PARTITION FUNCTIONS OF DETERMINANTAL AND PFAFFIAN COULOMB GASES WITH RADIALLY SYMMETRIC POTENTIALS SUNG-SOO BYUN NAM-GYU KANG AND SEONG-MI SEO

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PARTITION FUNCTIONS OF DETERMINANTAL AND PFAFFIAN COULOMB GASES

WITH RADIALLY SYMMETRIC POTENTIALS

SUNG-SOO BYUN, NAM-GYU KANG, AND SEONG-MI SEO

Abstract. We consider random normal matrix and planar symplectic ensembles, which can be interpreted

as two-dimensional Coulomb gases having determinantal and Pfaﬃan structures, respectively. For general

radially symmetric potentials, we derive the asymptotic expansions of the log-partition functions up to and

including the O(1)-terms as the number Nof particles increases. Notably, our ﬁndings stress that the formulas

of the O(log N)- and O(1)-terms in these expansions depend on the connectivity of the droplet. For random

normal matrix ensembles, our formulas agree with the predictions proposed by Zabrodin and Wiegmann up to

a universal additive constant. For planar symplectic ensembles, the expansions contain a new kind of ingredient

in the O(N)-terms, the logarithmic potential evaluated at the origin in addition to the entropy of the ensembles.

1. Introduction and main results

The Coulomb gas ensemble in the complex plane is governed by the law

(1.1) dP (β)

N(z1, . . . , zN) := 1

Z(β)

j>k=1 |zj−zk|β

j=1

e−βN

2Q(zj)dA(zj),

where Nis the number of particles, βis the inverse temperature and dA(z) := d2z/π is the area measure. Here,

Q:C→Ris called the conﬁning/external potential that satisﬁes suitable potential theoretic conditions. We

refer to [28,44,49] and references therein for recent developments of two-dimensional Coulomb gases. Contrary

to (1.1), the conﬁgurational canonical Coulomb gas ensemble in the upper-half plane [39] (cf. [14, Appendix

A]) has an additional complex conjugation symmetry (i.e. the particles come in complex conjugate pairs) and

is governed by the law

(1.2) de

P(β)

N(z1, . . . , zN) := 1

Z(β)

j>k=1 |zj−zk|β|zj−¯zk|β

j=1 |zj−¯zj|βe−βNQ(zj)dA(zj).

In (1.1) and (1.2), the normalization constants

Z(β)

N:= ZCN

j>k=1 |zj−zk|β

j=1

e−βN

2Q(zj)dA(zj),(1.3)

Z(β)

N:= ZCN

j>k=1 |zj−zk|β|zj−¯zk|β

j=1 |zj−¯zj|βe−βNQ(zj)dA(zj)(1.4)

that make (1.1) and (1.2) probability measures are called partition functions. Furthermore, the logarithm of a

partition function (divided by N2) is often called free energy.

For the special value β= 2, (1.1) and (1.2) represent joint probability distributions of the random normal

matrix and planar symplectic ensembles, respectively. In particular, if Q(z) = |z|2, these correspond to the

complex and symplectic Ginibre ensembles [31]. An important feature of this special value β= 2 is that, due

Date: October 7, 2022.

Sung-Soo Byun and Nam-Gyu Kang were partially supported by Samsung Science and Technology Foundation (SSTF-BA1401-

51) and by the National Research Foundation of Korea (NRF-2019R1A5A1028324). Sung-Soo Byun was partially supported by a

KIAS Individual Grant (SP083201) via the Center for Mathematical Challenges at Korea Institute for Advanced Study. Nam-Gyu

Kang was partially supported by a KIAS Individual Grant (MG058103) at Korea Institute for Advanced Study. Seong-Mi Seo was

partially supported by the National Research Foundation of Korea (2019R1A5A1028324, NRF-2022R1I1A1A01072052).

arXiv:2210.02799v1 [math.PR] 6 Oct 2022

2 SUNG-SOO BYUN, NAM-GYU KANG, AND SEONG-MI SEO

to the Vandermonde determinant terms, the ensembles (1.1) and (1.2) form determinantal and Pfaﬃan point

processes in the plane [28], respectively. In other words, all their correlation functions can be expressed in terms

of the (pre-)kernel of planar (skew-)orthogonal polynomials. In the sequel, for β= 2, we omit the superscript

(β)in (1.3) and (1.4), and simply write ZN≡Z(2)

Nand e

ZN≡e

Z(2)

We mention that the deﬁnition of partition functions (1.3) and (1.4) is more common in the statistical physics

community. On the other hand, in the random matrix theory community, another widely used convention for

the (canonical) partition functions is

(1.5) ZN:= 1

N!ZN,e

ZN:= 1

N!e

ZN,

see e.g. [28, Section 1.4]. The prefactor 1/N!in (1.5) allows writing ZNand e

ZNin terms of a structured

determinant and Pfaﬃan, respectively.

In this work, we study the asymptotic expansions of ZNand e

ZNas N→ ∞.

1.1. Summary of previous results. Before introducing our results, let us summarize some known results on

the asymptotics of Z(β)

Nfor general βand Q. Cf. the literature on e

Z(β)

Nis much more limited.

•(Zabrodin-Wiegmann prediction) In [54], it was predicted that the partition function Z(β)

Nhas an

asymptotic expansion of the form

(1.6) log Z(β)

N=C0N2+C1Nlog N+C2N+C3log N+C4+O(1

N).

Furthermore, they proposed explicit formulas for the constants Cj≡Cj(β, Q) (j= 0,...,4) depending

on βand Q, cf. (1.28). (See also [15,36] for a similar prediction in a diﬀerent setup.) Incidentally, the

formulas for C3and C4in [54] have been controversial as pointed out for instance in [47,50].

•(Asymptotic of the leading order O(N2)-term) It was shown in [33, Theorem 2.11] and [16,

Theorem 1.1] (among others) that as N→ ∞,

log Z(β)

N=−β

2N2IQ[µQ] + o(N2).

Here µQis Frostman’s equilibrium measure [46], a unique probability measure that minimizes the

weighted logarithmic energy

(1.7) IQ[µ] := ZZC2

log |z−w|dµ(z)dµ(w) + ZC

Q dµ.

•(Asymptotic up to the O(N)-term) It was shown by Leblé and Serfaty [42, Corollary 1.1] that as

N→ ∞,

(1.8) log Z(β)

N=−β

2N2IQ[µQ] + β

4Nlog N−C(β) + 1−β

4EQ[µQ]N+o(N),

where C(β)is a constant independent of the potential Qand

(1.9) EQ[µQ] := ZC

log(∆Q)dµQ

is the entropy associated with µQ. Here, ∆ := ∂¯

∂is the quarter of the usual Laplacian. We refer the

reader to [10,50] for the expansion (1.8) with quantitative error bounds.

Beyond the general cases mentioned above, for β= 2 with a speciﬁc (and fundamental from the random

matrix theory viewpoint) potential, there have been several works on the precise asymptotic expansion of the

partition functions, see e.g. [19,20] and references therein. This type of potential usually contains certain

singularities. As a result, the asymptotic expansions of the associated partition functions are more complicated

(for instance, some non-trivial O(√N)terms appear as well). Several topics in this direction will be discussed

in a separate remark at the end of the next subsection.

PARTITION FUNCTIONS OF DETERMINANTAL AND PFAFFIAN COULOMB GASES 3

1.2. Main results. We study asymptotic behaviors of ZNand e

ZNfor the exactly-solvable case where Qis

radially symmetric. Our main ﬁndings are summarized as follows.

(i) We derive the large-Nexpansions of log ZNand log e

ZNup to and including the O(1)-terms.

(ii) In the large-Nexpansions, the formulas of the O(log N)- and O(1)-terms depend on whether the limiting

spectrum is an annulus or a disc, see Theorems 1.1 and 1.2, respectively, cf. Figure 1. (This distinction

is crucial in the asymptotic analysis but seems not considered in [54].)

(iii) For the partition function ZNof random normal matrix ensembles, our expansions (1.17) and (1.23) up

to the O(N)-terms agree with the formula (1.8) with β= 2. Furthermore, we verify from (1.23) that

the asymptotic formula given in [54, Eqs.(1.2), (C.7)] holds up to an additive constant (1.34).

(iv) For the partition function e

ZNof planar symplectic ensembles, the asymptotic formulas (1.18) and (1.24)

are new to the best of our knowledge. Contrary to (1.8), the O(N)-terms in these expansions contain

not only the entropy but also the logarithmic potential (1.14).

Figure 1. Eigenvalues of complex Ginibre (left) and complex induced Ginibre (right) matrices

where N= 1000.

Let us be more precise in introducing our results. It is well known [11,33] that under some mild assumptions

on Q, as N→ ∞, the empirical measures 1

NPN

j=1 δzjof (1.1) and (1.2) weakly converge to µQ, which takes the

form

(1.10) dµQ= ∆Q·SdA.

Here S≡SQis a certain compact subset of Ccalled the droplet, see Figure 1.

We consider the case where the external potential Qis radially symmetric, i.e. Q(z) = q(|z|)for some

function qdeﬁned in [0,∞). We assume the basic growth condition

(1.11) lim inf

|z|→∞

Q(z)

2 log |z|>1,

which guarantees that ZN,e

ZN<+∞.Furthermore, we assume that Qis smooth, subharmonic in C, and

strictly subharmonic in a neighborhood of the droplet. In terms of the function q, the latter conditions are

equivalent to the requirements that rq0(r)is increasing on (0,∞), and strictly increasing in a neighborhood of

the droplet, cf. (2.3). Under the above assumptions, the droplet is given by

(1.12) S=Ar0,r1:= {z∈C:r0≤ |z| ≤ r1},

where r0is the largest solution to rq0(r) = 0 and r1is the smallest solution to rq0(r)=2, see [46, Section IV.6].

(We mention that the annular droplets often appear in non-Hermitian random matrix theory, see e.g. [32].) In

particular, if r0= 0, we denote Dr1=A0,r1.Henceforth, we keep the assumptions on Qdescribed above.

4 SUNG-SOO BYUN, NAM-GYU KANG, AND SEONG-MI SEO

For a radially symmetric potential Q, by using (1.10) and (1.12), one can show that the energy IQ[µQ]in

(1.7) is given by

(1.13) IQ[µQ] = q(r1)−log r1−1

4Zr1

rq0(r)2dr.

Similarly, in terms of the logarithmic potential

(1.14) Uµ(z) = Zlog 1

|z−w|dµ(w),

we have

(1.15) UµQ(0) = −ZS

log |w|dµQ(w) = −log r1+q(r1)−q(r0)

See [46, Section IV.6] for more details.

For the annular droplet case, we have the following.

Theorem 1.1. (Large-Nexpansion of the partition functions: annular droplet case) Suppose that

r0>0, i.e. the droplet Sin (1.12) is an annulus. Let

(1.16) FQ[Ar0,r1] := 1

12 log r2

0∆Q(r0)

1∆Q(r1)−1

16r1

(∂r∆Q)(r1)

∆Q(r1)−r0

(∂r∆Q)(r0)

∆Q(r0)+1

24 Zr1

r0∂r∆Q(r)

∆Q(r)2r dr.

Then as N→ ∞,the following holds.

(i) (Random normal matrix ensemble) We have

log ZN=−N2IQ[µQ] + 1

2Nlog N+log(2π)

2−1−1

2EQ[µQ]N

2log N+log(2π)

2+FQ[Ar0,r1] + O(N−1).

(1.17)

(ii) (Planar symplectic ensemble) We have

log e

ZN=−2N2IQ[µQ] + 1

2Nlog N+log(4π)

2−1−UµQ(0) −1

2EQ[µQ]N

2log N+log(2π)

2+1

2FQ[Ar0,r1] + 1

8log ∆Q(r0)

∆Q(r1)+O(N−1).

(1.18)

Using the convention (1.5) together with (2.21), our result can also be rewritten as

log ZN=−N2IQ[µQ]−1

2Nlog N+log(2π)

2−1

2EQ[µQ]N+FQ[Ar0,r1] + O(N−1).

(1.19)

and

log e

ZN=−2N2IQ[µQ]−1

2Nlog N+log(4π)

2−UµQ(0) −1

2EQ[µQ]N

2FQ[Ar0,r1] + 1

8log ∆Q(r0)

∆Q(r1)+O(N−1).

(1.20)

We mention that these formulas (1.19) and (1.20) as well as the formulas (1.25) and (1.26) below are more

convenient to compare with some asymptotic results in the previous literature [19,20].

Notice that the term log(r1/r0)is the extremal length of the annulus (1.12), see e.g. [29, p.142]. It is worth

pointing out that a characteristic diﬀerence between the expansions (1.17) and (1.18) is the appearance of the

logarithmic potential UµQ(0) in the O(N)-term of (1.18). This additional term can be interpreted as

(1.21) −ZS

log |w−¯w|dµQ(w)

after a proper renormalization. The interpretation (1.21) is natural from the perspective of the repulsion term

|zj−¯zj|βin (1.2) and is closely related to the notion of the next-order energy, see e.g. [43]. (We thank T. Leblé

for pointing out this.)

PARTITION FUNCTIONS OF DETERMINANTAL AND PFAFFIAN COULOMB GASES 5

In Subsection 4.1 we present an example of Theorem 1.1 for the Mittag-Leﬄer ensembles from which we

expect that the error terms O(N−1)are optimal.

For the disc droplet case, we have the following.

Theorem 1.2. (Large-Nexpansion of the partition functions: disc droplet case) Suppose that r0= 0,

i.e. the droplet Sin (1.12) is a disc. Let

FQ[Dr1] := 1

12 log 1

1∆Q(r1)−1

16r1

(∂r∆Q)(r1)

∆Q(r1)+1

24 Zr1

0∂r∆Q(r)

∆Q(r)2r dr.

(1.22)

Then as N→ ∞,the following holds.

(i) (Random normal matrix ensemble) We have

log ZN=−N2IQ[µQ] + 1

2Nlog N+log(2π)

2−1−1

2EQ[µQ]N+5

12 log N

+log(2π)

2+ζ0(−1) + FQ[Dr1] + O(N−1

12 (log N)3).

(1.23)

(ii) (Planar symplectic ensemble) We have

log e

ZN=−2N2IQ[µQ] + 1

2Nlog N+log(4π)

2−1−UµQ(0) −1

2EQ[µQ]N+11

24 log N

+log(2π)

2+1

2ζ0(−1) + 1

2FQ[Dr1] + 5

24 log 2 + 1

8log ∆Q(0)

∆Q(r1)+O(N−1

12 (log N)3).

(1.24)

Here ζis the Riemann zeta function.

Again, using the convention (1.5), we have

log ZN=−N2IQ[µQ]−1

2Nlog N+log(2π)

2−1

2EQ[µQ]N−1

12 log N

+ζ0(−1) + FQ[Dr1] + O(N−1

12 (log N)3)

(1.25)

and

log e

ZN=−2N2IQ[µQ]−1

2Nlog N+log(4π)

2−UµQ(0) −1

2EQ[µQ]N−1

24 log N

2ζ0(−1) + 1

2FQ[Dr1] + 5

24 log 2 + 1

8log ∆Q(0)

∆Q(r1)+O(N−1

12 (log N)3).

(1.26)

In Subsection 4.2, we provide an example of Theorem 1.2 for truncated unitary ensembles. Contrary to

Theorem 1.1, the error terms in Theorem 1.2 do not coincide with the expected optimal orders O(N−1). Our

error bounds originate from a decomposition of the analytic expressions of ZN,e

ZN(see Subsection 1.3), which

depends on suﬃciently large but seemingly arbitrary number mN>0.(Such a decomposition was not necessary

for the proof of Theorem 1.1.) Later, we choose mN=N1/6that gives rise to the control of the total error

bounds presented in Theorem 1.2. We mention that such error estimates also naturally appeared in similar

computations, see e.g. [13,19,20]. Nevertheless, we expect that the estimates can be improved with more

eﬀort.

In terms of the function χ:= 1

2log ∆Q, one can rewrite (1.22) as

(1.27) FQ[Dr1] = 1

12 log 1

1−1

12πI∂S

κ χ ds −1

4ZS

∆χ dA +1

12 ZS|∇χ|2dA,

where S=Dr1and κ= 1/r1is the curvature of the boundary, see [54, p.8960] and (1.33). Here, the third term

RS∆χ dA on the right-hand side of (1.27) is known as a “zero mode” of the loop operator (cf. [54, Eq.(5.26)]),

whereas the fourth term corresponds to the Dirichlet energy of χ.

We end this subsection by giving some crucial remarks on our theorems.

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PARTITIONFUNCTIONSOFDETERMINANTALANDPFAFFIANCOULOMBGASESWITHRADIALLYSYMMETRICPOTENTIALSSUNG-SOOBYUN,NAM-GYUKANG,ANDSEONG-MISEOAbstract.Weconsiderrandomnormalmatrixandplanarsymplecticensembles,whichcanbeinterpretedastwo-dimensionalCoulombgaseshavingdeterminantalandPfaanstructures,respectively.Forgen...

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PARTITION FUNCTIONS OF DETERMINANTAL AND PFAFFIAN COULOMB GASES WITH RADIALLY SYMMETRIC POTENTIALS SUNG-SOO BYUN NAM-GYU KANG AND SEONG-MI SEO

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