a more recent study of finer aspects and more quantitative results). We state the result as (see the appendix in [40]):
Theorem 4.1
In particular, for \(r=R\) and \(R\gg 1\), we have \(\rho (R)\sim \frac{1}{2}\pi R^2\) (see equation (2.13) in [40]). This roughly corresponds to the appearance of a delta measure at the edge of the hole under appropriate renormalization.
In [59], Shirai described the complete behavior of the conditional intensity of eigenvalues for the more general “Ginibretype” ensembles. We mention here a version of Theorem 1.4 therein, adapted to our specific context and using our notation. For a Borel set \(D\subset \mathbb {C}\), let \(\xi (D)\) denote the number of Ginibre eigenvalues in D. Let A(x, y) denote the annulus \(\{z \in \mathbb {C}: x \le z < y\}\). With this notation, we can state:
Theorem 4.2
The discontinuity in the limit at \(a=1\) captures the delta measure at the edge of the hole. The above limit aslo shows clearly that asymptotically, beyond the hole, the conditional intensity converges to the equilibrium intensity.
We point out that only the specific situation of a “round” hole was considered in this approach  that is, the hole consisted of no particles present in the disk of radius R, as opposed to, say, a hole in the form of a particlefree square of side length R, with \(R \rightarrow \infty \). This is crucial for obtaining the above results (as previously mentioned, the set of the squares of the moduli of the eigenvalues is distributed like a set of independent Gamma random variables).
A crucial deficiency of this approach is the dependence on the above explicit description of the radii of the Ginibre points, which (or any substitute thereof) is not available for the other point processes. Even for the Ginibre ensemble, this approach depends crucially on the radial symmetry of the hole, and thus precludes any understanding of holes of any shape other than a disk.
4.2 GEF Zeros
Very recently, progress has been achieved ([1, 33]) in understanding the conditional intensity around a large hole for point processes other than the Ginibre ensemble and for noncircular holes. The new progress relies on a novel large deviation based approach. As an example, we state the following description for the (limiting) conditional intensity measure around a “round” hole for the GEF zeros process ([33]):
Theorem 4.3
The paper [33], in fact, provides quantitative estimates of the typical number of zeros in the annulus between R and \(\sqrt{e} R\). In what follows, we denote by \(N_{F}(A)\) the number of zeros of the GEF in the set \(A \subset \mathbb {C}\).
Theorem 4.4
The proofs of Theorem 4.3 and Theorem 4.4 are based on a certain deviations inequality for linear statistics of the GEF zeros. We provide more details in Sect. 7.
It is an interesting problem to establish fine asymptotics, on the lines of Theorem 4.1, for the GEF zeros process. This would involve, in the best case scenario, an explicit expression for the conditional density function. At a more modest level, it can also envisage asymptotics of the conditional intensity function in various regimes, an important example of which is the rate of blowup of this function at the edge of the hole. It is also of interest to find the asymptotics of the (conditional) expected value of \(N_F(A(R(1+\epsilon ), \sqrt{e}R(1\epsilon )) )\), as \(R\rightarrow \infty \).
4.3 Ginibre Ensemble: General Holes and Weighted Fekete Points
Theorem 4.5

(Balayage condition) There exists a sequence of open sets \(U_n\) such that \(\overline{U} \subset U_n \subseteq D\) for all n, and the balayage measure \(\nu _n\) on \(\partial U_n\) converges weakly to the balayage measure on \(\partial U\).
 (Exterior ball condition) There exists \(\epsilon >0\) such that for every \(z \in \partial U\), there exists a \(\eta \in U^\complement \) such that$$\begin{aligned} B(\eta , \epsilon ) \subset U^\complement \text { and } z\eta =\epsilon . \end{aligned}$$
Note that all convex domains satisfy the exterior ball condition.
For the infinite Ginibre ensemble, we have:
Theorem 4.6
5 Large Fluctuations in the Number of Points and the Jancovici–Lebowitz–Manificat Law
Closely related to the hole event are the phenomena of “deficiency” and “overcrowding” in the number of particles, which entail that the number of particles in \(D_r\) is very far from its typical value of about \(r^2\) particles (with the standard normalization for the Ginibre ensemble and the GEF zeros). This has been extensively studied both for the Ginibre ensemble and the GEF zeros ([40, 41, 49]), with the discovery that the fluctuations in both cases obey the Jancovici–Lebowitz–Manificat law (in short, the JLM law), that was first introduced in the context of large charge fluctuations for the 2D Coulomb gas ([40], see Conjecture 5.1 for the statement).
5.1 The Jancovici–Lebowitz–Manificat Law
Compared with (5), one can certainly consider a wider range of fluctuations in the number of particles and also examine other ensembles. We find it rather surprising that the asymptotic decay of the probability of large fluctuations is described by a common ‘law’, both for the Ginibre ensemble and the GEF zeros process (this law also appears in other ensembles, such as certain randomly perturbed lattices, see [49]).
5.1.1 Finite \(\beta \)Ginibre Ensemble
For N large, the particles tend to be asymptotically uniformly distributed inside the disk of radius \(\sqrt{N}\) centered at the origin. Let us denote by n(R) the number of particles in the disk \(D(0,R) = \left\{ z \le R \right\} \). For N large compared with \(R^2\), we have that n(R) is typically about \(R^2\).
Conjecture 5.1
Remark 5.1
See [40] for the precise expression for \(\psi (\beta ;a,b, \gamma )\) in the case \(a = 2\) (cf. (5)). It seems that no such expression is known for \(a = 1\) (even when \(\beta = 2\)).
The constant \(c_\beta \) is derived from the (conjectured) central limit theorem (CLT) for n(R). Recently the CLT for smooth linear statistics was proved in [44] (in this case the dependence on \(\beta \) is explicit).
In the case of the (infinite) Ginibre ensemble (\(\beta = 2\)), the arguments of [40] are essentially mathematically rigorous (for proofs in the case \(a = 2\), see the aforementioned [58]). For other values of \(\beta \), it is not even known if a limiting object for the \(\beta \)Ginibre ensemble exists when the number of particles goes to infinity.
5.1.2 Fluctuations for the GEF Zeros Process
Nazarov, Sodin, and Volberg ([49]) confirmed that some of the predictions of [40] hold also for large fluctuations in the number of zeros of the GEF. More precisely, they proved the following result:
Theorem 5.1
Using the results of [33] together with the approach of [41], it is possible to establish finer asymptotics for fluctuations in the GEF zeros process that are analogous to the JLM law, in a restricted range of exponents. As an example we mention:
Theorem 5.2
The lower bound in the above asymptotics can, in fact, be shown to hold for \(a \in (1,2)\), and it is plausible that the results hold in this range.
5.2 Deficiency and Overcrowding: conditional distribution
Denote by \(n_F(R)\) the number of zeros of the GEF inside the disk \(\{z < R\}\). Recall that the zero counting measure \([\mathcal {Z}_R]\) denotes the GEF zero counting measure, conditioned on the hole event in \(\{n_F(R) = 0\}\). We now denote by \([\mathcal {Z}^p_R]\) the GEF zero counting measure, conditioned on the event \(\{n_F(R) = \lfloor pR^2 \rfloor \}\), with \(p\ge 0, p \ne 1\).
Notation: If \(p = 0\), we set \(q = e\). Otherwise, for \(0< p < e\), let \(q = q(p)\) be the nontrivial solution of the equation \(p(\log p  1) = q(\log q  1)\).
6 Analysis of an Illustrative Model: 2D \(\beta \)Ensembles
In this section, we will provide proof sketches for the convergence of the conditional distributions of the particles and of the JLM law for the finite \(\beta \)Ginibre ensembles. In the next section, we will briefly describe our proof from [33] for the zeros of the GEF. We recall that the twodimensional \(\beta \)Ginibre ensemble consists of N particles, whose joint probability density, with respect to the Lebesgue measure on \(\mathbb {C}^N\), is given by (6).
Since, at the moment, a limiting object for the finite \(\beta \)Ginibre ensemble is not known to exist, we choose a different limiting procedure than the one in the paper [40] (see Sect. 5.1). We fix a scaling parameter \(\alpha \ge 1\) and consider the asymptotics in terms of the large parameter \(R = \sqrt{\frac{N}{\alpha }}\). Heuristically, with the parameter \(\alpha \), the finite system resembles the (hypothetical) infinite system up to distances less than \(\sqrt{\alpha } R\) from the origin. In this setting, there are typically about \(R^2\) particles in the disk \(D(0,R) = \left\{ z \le R \right\} \). We again denote the number of particles in this disk by n(R).
We will illustrate below the proof of some of the predictions in Conjecture 5.1 above (using the different scaling procedure above). The proof in the case of overcrowding is similar. For \(a\ge 2\), one has to choose the value of the scaling parameter \(\alpha \) depending on a, b, and also R (if \(a > 2\)).
Theorem 6.1
Our proof of Theorem 6.1 proceeds via large deviation type estimates, and for \(a\le 4/3\), the error in our estimates (see Proposition 6.1) overwhelms the leading term. In the range \(a \in (1,\frac{4}{3}]\), establishing the JLM law for the \(\beta \)Ginibre ensembles with \(\beta \ne 2\) is an interesting open problem.
The following result describes the conditional limiting distribution, where m is the Lebesgue measure on \(\mathbb {C}\) and \(m_\mathbb {T}\) is the uniform probability measure on the unit circle \(\{z = 1\}\).
Theorem 6.2
Remark 6.1
We write \(\mathbb {P}_F\) (resp. \(\mathbb {E}_F\)) for the conditional probability (resp. expectation) on the event F.
Remark 6.2
6.1 Deviation Inequality for Linear Statistics
Theorem 6.2 follows from the following deviation inequality:
Proposition 6.1
Remark 6.3
The proposition is nontrivial only for \(\lambda \ge C R \sqrt{\log R}\); hence we may assume this holds below.
Claim 1
Proof
See the similar [33, Claim 11]. \(\square \)
6.2 Approximation of the Joint Density
We start with an asymptotic estimate for the normalizing constant \(Z_N^\beta \) (see [43, Corollary 1.5]).
Proposition 6.2
6.2.1 Smoothed Empirical Measure
Claim 2
6.2.2 Upper Bound for the Joint Density
6.3 Proof of Proposition 6.1
6.4 Outline of the Proof of Theorem 6.1
Remark 6.4
6.4.1 Lower Bound

There are at most \(p R^2\) points inside D (the open unit disk).

Pointtopoint separation: \(w_j^0  w_k^0 \ge C t\) for all \(j \ne k\).

Separation from boundary: \(\bigcup _{w_j^0 \in D} D(w_j^0, t) \subset D\).
 The points approximate the minimizing measure$$\begin{aligned} \left U_{\mu _{\underline{w}^0}^t} (z)  U_{\overline{\mu }_p^\alpha } (z)\right \le \frac{C \log N}{N} ,\quad \forall z \in \mathbb {C}. \end{aligned}$$(12)
Remark 6.5
One can construct such a ‘good’ set of points directly, using the radial symmetry of the problem. Another possibility is to use Fekete points for (essentially) the measure \(\overline{\mu }_p^\alpha \) (that is, weighted Fekete points with respect to the weight given by \(U_{\overline{\mu }_p^\alpha }\)). A small difficulty with the second approach is that p can depend on N (the number of points).
6.5 Minimizing Measures
The following lemma gives a characterization of the (constrained) minimizers of the functional \(I_\alpha \) (cf. [33, Lemma 10]).
Lemma 6.1
Proof
Let \(\mu \in \mathcal {C}\) be a probability measure such that \(\mu \ne \mu _0\). Without loss of generality, we assume \(\mu \) has compact support and finite logarithmic energy. It is known ([63, Lemma I.1.8]) that \( \Sigma (\mu  \mu _0) > 0\).
Claim 3
Remark 6.6
The above result also holds in the range \(p>1\).
7 Analysis of the GEF Zeros Process
Roughly speaking, with this choice of the parameters, the scaled GEF \(F(Rz)\) (defined in (1)) and the polynomial \(P_{\alpha ,R}\) have a very similar bevavior inside a disk of radius \(\sqrt{\beta }\), as long as \(\beta \ll \alpha \). Therefore, by taking \(\alpha \) large, we can obtain an understanding of the conditional intensity of the Gaussian zeros (under conditioning by \(\mathcal {H}_R\)) by analyzing the same problem for the polynomials.
Remark 7.1
It turns out that in order to carry out this approximation scheme, we need to let \(\alpha \rightarrow \infty \) logarithmically with R. A drawback of this is that we cannot use ‘offtheshelf’ large deviation principles for empirical measures of random polynomial zeros such as [67].
7.1 Joint Probability Density and the Limiting Functional
Remark 7.2
The global minimizer of the functional \(I^Z_\alpha \) is the uniform probability on the disk \(\{z \le \sqrt{\alpha }\}\), which we denoted by \(\overline{\mu }^\alpha _{\mathrm {eq}}\).
7.1.1 Deviation Inequality
The analysis is done in a similar way to the case of the \(\beta \)Ginibre ensembles (Sects. 6.1 and 6.2).
Remark 7.3
The actual proof is technically more involved, in large part because the choice of the value of N has to be random.
7.1.2 Lower Bound for \(\mathbb {P}\left[ F^Z_p(R)\right] \)
Because of the circular symmetry of the problem, one can use analytic techniques to obtain the lower bound for the probability of the event \(F^Z_p(R)\), which are not available in the case of the \(\beta \)Ginibre ensembles.
7.1.3 The Minimizing Measures
In order to find the limiting conditional measures for the GEF zeros process, one has to identify the (probability) measure \(\overline{\nu }_p^\alpha \) which minimizes the functional \(I^Z_\alpha \) over the set \(\mathcal {F}_p\). The interested reader can find the details in [33, Section 5].
8 Conditional Intensities in 1D
The conditional intensity around a hole has been studied in 1D (where it is usually called a ‘gap’), in the context of the GUE (Gaussian Unitary Ensemble) point process in [46]. In this section, we briefly describe the approach in [46] and compare and contrast the results therein with the situation we already discussed in 2D.
In [46], the authors approach this problem by obtaining a singular integral equation for \(f_w\), which is deduced essentially from a variational perturbation of the LDP rate functional around the minimizing measure. The authors then illustrate two approaches to solving this singular integral equation  one of them being a Riemann–Hilbert approach and the other being via an application of Tricomi’s theorem ([66]).
In [46], the authors also study the particle distribution for atypical indices for the GUE. The index \(N_+\) of a configuration of N particles on \(\mathbb {R}\) is the number of particles on \(\mathbb {R}_+\). By symmetry, the typical value of \(N_+/N=1/2\). Using similar variational techniques (as discussed above) on the LDP rate functional for the GUE ensemble, in [46] the authors obtain the asymptotics of the probability of an atypical index \(N_+/N = c \ne 1/2\), as well as the typical particle profile given such an atypical index. This can be compared with [6], where a similar problem has been studied for the Ginibre ensemble using free boundary techniques.
9 Simulation of the Hole Event and Numerical Aspects
Numerical methods to effectively simulate the distribution of zeros (or eigenvalues) conditioned on a large hole is a challenging problem, because of the rarity of the hole event and the strong correlation among the particles. E.g., for the Ginibre ensemble, the asymptotics of the hole probabilities can be understood via the statistical independence of their absolute values, but this approach is not useful for simulations, because it carries no information about the correlations between the particles.
In the present paper, Fig. 1 is obtained by simulating the GEF (and then the zeros thereof) under the conditions (on the coefficients) that produce the tight lower bound for the hole probability (as alluded to in Sect. 7.1.2). Figure 2 is obtained by manually moving the eigenvalues of a Ginibre matrix (that are inside the disk) to the disk’s boundary. In [46], a modified Metropolis Hastings algorithm was studied for simulating such conditional distributions (conditioned on hole, overcrowding, or deficiency events) for the GUE process in 1D.
To our knowledge, very little is known about this problem in dimensions greater than one. In 1D, a similar numerical problem has been addressed by [18] in the weighted case (using an approach involving iterated balayage), and by [19, 36] in the unweighted case. Even in the 1D situation, there are various assumptions on the Riesz measure corresponding to the weight V, which would be of interest to relax.
It remains a nontrivial and highly interesting question to devise efficient numerical techniques to simulate the particle configurations for the hole (and, in the same vein, for overcrowding and deficiency) events. In the case of the hole event for the Ginibre ensemble, one can use weighted Leja points (see [63, Chapter V]) to approximate the most likely eigenvalue configurations (given by the weighted Fekete points, mentioned in Sect. 4.3). Finding a similar method for the GEF zeros process seems to be an interesting problem.
Notes
Acknowledgements
We thank the authors of the paper [46] for allowing us to use the picture in Fig. 7. We thank Diego Ayala for allowing us to use the picture in Figure 8. We thank the anonymous referee for numerous helpful suggestions. The work of S.G. was supported in part by the ARO Grant W911NF1410094, the NSF Grant DMS1148711 and the NUS Grant R146000250133.
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