Disk counting statistics near hard edges of random normal matrices the multi-component regime Yacin Ameur Christophe Charlier Joakim Cronvalland Jonatan Lenells

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Disk counting statistics near hard edges of random normal

matrices: the multi-component regime

Yacin Ameur∗, Christophe Charlier∗, Joakim Cronvall∗and Jonatan Lenells†

October 26, 2022

Abstract

We consider a two-dimensional point process whose points are separated into two disjoint

components by a hard wall, and study the multivariate moment generating function of the

corresponding disk counting statistics. We investigate the “hard edge regime” where all disk

boundaries are a distance of order 1

naway from the hard wall, where nis the number of points.

We prove that as n→+∞, the asymptotics of the moment generating function are of the form

exp C1n+C2ln n+C3+Fn+C4

√n+O(n−3

5),

and we determine the constants C1,...,C4explicitly. The oscillatory term Fnis of order 1 and

is given in terms of the Jacobi theta function. Our theorems allow us to derive various precise

results on the disk counting function. For example, we prove that the asymptotic ﬂuctuations

of the number of points in one component are of order 1 and are given by an oscillatory discrete

Gaussian. Furthermore, the variance of this random variable enjoys asymptotics described by

the Weierstrass ℘-function.

AMS Subject Classification (2020): 41A60, 60B20, 60G55.

Keywords: Oscillatory asymptotics, Moment generating functions, Random matrix theory.

1 Introduction and statement of results

In recent years there have been a lot of works on counting statistics of two dimensional point processes,

see e.g. [23,42,43,53,19,24,54,3,27,18,9] and references therein. The common feature of these

works is that they all deal exclusively with models for which the points condensate on a single

connected component (“the one-component regime”). In this paper we deviate from these earlier

works in that we study the disk counting statistics of a Coulomb gas (at inverse temperature β= 2)

whose points are separated into two disjoint components by a hard wall. Let us now introduce the

Coulomb gas model investigated in this work.

The Mittag-Leﬄer ensemble is the following joint probability density function

n!ZnY

1≤j<k≤n|ζk−ζj|2

j=1 |ζj|2αe−n|ζj|2b, ζ1, . . . , ζn∈C,(1.1)

where b > 0 and α > −1 are ﬁxed parameters and Znis the normalization constant. As n→+∞,

with high probability the random points ζ1, . . . , ζnaccumulate on the disk centered at 0 of radius b−1

according to the probability measure µ(d2z) = b2

π|z|2b−2d2z[39,50]. This determinantal point process

∗Centre for Mathematical Sciences, Lund University, 22100 Lund, Sweden. e-mails: yacin.ameur@math.lu.se,

christophe.charlier@math.lu.se, joakim.cronvall@math.lu.se

†Department of Mathematics, KTH Royal Institute of Technology, 10044 Stockholm, Sweden. e-mail:

jlenells@kth.se

arXiv:2210.13962v1 [math-ph] 25 Oct 2022

generalizes the complex Ginibre process (which corresponds to (b, α) = (1,0)) and has attracted a

lot of attention over the years, see e.g. [7,10,22].

In this paper we focus on the Mittag-Leﬄer ensemble with a hard wall that separates the random

points into two disjoint components. To be precise, let 0 < ρ1< ρ2< b−1

2b. We consider the

probability density

n!ZnY

1≤j<k≤n|zk−zj|2

j=1

e−nQ(zj), z1, . . . , zn∈C,(1.2)

where Znis the normalization constant and

Q(z) = (|z|2b−2α

nln |z|,if |z| ∈ [0, ρ1]∪[ρ2,+∞),

+∞,otherwise. (1.3)

Because ρ1, ρ2< b−1

2b, the macroscopic behavior of (1.2) is diﬀerent from that of (1.1); it is described

by a probability measure µhwhich is supported on {z∈C:|z| ∈ [0, ρ1]∪[ρ2, b−1

2b]}and has a singular

component on the circles of radii ρ1and ρ2. This measure can be computed using standard balayage

techniques [50] (we provide the details of this computation in Appendix A) and is given by

µh(d2z)=2b2r2b−1χ[0,ρ1]∪[ρ2,b−1

2b](r)dr dθ

2π+σ1δρ1(r)dr dθ

2π+σ2δρ2(r)dr dθ

2π,(1.4)

where z=reiθ,r > 0, θ∈(−π, π] and

σ1=σ?−bρ2b

1, σ2=bρ2b

2−σ?, σ?:= ρ2b

2−ρ2b

2 ln( ρ2

ρ1).(1.5)

The assumption 0 < ρ1< ρ2< b−1

2bimplies that bρ2b

1< σ?< bρ2b

2and hence σ1, σ2>0. The

quantity σ?is the mass of µhon {|z| ≤ ρ1}. Indeed, straightforward calculations show that

Z|z|≤ρ1

µh(d2z) = σ?,Z|z|≥ρ2

µh(d2z) = 1 −σ?,(1.6)

which means that for large n, the number of zj’s on {|z| ≤ ρ1}is roughly σ?nwith high probability

(see also Corollary 1.6 and the asymptotics (1.29) and (1.32) below). The point process (1.2) is

an example of a two-dimensional Coulomb gas that is rotation invariant (meaning that the density

(1.2) remains unchanged if all zj’s are multiplied by eiβ ,β∈R). This ensemble can be seen as a

conditional process where the points from (1.1) are conditioned on the hole event Hthat no ζj’s lie in

the annulus centered at 0 of radii ρ1and ρ2. The partition function Znof (1.2) is precisely equal to

ZnP(H), and its large nasymptotics were investigated in [1,2,25]; see also [38,35,40,5,6,4,36,43]

for related works on the hole event. The process (1.2) can also be realized as the eigenvalues of an

n×nrandom normal matrix Mtaken at random according to the probability density proportional

to e−ntrQ(M)dM, where “tr” is the trace and dM is the measure on the set of n×nnormal matrices

induced by the ﬂat Euclidian metric of Cn×n[47,28,33]. Correlation kernels near hard edges have

been studied in [60,48,51].

We will focus on the “the hard edge regime”, i.e. when all disk boundaries are a distance of order

naway from the hard edges {|z|=ρ1}and {|z|=ρ2}. (Further away from the hard edges, it turns

out that there are no oscillations and the situation gets similar to the one-component semi-hard edge

regime considered in [9].) Let us now be more speciﬁc. Let N(y) := #{zj:|zj|< y}be the random

b=1

2b= 1 b= 2

Figure 1: Illustration of the point processes corresponding to (1.2) with n= 4096, ρ1=3

5b−1

2b,

ρ2=4

5b−1

2b,α= 0 and the indicated values of b.

variable that counts the number of points of (1.2) in the disk centered at 0 of radius y. Our main

result is a precise asymptotic formula as n→+∞for the multivariate moment generating function

(MGF)

E2m

j=1

eujN(rj)(1.7)

where m∈N>0is arbitrary (but ﬁxed), u1, . . . , u2m∈R, and the radii r1, . . . , r2msatisfy r1<··· <

r2mand are merging at a critical speed in the following way

r`=ρ11−t`

n1

, t`≥0, ` = 1, . . . , m, (1.8)

r`=ρ21 + t`

n1

, t`≥0, ` =m+ 1,...,2m. (1.9)

As n→+∞, we prove that EQ2m

j=1 eujN(rj)enjoys asymptotics of the form

exp C1n+C2ln n+C3+Fn+C4

√n+O(n−3

5),(1.10)

and we determine C1, . . . , C4,Fnexplicitly. As corollaries of our various results on the generating

function (1.7), we also provide a central limit theorem for the joint ﬂuctuations of N(r1),...,N(r2m),

and precise asymptotic formulas for all cumulants of these random variables. Even for m= 1 our

results are new.

We now introduce the necessary material to present our results. For j≥0, deﬁne

Tj(x;~

t, ~u) =

`=1

ω`tj

`e−t`

b(x−bρ2b

1),ˆ

Tj(x;~

t, ~u) =

`=m+1

ω`tj

`e−t`

b(bρ2b

2−x),(1.11)

where

ω`=









eu`+···+u2m−eu`+1+···+u2m,if ` < 2m,

eu2m−1,if `= 2m,

1,if `= 2m+ 1,

(1.12)

and ~

t= (t1, . . . , t2m), ~u = (u1, . . . , u2m). Since the quantities T0(bρ2b

1;~

t, ~u) and ˆ

T0(bρ2b

2;~

t, ~u) are

independent of ~

t, we will simply write T0(bρ2b

1;~u) and ˆ

T0(bρ2b

2;~u) instead. Deﬁne

f(x;~

t, ~u) = −bρ2b

x−bρ2b

+α

bT1(x;~

t, ~u)−x

2bT2(x;~

t, ~u)

1 + T0(x;~

t, ~u) + ˆ

T0(bρ2b

2;~u),(1.13)

f(x;~

t, ~u) = bρ2b

bρ2b

2−x−α

bˆ

T1(x;~

t, ~u) + x

2bˆ

T2(x;~

t, ~u)

1−ˆ

T0(x;~

t, ~u) + ˆ

T0(bρ2b

2;~u).(1.14)

Let Ω := eu1+···+u2mand let

Q(~

t, ~u) := 1 + T0(σ?;~

t, ~u) + ˆ

T0(bρ2b

2;~u)

1−ˆ

T0(σ?;~

t, ~u) + ˆ

T0(bρ2b

2;~u).(1.15)

Our main result involves ln Q(~

t, ~u) and the next lemma, whose proof is given in Appendix B, shows

that this logarithm is well-deﬁned. It also shows that fand ˆ

fare smooth functions of x∈(bρ2b

1, bρ2b

2).

Lemma 1.1. Suppose ~u ∈R2m,t1>··· > tm≥0, and 0≤tm+1 <··· < t2m. For x∈[bρ2b

1, bρ2b

2],

it holds that

1 + T0(x;~

t, ~u) + ˆ

T0(bρ2b

2;~u)>0,1−ˆ

T0(x;~

t, ~u) + ˆ

T0(bρ2b

2;~u)>0.

The Jacobi theta function θ(z;τ) is deﬁned for z∈Cand Im τ > 0 by

θ(z;τ) = ∞

`=−∞

eπi`2τ+2πi`z.

This function satisﬁes

θ(z+ 1; τ) = θ(z;τ), θ(z+τ;τ) = e−2πize−πiτ θ(z), θ(−z) = θ(z),for all z∈C,(1.16)

see also [49, Chapter 20] for further properties. We are now ready to state our main result.

Theorem 1.2 (Merging radii at the hard edge: the multi-component regime).Let m∈N>0,b > 0,

0< ρ1< ρ2< b−1

2b,t1, . . . , t2m≥0, and α > −1be ﬁxed parameters such that t1>··· > tm≥0

and 0≤tm+1 <··· < t2m. For n∈N>0, deﬁne

r`=(ρ11−t`

n1

2b, ` = 1, . . . , m,

ρ21 + t`

n1

2b, ` =m+ 1,...,2m. (1.17)

For any ﬁxed x1, . . . , x2m∈R, there exists δ > 0such that

E2m

j=1

eujN(rj)= exp C1n+C2ln n+C3+Fn+C4

√n+On−3

5,as n→+∞(1.18)

uniformly for u1∈ {z∈C:|z−x1| ≤ δ}, . . . , u2m∈ {z∈C:|z−x2m| ≤ δ}, where

C1=bρ2b

j=1

uj+Zσ?

bρ2b

ln(1 + T0(x;~

t, ~u) + ˆ

T0(bρ2b

2;~u))dx +Zbρ2b

σ?

ln(1 −ˆ

T0(x;~

t, ~u) + ˆ

T0(bρ2b

2;~u))dx,

C2=−bρ2b

T1(bρ2b

1;~

t, ~u)

Ω+bρ2b

2ˆ

T1(bρ2b

2;~

t, ~u),

C3=−1

j=1

uj−α−2 ln(σ2/σ1) + ln Q(~

t, ~u)

4 ln(ρ2/ρ1)ln Q(~

t, ~u)

+Zσ?

bρ2b

1f(x;~

t, ~u) + bρ2b

1T1(bρ2b

1;~

t, ~u)

Ω(x−bρ2b

1)dx +Zbρ2b

σ?ˆ

f(x;~

t, ~u)−bρ2b

2ˆ

T1(bρ2b

2;~

t, ~u)

bρ2b

2−xdx

+bρ2b

T1(bρ2b

1;~

t, ~u)

Ωln bρb

√2π(σ?−bρ2b

1)−bρ2b

2ˆ

T1(bρ2b

2;~

t, ~u) ln bρb

√2π(bρ2b

2−σ?),

C4=√2Ibρ3b

T2(bρ2b

1;~

t, ~u)

Ω−ρb

T1(bρ2b

1;~

t, ~u)

Ω−ρ3b

T1(bρ2b

1;~

t, ~u)2

Ω2

−ρ3b

2ˆ

T2(bρ2b

2;~

t, ~u)−ρb

2ˆ

T1(bρ2b

2;~

t, ~u)−ρ3b

2ˆ

T1(bρ2b

2;~

t, ~u)2,

Fn= ln θ(σ?n+1

2−α+ln(σ2/σ1)+ln Q(~

t,~u)

2 ln(ρ2/ρ1);πi

ln(ρ2/ρ1))

θ(σ?n+1

2−α+ln(σ2/σ1)

2 ln(ρ2/ρ1);πi

ln(ρ2/ρ1)),(1.19)

and the constant I ∈ Ris given by

I=Z+∞

−∞ y e−y2

√πerfc(y)−χ(0,+∞)(y)y2+1

2dy ≈ −0.81367.(1.20)

In particular, since EQ2m

j=1 eujN(rj)is analytic in u1, . . . , u2m∈Cand is positive for u1, . . . , u2m∈

R, the asymptotic formula (1.18)together with Cauchy’s formula shows that

∂k1

u1. . . ∂k2m

u2mln E2m

j=1

eujN(rj)−C1n+C2ln n+C3+C4

√n=On−3

5,as n→+∞,

(1.21)

for any k1, . . . , k2m∈N, and u1, . . . , u2m∈R.

Remark 1.3. It is well-known that the theta function is a universal object of one-dimensional

point processes in the multi-cut regime, see e.g. [31,32,16,37,52,29,17,13,14,34,15,41,

26]. In dimension two, the emergence of this function was conjectured in [45, Section 1.5] and

proved in [25] in the context of large gap problems (or equivalently, partition function asymptotics

with hard edges). This function also appears in [8] in the study of microscopic correlations and

smooth macroscopic statistics of two-dimensional rotation-invariant ensembles with soft edges. To

our knowledge, Theorem 1.2 is the ﬁrst result on counting statistics of a two-dimensional point process

involving the θ-function.

Remark 1.4. (Periodicity of Fn.) In the multi-cut regime of one-dimensional point processes,

asymptotic formulas are, in general, only quasiperiodic in n, see e.g. [58,32,52,17]. However,

in the special case where the mass of the equilibrium measure on each interval of the support is a

rational number, these asymptotic formulas become periodic, see e.g. [46] and [26, Corollary 2.2].

Interestingly, Theorem 1.2 shows that an analogous phenomenon holds in our two-dimensional

setting. To see this, recall from (1.16)that θis periodic of period 1. Since nruns over the integers,

the function n7→ Fnis, in general, only quasiperiodic in n. However, it follows from (1.19)and

(1.6)that if the mass σ?=R|z|≤ρ1µh(d2z)is rational, then n7→ Fnis periodic in n.

More generally, asymptotic formulas related to a given two-dimensional point process are expected

to be periodic in nwhenever the masses of the components of the equilibrium measure are all rational.

This holds true in the setting of this paper, as well as in the two-dimensional soft edge setting of [8].

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Diskcountingstatisticsnearhardedgesofrandomnormalmatrices:themulti-componentregimeYacinAmeur,ChristopheCharlier,JoakimCronvall*andJonatanLenellsOctober26,2022AbstractWeconsideratwo-dimensionalpointprocesswhosepointsareseparatedintotwodisjointcomponentsbyahardwall,andstudythemultivariatemomentgene...

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Disk counting statistics near hard edges of random normal matrices the multi-component regime Yacin Ameur Christophe Charlier Joakim Cronvalland Jonatan Lenells

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