Hole probabilities for β -ensembles and determinantal point processes in the complex plane

We compute the exact decay rate of the hole probabilities for β -ensembles and determinantal point processes associated with the Mittag-Lefﬂer kernels in the complex plane. We show that the precise decay rate of the hole probabilities is determined by a solution to a variational problem from potential theory for both processes.


Introduction and main results
Let U be an open subset of the complex plane. The probability that U contains no points of X , a point process (see [DVJ08,p. 7]) in the complex plane, is called hole/gap probability for U . The hole probability for various point processes in the complex plane has been studied extensively in the literature.
The hole probabilities for zeros of Gaussian analytic functions have been considered in [BNPS16,GN16,Nis10,Nis11,Nis12,ST05]. The asymptotics of the hole probabilities for the eigenvalues of the product of finite matrices with i.i.d. standard complex Gaussian entries have been calculated in [AS13]. For the asymptotics of the hole probabilities for the finite and infinite Ginibre ensembles, we refer to [AR16] and [Shi06]. In this article, we calculate the asymptotics of the hole probabilities for finite β-ensembles and determinantal point processes associated with the Mittag-Leffler kernels in the complex plane, which we describe in the next sections.
Observe that, for α > 0, g(r) = r α satisfies Assumption 1.1. The joint density of the set of points of X These ensembles appear in physics to explain the 2-dimensional Coulomb gas models (see, [HM13,Blo14]). In this model the coordinates of a point are the positions of particles, the parameter β corresponds to the inverse temperature and g corresponds to the external potential. If β = 2 and g(|z|) = |z| 2 , then the set of points of X (g) n,β has the same distribution as the eigenvalues of n × n random matrix with i.i.d. complex Gaussian entries with mean zero and variance 1 n , known as n-th Ginibre ensemble (see [Gin65], [HKPV09,p. 60]).
We calculate the hole probabilities for X (g) n,β and X (α) n for two classes of domains.
Before stating the result, we introduce a few notation and definitions. Let D(0, T ) denote the disk of radius T centered at the origin. T β denotes the unique solution, by (A3), of tg (t) = β. The weighted energy R (g) µ,β associated to µ and the minimum weighted energy R (g) U,β for U c , where U is an open subset of C, with the external field 2 β g are defined by where P(U c ) denotes the set of all compactly supported probability measures with support in U c . A probability measure µ with support in U c is said to be weighted EJP 23 (2018), paper 48. equilibrium measure of U c with the external field 2 U,β . For simplicity, we write equilibrium measure and minimum energy instead of weighted equilibrium measure and weighted minimum energy.
We consider the following two classes of domains. Let U be an open subset of D(0, T β ) such that (C1) there exists a sequence of open sets U m such that U ⊂ U m ⊆ D(0, T β ) with U m+1 ⊆ U m for all m and the equilibrium measure µ m of U m with the external field 2 β g converges weakly to the equilibrium measure µ of U with the external field 2 β g. (C2) there exists > 0 such that for every z ∈ ∂U there exists a η ∈ U c such that U c ⊃ B(η, ) and |z − η| = .
(1.2) Disks and annuli satisfy (C1) for general g (see Example 3.3 and Example 3.4). For g = t α , if U is an open set such that U ⊂ rU := {rz : z ∈ U } for all r > 1, then U satisfies (C1) (see Remark 3.1 (3)). All convex domains satisfy (C2) with any > 0. Annulus is not a convex domain but it satisfies (C2). In general verifying (C1) is much harder than verifying (C2). The following two results gives the hole probabilities for X (g) n,β . Theorem 1.2. Let U be an open subset of D(0, T β ). The following statements are true: As a corollary of the above result we get the hole probabilities for X for α > 0 (or α ≥ 1 respectively).

Determinantal point processes with Mittag-Leffler kernels
∞ denote the determinantal point process in the complex plane with with respect to Lebesgue measure on the complex plane, where E a,b (z) denotes the Mittag-Leffler function (see [HMS11]), an entire function when a > 0 and b > 0, defined by .
∞ is the determinantal point process with kernel 1 π e zw− |z 2 | 2 − |w| 2 2 with respect to Lebesgue measure in the complex plane, known as infinite Ginibre ensemble (see [AR16], [HKPV09,p. 61 ]). The point process X    R k are independent and R α k ∼ Gamma( 2 α k, 1). Using this fact it can be shown, for The calculations for proving (1.4) in this method is similar to the proof of Theorem 1.1 in [AR16], we skip the calculations. But we obtain (1.4) using Theorem 1.4 (Remark 5.1). But the above method can not be applied to calculate the hole probabilities for general sets. We use a technique from potential theory, developed in [AR16], to calculate the hole probabilities for general sets. The next result gives the hole probabilities for for α > 0 (or α ≥ 1 respectively).

Weighted equilibrium measure and weighted minimum energy
Let U be an open subset of C. For the ease of notation, we use the terms R (g) µ,2 and T 2 respectively. The equilibrium measure is the uniform probability measure on the unit disk and R (g) φ is 3 4 when g(r) = r 2 and U = ∅ (see [AR16,ATW14,ASZ14]). In general, when U = ∅ and g satisfies Assumption 1.1, the equilibrium measure µ is given by (1.5) where T > 0 such that T g (T ) = 2, and the minimum energy is R (g) See [ST97, Theorem IV.6.1] for the proof of (1.5). The equilibrium measure for U c , where U is an open subset of D, has been studied in [AR16,ASZ14] when g(r) = r 2 .
In the next result we calculate the equilibrium measure and R A measure µ bal is said to be the balayage measure associated to a finite Borel measure µ on a bounded open set U if µ bal (∂U ) = µ(U ), µ bal (B) = 0 for every Borel polar set B ⊂ C and p µ bal (z) = p µ (z) for quasi-everywhere z ∈ U c , where p µ (z) := − log |z − w|dµ(w) denotes the logarithmic potential of µ at the point z ∈ C. A property is said to hold quasi-everywhere (q.e.) on a set E ⊂ C if it holds everywhere on E except some polar set. A set E is said to be polar if the energy is infinite, i.e., − log |z − w|dµ(z)dµ(w) = ∞ for all compactly supported probability measures µ with support in E.
be an open set, where T g (T ) = 2. Then the equilibrium measure for U c is ν = µ out + µ bal and R (g) (1.6) where µ out and µ in are restrictions of the measure µ, as in (1.5), on to the sets D(0, T )\U and U respectively, i.e., otherwise and µ bal is the balayage measure on ∂U associated to µ in .
Note that to calculate R (g) U we need to compute the balayage measure. In general computing balayage measure is not easy. In Section 3.1 we compute the balayage measure associated to µ in when U is a disk or an annulus centered at the origin.
The rest of the article is organized as follows. In Section 2 we shall recall a few basic definitions and facts from the potential theory. In Section 3 we present the proof of Theorem 1.5. In Section 4 we give the proofs of Theorem 1.2 and Corollary 1.3. We prove Theorem 1.4 in Section 5. In the final section we prove Lemmas 4.2 and 4.1.

Preliminaries
2 is an admissible weight function when g satisfies Assumption 1.1 ((A4) is not required).
The following fact, which characterizes the equilibrium measure uniquely, will be used repeatedly.
2 is an admissible weight function and U is an open subset of the complex plane, then there exists an unique equilibrium measure ν, for U c with external field g(|z|). The equilibrium measure ν has compact support and R (g) ν is finite.
Further, ν satisfies the following conditions for some constant C. Also, the above conditions uniquely characterize the equilibrium measure, i.e. a probability measure with compact support in U c and finite energy, which satisfies the conditions (2.1) and (2.2) for some constant C, is the equilibrium measure for U c with external field g(|z|).
Fekete points: Let E ⊂ C be a closed set and ω = e − g(|z|) 2 be an admissible weight function. Define . The points z * 1 , z * 2 , . . . , z * n in a n-th weighted Fekete set F n are called n-th weighted Fekete points. We write Fekete points instead of weighted Fekete points. The set {z * 1 , z * 2 , . . . , z * n } always exists, since E is a closed set and w is an upper semi-continuous. But the sets need not be unique. Observe that, for η ∈ E, Which implies that, for η ∈ E, (2.4) Moreover, the discrete uniform probability measures on n-th Fekete sets converge weakly to equilibrium measure ν, i.e. lim n→∞ ν Fn = ν, where ν Fn is the discrete uniform measure on F n . For the proofs of (2.4) and (2.5), see

Proof of Theorem 1.5
In this section give the proof of Theorem 1.5 and some examples of the balayage measures. Before that we make some remarks, which will be used in calculating the hole probabilities for X (α) ∞ and X (g) n,β . Remark 3.1.
In particular if α = 2 then T = 1. The equilibrium measure µ is uniform measure on D, i.e., dµ(z) = 1 π dm(z) on D and R 3. Theorem 1.5 implies that the equilibrium measures µ m of U m converge to the equilibrium measure µ of U iff the balayage measures µ bal m associated to µ m Um converge to the balayage measure µ bal associated to µ U and µ out m converges to µ out . In particular, for g = t α , if U is an open set such that U ⊂ rU for all r > 1, then U satisfies (C1). Because the balayage measure µ bal r on ∂(rU ) is given in terms of the balayage measure µ bal on ∂U as µ bal r (rB) = r α µ bal (B) for any measurable set B ⊂ ∂(U ). Therefore µ bal r converges weakly to µ bal as r → 1. Remark 3.2. Replacing g by 2 β g in (1.5) and Theorem 1.5, we have 1. The equilibrium measure for C associated to the external field g(|z|) β is supported on D(0, T β ) and given by The minimum energy is given by 2. Let U be an open subset of D(0, T β ). Then from Theorem 1.5, we have where µ bal is the balayage measure on ∂U associated to µ in = µ β U .
3. If g(t) = t α . Then T β = β α 1 α is the radius of the support of the equilibrium measure. In particular for α = 2, β = 2 we have T 2 = 1, the radius of the support of the equilibrium measure associated to the quadratic external field.
Proof of Theorem 1.5. Let µ be the equilibrium measure for C, as in (1.5). Let µ = µ out + µ in , where µ out and µ in are µ restricted to U c and U respectively. Fact 2.2 implies that there exists a measure µ bal on ∂U such that µ bal (∂U ) = µ in (U ), µ bal (B) = 0 for every Borel polar set and Define ν = µ out + µ bal . Then we have that the support of ν is D(0, T )\U and where the last equality follows from the facts that lim r→0 + rg (r) = 0 and T g (T ) = 2.
Therefore we get On other hand, for |z| > T we get last equality by using the facts lim r→0 + rg (r) = 0 and T g (T ) = 2. The function f (r) = log 1 The energy of the measure ν, is finite. The second equality follows from the fact that ν(B) = 0 for all Borel polar sets B. So, ν has finite energy and satisfies conditions (2.1) and (2.2). Therefore, by Fact 2.1, ν is the equilibrium measure for U c .
Value of R (g) Therefore, by (3.4), we have The result follows from the fact that R (g) EJP 23 (2018), paper 48.

Examples of balayage measures
We calculate the balayage measures for disks and annuli centered at the origin associated to µ U , where µ as in (1.5).
Example 3.3. Let U = D(0, a) be a disk of radius a < T centered at origin. Then the balayage measure on ∂U associated to µ U , where µ as in (1.5), is Example 3.4. Let U = {z : 0 < a < |z| < b < T } be an annulus centered at the origin with the inner radius a and the outer radius b. Then the balayage measure on ∂U associated to µ U , where µ as in (1.5), is where λ is given by (3.5) Example 3.6. Suppose g(t) = t α , for α > 0 and U = {z : 0 < a < |z| < b < ( 2 α ) 1 α } is an annulus centered at the origin with the inner radius a and the outer radius b. Then the balayage measure on ∂U is The minimum energy is given by where the last equality follows from the Fact 2.3. By equating p µ in (z) = p µ bal (z) for |z| ≤ a, we get Similarly, it can be shown that p µ in (z) = p µ bal (z) for |z| ≥ b for all choice of λ. Therefore p µ in (z) = p µ bal (z) if z ∈ U c for the above particular choice of λ. Hence the result.

Proofs of Theorem 1.2 and Corollary 1.3
In this section we give the proofs of Theorem 1.2 and Corollary 1.3.
Proof of Corollary 1.3. Recall X (α) n is the determinantal point process with kernel K (α) n (z, w) with respect to Lebesgue measure in the complex plane. Equivalently, the vector of points of X (α) n (in uniform random order) has density α n n!(2π) n n−1 k=0 Γ( 2 α (k + 1)) with respect to Lebesgue measure on C n . Therefore we have where Z (α) n denotes the normalizing constant, for all α > 0. On the other hand, for g(t) = t α , g is bounded on [0, T 2 + 1] only when α ≥ 1. Therefore if U satisfies (C2), then the last equality holds for α ≥ 1.  n,β has the following asymptotics In the next two subsections we give the proofs of (I), (II), (III) and (IV).
Therefore we have . Therefore from (4.3), we have where a = U c e −g(|z|) dm(z). (A4) implies that a is finite. By taking logarithm and diving by n 2 in both sides, we get lim sup n→∞ 1 n 2 log P[X Note that, by the same arguments it can be shown that n,β ≤ a n (δ ω n (C)) β 2 n(n−1) , where a = C e −g(|z|) dm(z). Therefore by (2.4), we have lim sup (4.4)

Lower bound
We prove (II) using the following lemma, we give the proof of the lemma in Appendix.
The last inequality follows from the facts that U ⊂ U m and A m ⊂ A. We have where the last equality follows from (2.1). Observe that the constant C β,g does not depend on the domain U m (see the proof of Theorem 1.5 for the details). Since µ m converges weakly to µ and g is continuous function, we have g(|z|)dµ m (z) → g(|z|)dµ(z) as g(|z|) is a bounded continuous function on D(0, T β )\U m . Therefore, from (4.5), R Hence the result. Now we prove (III) using the following lemma, which provides separation between n-th Fekete points. The lemma says that two Fekete points cannot be too close. This is not the tightest separation result but it suffices for our purpose. The separation of Fekete points has been studied by many authors, e.g., see [AR16, AOC12, BLW08] and references therein.
1 n 3 for some constant C > 0 (which does not depend on n). Proof of (III). Let z * 1 , z * 2 , . . . , z * n be n-th Fekete points for U c with the weight function ω(z) = e − g(|z|) β . Since the support of the Fekete points is contained in support of equilibrium measure (see [ST97, Chapter III Theorem 2.8]), it follows that |z * | ≤ T β for = 1, 2, . . . , n. Let B = U c ∩ B(z * , C n 4 ) for = 1, 2, . . . , n. Then, for large n, we have By Lemma 4.2, for large n, we have |z * i − z * j | ≥ C n 3 for i = j, for some constant C independent of n. Suppose z i ∈ B(z * i , C n 4 ) and z j ∈ B(z * j , C n 4 ) for i = j, then for large n Therefore we have Since g is bounded on [0, T β + 1], therefore |g(|z|) − g(|w|)| ≤ K.|z − w| for some constant K, for all z, w ∈ D(0, T β + 1). Therefore for large n, for z i ∈ B(z * i , C n 4 ), i = 1, 2, . . . , n, where C = CK/2. Hence for large n, we have For large n, we have Bi dm(z i ) ≥ π( C 2n 4 ) 2 , i = 1, 2, . . . , n (condition (1.2) implies that B i contains at least a ball of radius C 2n 4 ). Hence we have Proof of (IV). By the same arguments as in the proof of Lemma 4.1 it can be shown that lim inf for all compactly supported probability measures µ in the complex plane. The result follows from (4.4) and (4.6).

Proof of Theorem 1.4
Before proving the theorem we have following remarks.
where U = D(0, a). Therefore by (3.5) we get where bU c = {z : 0 < cb < |z| < b} is an annulus with the inner radius cb and the outer radius b. Therefore by (3.6), for a = cb, we get 3. In particular α = 2 gives the asymptotics of the hole probabilities for infinite Ginibre ensemble X n (z, w) with respect to Lebesgue measure. The kernel K (α) n (z, w) can be expressed as EJP 23 (2018), paper 48.