DETERMINANTAL COULOMB GAS ENSEMBLES WITH A CLASS OF DISCRETE ROTATIONAL SYMMETRIC POTENTIALS SUNG-SOO BYUN AND MENG YANG

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DETERMINANTAL COULOMB GAS ENSEMBLES WITH A CLASS OF

DISCRETE ROTATIONAL SYMMETRIC POTENTIALS

SUNG-SOO BYUN AND MENG YANG

Abstract. We consider determinantal Coulomb gas ensembles with a class of discrete rotational

symmetric potentials whose droplets consist of several disconnected components. Under the insertion

of a point charge at the origin, we derive the asymptotic behaviour of the correlation kernels both in

the macro- and microscopic scales. In the macroscopic scale, this particularly shows that there are

strong correlations among the particles on the boundary of the droplets. In the microscopic scale,

this establishes the edge universality. For the proofs, we use the nonlinear steepest descent method

on the matrix Riemann-Hilbert problem to derive the asymptotic behaviours of the associated planar

orthogonal polynomials and their norms up to the ﬁrst subleading terms.

1. Introduction and main results

We consider a conﬁguration {zj}N

1of Npoints in Cwith joint probability distribution

(1.1) dPN=1

ZNY

j>k |zj−zk|2

j=1

e−NQ(zj)dA(zj), dA(z) := d2z

π,

where ZNis the normalisation constant and Q:C→Ris a suitable function called external potential.

The ensemble (1.1) corresponds to the eigenvalue system of the random normal matrix model, which

can be interpreted as the two-dimensional Coulomb gas ensemble at a speciﬁc inverse temperature

β= 2. For a recent account of the theory and various topics on the Coulomb gas ensemble, we refer

the reader to [41] and references therein.

By deﬁnition, the k-point correlation function RN,k of the system (1.1) is given by

(1.2) RN,k(z1,··· , zk) := N!

(N−k)! ZCn−k

j=k+1

dA(zj).

The normalised 1-point function 1

NRN,1corresponds to the macroscopic density of the model. It is

well known that as N→ ∞, the empirical measure of {zj}N

1converges to Frostman’s equilibrium

measure, see e.g. [5,25]. In particular, the system {zj}N

1tends to occupy certain compact set Scalled

the droplet.

The k-point function RN,k can be eﬀectively analysed in terms of the correlation kernel. To be more

concrete, let pk≡pk,N be the k:th orthonormal polynomial with respect to the weighted Lebesgue

measure e−NQ dA:

(1.3) ZC

pj(z)pk(z)e−NQ(z)dA(z) = δjk,

Date: October 11, 2022.

Sung-Soo Byun was supported by Samsung Science and Technology Foundation (SSTF-BA1401-51), by the National

Research Foundation of Korea (NRF-2019R1A5A1028324) and by a KIAS Individual Grant (SP083201) via the Center

for Mathematical Challenges at Korea Institute for Advanced Study. Meng Yang was supported by VILLUM FOUNDEN

research grant no. 29369 and the NNF grant NNF20OC0064428.

arXiv:2210.04019v1 [math-ph] 8 Oct 2022

2 SUNG-SOO BYUN AND MENG YANG

where δjk is the Kronecker delta. We write

(1.4) KN(z, w) = e−N

2(Q(z)+Q(w))

N−1

j=0

pj(z)pj(w)

for the weighted reproducing kernel of analytic polynomials (of degree less than N−1) in L2(e−N Q dA).

Then the k-point function RN,k in (1.2) is expressed as

(1.5) RN,k(z1,··· , zk) = det hKN(zj, zl)ik

j,l=1.

We mention that the correlation kernel can be deﬁned up to a sequence of cocycles, i.e.

det hKN(zj, zl)ik

j,l=1 = det hgN(zj)gN(zl)·KN(zj, zl)ik

j,l=1,

where gNis a continuous unimodular function.

Due to the property (1.5), the system (1.1) is also called the determinantal Coulomb gas ensemble.

Moreover, this naturally calls for the investigation of various asymptotic behaviours of KNas N→ ∞.

Here, one has to distinguish two cases, the macroscopic scale and the microscopic scale.

The asymptotic behaviour in the microscopic scale is closely related to the universality principle

in random matrix theory. To describe the local statistics of the model at a given base point p∈S,

one needs to investigate the asymptotic behaviour of the function

(1.6) (z, w)7→ KNp+eiθ z

pN∆Q(p), p +eiθ w

pN∆Q(p).

Here if p∈∂S, the angle θ∈[0,2π) is chosen so that eiθ is outer normal to ∂S at p, and otherwise

θ= 0.We remark that the speciﬁc choice of the rescaling factor pN∆Q(p) in (1.6) (which is often

called the “unfolding”) comes from the fact that 1

NRN,1(p)∼∆Q(p).

For the bulk case when p∈Int S, it was shown in [8] that for a general external potential Q,

(1.7) KNp+z

pN∆Q(p), p +w

pN∆Q(p)→G(z, w) := ez¯w−|z|2

2−|w|2

Here Int Sstands for the interior of S, the largest open set of S, and the universal scaling limit Gin

(1.7) is called the Ginibre kernel [32]. For the edge case when p∈∂S, it was shown in a fairly recent

work [35] that for a general external potential Q,

(1.8) KNp+eiθ z

pN∆Q(p), p +eiθ w

pN∆Q(p)→G(z, w)1

2erfc z+ ¯w

√2.

The class of potentials Qcovered in [35] is quite general but dependent on the topology of the associated

droplet.

Turning to the macroscopic scale, recently, Ameur and Cronvall [7] made signiﬁcant results on the

asymptotic behaviour of KN(z, w). For the Ginibre ensemble with Q(z) = |z|2, they obtained a precise

asymptotic result. Namely, it was obtained in [7, Theorem 1.1] that

(1.9) KN(z, w) = rN

2π

z¯w−1(z¯w)NeN−N

2(|z|2+|w|2)·1 + O(1

N),

where z6=wand z¯wis outside the Szeg˝o curve

(1.10) S1:= {z∈C:|z| ≤ 1,|z e1−z|= 1}.

DETERMINANTAL COULOMB GAS ENSEMBLES OF LEMNISCATE ARCHIPELAGO TYPE 3

Here, we intentionally add the subscript 1 since (1.10) can be realised as a special case of Sain (1.12)

below with a= 1. We stress that [7, Theorem 1.1] indeed provides a closed form of large-Nexpansions

of KN.Let us also mention that (1.9) can also be interpreted as an asymptotic result of the incomplete

gamma function with complex argument, see [7, Section 1.4] and (A.4). (Cf. this was crucially used

in a recent work [20].)

Beyond the Ginibre ensemble, Ameur and Cronvall considered general external potential Qand

derived the uniform asymptotic behaviour of KN(z, w) for z, w outside the droplet, see [7, Theorem

1.3]. (We also refer to [3,31,42] for similar results on the elliptic Ginibre ensemble.) In particular, they

showed that there are strong correlations among the particles on the boundary of the droplet. One

of the main ingredients in their proof is the asymptotic behaviour of planar orthogonal polynomials

(1.3) due to Hedenmalm and Wennman [35].

The above-mentioned results were mainly obtained for the case where the external potential Q

is ﬁxed, i.e. independent of N. Nevertheless, the case when Qdepends on Nis also interesting in

particular in the context of the insertion of point charges [11] also known as the induced ensembles [30]

or spectral singularities [36]. (Another important example that N-dependence of the potential being

crucial is the almost-Hermitian regime, see e.g. [6].)

Furthermore, in [35] (and also in the follow-up paper [34]), the asymptotic behaviours of planar

orthogonal polynomials were constructed in terms of a conformal map from the outside the droplet

onto the outside the unit disc. Accordingly, the asymptotic result in [35] was obtained for the potential

Qwhose associated droplet is simply connected as a domain on the Riemann sphere. As a consequence,

the edge universality (1.8) in [35] as well as the Szeg˝o type asymptotic behaviour in [7] were obtained

under the assumption that the associated droplet does not have several disconnected components.

In this work, we aim to provide concrete examples of asymptotic results for the ensembles with a

class of N-dependent potentials associated with disconnected droplets, see Figure 1.

Figure 1. Illustration of the lemniscate archipelago and zooming process

1.1. Main results. We now precisely introduce our models. It is more convenient to begin with a

special case when removing the discrete rotational symmetry. In this case, the model corresponds to

the induced Ginibre ensemble [30] with the potential

(1.11) Qc(z)≡QN,c(z) := |z|2−2c

Nlog |z−a|,

4 SUNG-SOO BYUN AND MENG YANG

where c > −1 and a≥0. From the statistical physics point of view, we insert a point charge cat a

given point a. When cis an integer, the ensemble (1.1) with the potential (1.11) can also be realised

as the Ginibre ensemble conditioned to have eigenvalue awith multiplicity c.

The orthogonal polynomials associated with (1.11) reveal a discontinuity at c= 0. Namely, if

c= 0, since the orthogonal polynomials are simply given by monomials, all the zeros are located at

the origin. On the other hand, in [38], it was shown that for any c6= 0 and a > 1, the zeros of

orthogonal polynomials tend to occupy the limiting skeleton (also known as mother body, cf. [33])

(1.12) Sa:= nz∈C: log |z| − aRe z= log 1

a−1,Re z≤1

ao.

Note that Sacrosses the point 1/a. The limiting skeleton Saplays an important role in the asymptotic

behaviours of the orthogonal polynomials. See Figure 2 for the shape of Sa.

(a) a= 4 (b) a= 2 (c) a= 4/3

Figure 2. The plots display Sa. The red dots show the origin and 1/a. The green

dashed lines indicate the branch cuts in Theorem 1.1.

In our ﬁrst result, we obtain the following asymptotic behaviour of KNin the macroscopic scaling.

Theorem 1.1. (Macroscopic asymptotic of the induced Ginibre ensemble) Let Qbe the

induced Ginibre potential (1.11) with a > 1and c > −1 (c6= 0). Suppose that zand ware outside Sa,

and |z−w|> δ for some δ > 0. Then we have

(1.13) KN(z, w) = rN

2π

z¯w−1z

1−az

¯w

1−a¯wc(z¯w)N|(z−a)(w−a)|ceN−N

2(|z|2+|w|2)·1+O(1

N).

Here the branch cuts for the variables zand ¯ware the line segment [0,1/a].

Note that if we formally put c= 0, the formula (1.13) corresponds (1.9). We mention that the

condition zand wbeing outside the limiting skeleton was also considered in [3] for the elliptic Ginibre

ensemble. (In this case, the limiting skeleton is a line segment connecting two foci of the ellipse.)

In the spirit of the edge universality (1.8), we obtain the following.

Theorem 1.2. (Boundary scaling limits of the induced Ginibre ensemble) Let Qbe the

induced Ginibre potential (1.11) with a > 1and c > −1. Let pbe a point on the unit circle. Then as

N→ ∞, we have

NKNp+p z

√N, p +p w

√N→G(z, w)1

2erfc z+ ¯w

√2,

uniformly for z, w on compact subsets of C.

DETERMINANTAL COULOMB GAS ENSEMBLES OF LEMNISCATE ARCHIPELAGO TYPE 5

We now discuss the ensemble with discrete rotational symmetry. For a≥0 and d∈N, let

(1.14) V(z) = 1

d|zd−a|2, Vc(z)≡VN,c := V(z)−2c

Nlog |z|.

We refer to [10,21,27] and references therein for recent studies on such models. Note that the induced

Ginibre potential (1.11) corresponds to (1.14) with d= 1 up to a translation. It is well known that

the droplet SVassociated with the potential Vis given by

(1.15) SV:= {z∈C:|zd−a| ≤ 1},

see e.g. [14, Lemma 1]. The density with respect to dA is given by

(1.16) ∆V(z) = d|z|2d−2.

Due to the explicit formula (1.15), one can easily notice that if a < 1, SVis connected. On the

other hand, if a > 1, SVconsists of d-connected components that we call the lemniscate archipelago

following [7], see Figure 1.

We denote by qc

j,N the orthonormal polynomials associated with the weighted measure e−NVcdA:

(1.17) ZC

j,N (z)qc

k,N (z)|z|2ce−N V (z)dA(z) = δjk.

For a > 1, it was shown in [13,38] that as j→ ∞, the (non-trivial) zeros of qc

j,N tend to accumulate

on the curve

(1.18) Sd

a:= nz∈C: log |zd−a|+aRe zd= log 1

a−1 + a2,Re zd≥a−1

ao.

Notice that (1.18) and (1.12) are related by the mapping z7→ a−zd. See Figure 3 for the shape of

(a) d= 2 (b) d= 3 (c) d= 4

Figure 3. The plots display ∂SV(black) and Sd

a(blue), where a= 1.1. The red dots

indicate (a−1

a)1/dωkand a1/dωk. The green dashed lines are the branch cuts in (1.20).

Let us consider the associated correlation kernel

(1.19) Kc

N(z, w) := |zw|ce−N

2(V(z)+V(w))

N−1

j=0

j,N (z)qc

j,N (w).

The kernel (1.19) corresponds to the reproducing kernel (1.4) associated with the potential Q=Vc.

We derive the asymptotic behaviours of Kc

Nin the macroscopic scale.

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DETERMINANTALCOULOMBGASENSEMBLESWITHACLASSOFDISCRETEROTATIONALSYMMETRICPOTENTIALSSUNG-SOOBYUNANDMENGYANGAbstract.WeconsiderdeterminantalCoulombgasensembleswithaclassofdiscreterotationalsymmetricpotentialswhosedropletsconsistofseveraldisconnectedcomponents.Undertheinsertionofapointchargeattheorigin,w...

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DETERMINANTAL COULOMB GAS ENSEMBLES WITH A CLASS OF DISCRETE ROTATIONAL SYMMETRIC POTENTIALS SUNG-SOO BYUN AND MENG YANG

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