Asymptotic Results in Solvable Two-charge Models

# Asymptotic results in solvable two-charge models

## Abstract.

In [16], a solvable two charge ensemble of interacting charged particles on the real line in the presence of the harmonic oscillator potential is introduced. It can be seen as a special form of a grand canonical ensemble with the total charge being fixed and unit charge particles being random. Moreover, it serves as an interpolation between the Gaussian orthogonal and the Gaussian symplectic ensembles and maintains the Pfaffian structure of the eigenvalues. A similar solvable ensemble of charged particles on the unit circle was studied in [17, 10].
In this paper we explore the sharp asymptotic behavior of the number of unit charge particles on the line and on the circle, as the total charge goes to infinity. We establish extended central limit theorems, Berry-Esseen estimates and precise moderate deviations using the machinery of the mod-Gaussian convergence developed in  [6, 15, 14, 4, 7]. Also a large deviation principle is derived using the Gärtner-Ellis theorem.

MSC 2010 subject classifications: Primary 60B20, 82B05, 15B52, 15A15; Secondary
33C45, 60G55, 60F05, 60F10, 62H10.
Keywords: Random matrices, two-charge ensembles, large deviation principle, moderate deviations, central limit theorem, Berry-Esseen estimate, local limit theorem, Gaussian orthogonal ensemble, circular orthogonal ensemble.

## 1. Introduction

### 1.1. A general two-charge ensemble

We first present a general model of charged particles with charge ratio interacting via a logarithmic potential. Let be non-negative integers such that The two-component log-gas to be considered consists of particles with unit charge and particles with charge two, located either on the line or on the unit circle and interacting via the logarithmic potential

 UL,M(ξ,ζ):=−∑1≤i

where the charge one particles are located at and charge two particles at respectively.
The system may be in the presence of an external field, and the interaction energy between the charges and the field is for some potential . The total energy is then

 EL,M(ξ,ζ)=UL,M(ξ,ζ)+L∑i=1V(ξi)+2M∑k=1V(ζk).

The partition function of the system is

 ZL,M=1L!M!∫EL∫EMe−EL,M(ξ,ζ)dμL(ξ)dμM(ζ) ,  (E=R or T).

We study a special form of the grand canonical ensemble, where the number of charges and are random but they sum up to the total charge for some Moreover, the probability of the system having particles with unit charge and particles with charge two is equal to

 XLZL,MZN(X),

where

 ZN(X)=∑L+2M=NXLZL,M

and is a parameter, called fugacity, which determines the probability of the system having a particular population vector .
In what follows we distinguish two models of two-charge ensemble, on the real line and on the unit circle. For both models, we study the distribution of the number of charge one particles for the proper scaling of the parameter , when the total charge tends to infinity.
The structure of the paper goes as follows. In Subsections 1.1.1 and 1.1.2 we discuss separately two-charge models on the line and on the circle, respectively. Thereafter, in Subsection 1.2 we recall the definition of mod-Gaussian convergence and the limiting theorems this convergence implies. In Section 2, the limiting results for the two-charge models on the line are stated for two different scaling of the fugacity parameter. In Section 3, we introduce the limiting theorems for the two charge ensemble on the circle.

#### A two-charge ensemble on the real line

The charged particle model on the line, was introduced in [16]. A generalization of these models to more than two types of charges recently has been studied in [12]. The motivation to study the two charged particle model on the line comes from its analogy with the real Ginibre ensemble (instead of complex conjugate pairs of eigenvalues, we have particles of charge on the real line). For more details on the complex and real eigenvalues of the real Ginibre ensembles and the existence of Pfaffian structure on the particles in these ensembles, see [13, 11, 2].
We study the two charge models on the line in the presence of the harmonic oscillator potential As a result, the case gives exactly the Gaussian orthogonal ensemble (). In contrast, as the above model corresponds to the Gaussian symplectic ensemble (). Therefore, the two-charge ensemble on the line can be interpreted as an interpolation between these two point processes.
We denote by the number of particles with unit charge with the parameter The aim is to investigate the limiting behavior of the sequence depending on the fugacity parameter Typically, we will focus on two cases and for some The case is the analog of the real Ginibre ensemble, i.e. the charge 1 particles play the role of real eigenvalues in real Ginibre and the charge 2 particles the role of pairs of complex conjugate eigenvalues.
For the case Rider, Sinclair and Xu, using Laplace method, provide asymptotic formulas for the quantities ([16]). From our side, we will show large deviation principle (LDP) for the sequence as well as mod-Gaussian convergence for the normalized sequence Note that the central limit theorem (CLT) will be one of the consequences of the mod-Gaussian convergence mentioned above.
The case has been studied in details in [16] and the limiting results such as CLT and LDP have been derived. We extend their results by proving mod-Gaussian convergence for the sequence and deducing all its consequences such as precise moderate deviations and extended central limit theorem. In order to derive these limiting results, we will use the following characterization of the partition function, shown in Theorem 2.1 in [16] and following from the solvability of the ensemble

 ZN(X)=n−1∏j=0rj(X),

where

 rj(X)=4(j+1)Γ(j+12)Lj+1(−X2)Lj(−X2),

with being the generalized th Laguerre polynomial with parameter defined as

 Lαj(x)=exx−αj!djdxj(e−xxj+α),

and orthogonal with respect to the measure on with density .

Thus, the probability generating function of can be written

 Pn(t)=ZN(tX)ZN(X)=Ln(−t2X2)Ln(−X2), (2)

and then the moment generating function of is given by

 E[ezUn]=ZN(ezX)ZN(X)=Ln(−X2e2z)Ln(−X2). (3)

#### A two-charge ensemble on the circle

This model represents the circular version of the above model and has been introduced and studied in [8, 9, 10, 17]. We have again particles with unit charge and particles with charge two that are located on the unit circle and interact logarithmically in the absence of an external field. This ensemble forms a Pfaffian point process that interpolates between the circular and symplectic ensembles.
We denote by the number of particles with unit charge when with Following Section 6.7.1 of [10], the probability generating function of is

 Pn(t)=E[tVn]=n∏k=1(2nρt)2+(2k−1)2(2nρ)2+(2k−1)2. (4)

Taking into account this polynomial structure, Forrester used an argument of Bender [1] to prove the local limit theorem. In [17], the central limit theorem has been derived for the sequence by a direct computation using the moment generating function,

 E[ezVn]=n∏k=1(2nρez)2+(2k−1)2(2nρ)2+(2k−1)2. (5)

We will deduce the LDP for the sequence and the mod-Gaussian convergence with its consequences for the sequence

### 1.2. The framework of mod-Gaussian convergence

The framework of mod-Gaussian convergence for a sequence of random variables has been developed by Delbaen, Féray, Jacod, Kowalski, Méliot and Nikeghbali in [6, 15, 14, 4, 7]. For , set

 S(c,d)={z∈C,c
###### Definition 1.1.

Let be a sequence of real valued random variables, and be their moment generating functions, which we assume to all exist in a strip . We assume that there exists an analytic function not vanishing on the real part of , such that locally uniformly on ,

 limn→∞E[ezXn]e−tnz22:=limn→∞ψn(z)=ψ(z),

where is some sequence going to infinity. We then say that converges mod-Gaussian on with parameters and limiting function .

To establish the mod-Gaussian convergence in two charge models, the following theorem will be useful. It is a straightforward consequence of Theorem 8.2.1 in [6].

###### Theorem 1.2.

Let be a sequence of bounded random variables with non-negative integer values, with mean variance and third cumulant . Suppose that the probability generating function of can be factorized as

 Pn(t)=∏1≤j≤nPn,j(t)

where each is a polynomial with non-negative coefficients and maximal degree not depending on . If

 σ2nn→σ2>0 , κ(3)nn→κ, (6)

then the sequence

 ~Xn:=Xn−μnn1/3

converges in the mod-Gaussian sense with limiting function and parameter . The convergence takes place of the whole complex plane.

###### Remark 1.3.

If , we can take .

Although mod-Gaussian convergence entails much more information, for the aim of this paper, it will be an instrument to deduce asymptotic results, such as a central limit theorem, precise deviations etc. In what follows we summarize some of the limiting results that mod-Gaussian convergence implies (see [6], [7] for the results below). In the remaining part of this subsection, is a sequence converging mod-Gaussian on with parameters and limiting function .

###### Theorem 1.4 (Precise deviations at the scale O(tn), Theorem 4.2.1 in [6]).

For ,

 P[Xn≥tnx]=e−tnx22x√2πtnψ(x)(1+o(1)),

and for ,

 P[Xn≤tnx]=e−tnx22|x|√2πtnψ(x)(1+o(1)).
###### Theorem 1.5 (Central limit theorem at the scale o(tn), Theorem 4.3.1 in [6]).

For ,

 P[Xn√tn≥y]=P[N(0,1)≥y](1+o(1))=e−y22y√2π(1+o(1)).

Note that Theorem 1.5 immediately implies a central limit theorem for the rescaled sequence (taking ).
Under additional assumptions, it is possible to deduce the speed of convergence of the above mentioned CLT and a local limit theorem as well.

###### Definition 1.6.

Let be a sequence of real random variables, and a sequence growing to infinity. Consider the following assertions:

1. Fix and . There exists a zone , , such that, for all in this zone, if , then

 |ψn(iξ)−1|≤K1|ξ|veK2|ξ|w

for some positive constants and , that are independent of .

2. One has

 w≥2;−12≤γ≤1w−2;D≤(14K2)1w−2.

If Conditions 1 and 2 are satisfied, we say that we have a zone of control with index .

###### Theorem 1.7 (Speed of convergence, Theorem 2.16 in [7]).

Let be a sequence converging in mod-Gaussian sense with a zone of control of index . In addition, we assume . Then,

 dKol(Xn√tn,N(0,1))≤C(D,v,K1)1tγ+12n,

where is the Kolmogorov distance and

 C(D,v,K1)=minλ>0(1+λ√2π(2v−12Γ(v2)K1+π1/6D(43√1+1λ+33√3))). (7)
###### Theorem 1.8 (Local limit theorem, see [3]).

Let and be a fixed interval, with . Assume that conditions (Z1) and (Z2) hold. Then for every exponent ,

 limn→∞(tn)δP[Xn√tn−x∈1tδn(a,b)]=b−a√2π.

In particular, assuming , with one obtains

## 2. The limiting results for the two-charge ensemble on the line

### 2.1. The fugacity parameter increasing with n

In this subsection we establish various asymptotic results in the case The first theorem is the large deviation principle for the random sequence (as before is the number of particles of unit charge). We use the notation of Dembo and Zeitouni (see [5]). In particular, we denote the limiting cumulant generating function and its Legendre-Fenchel transform, by and , respectively.

###### Theorem 2.1 (Ldp).

The sequence of random variables satisfies the LDP in at speed with good rate function

 Λ∗(x)=xlogx+1−x2log(1−x)+a(γ)x+b(γ) (8)

where and is such that .

###### Proof.

We derive the above LDP as a consequence of the Gärtner-Ellis theorem (see Theorem 2.3.6 in [5]). Therefore, we need to estimate the limiting behavior of the normalized cumulant generating function . We know that

 E[ezUn]=Ln(−X2e2z)Ln(−X2). (9)

For this propose, we use the relation between Legendre and Hermite polynomials:

 Ln(−X2)=(−1)n2−2n(n!)−1H2n(iX)

([18] formula 5.6.1) and the representation by means of the Fourier transform

 Hn(t)=et2√2π∫∞−∞(−is)neits−s24ds,

to get

 Ln(−X2)=(−1)n2−2n(n!)−1√2πe−X2∫∞−∞s2ne−sX−s24ds. (10)

With the change of variables , we obtain

 Ln(−X2)=2(−1)nX2n+1(n!)−1√2πe−X2∫∞−∞s2ne−2sX2−X2s2ds.

Hence, since and

 E[ezUn]=e(N+1)ze−Nγ(e2z−1)IN(γe2z)IN(γ), (11)

where

 IN(γ)=∫∞−∞sNe−Nγ(2s+s2)ds. (12)

To apply the Laplace method, we split the integral in two parts:

 IN(γ)=I+N(γ)+I−N(γ),

with

 I±N(γ)=∫∞0e−Ng±(γ,s)ds,g±(γ,s)=−logs±2γs+γs2.

Let be the positive solution of

 2γt2±2γt−1=0.

The function is convex. Its minimum, reached at , is

 g±(γ,t∗±(γ))=−logt∗±(γ)±2γt∗±(γ)+γt∗±(γ)2.

A simple look shows that , so that if we set

 t(z):=t∗−(γe2z)=12(1+√1+2e−2zγ), (13)

we get, for every ,

 12nΛn(z)→ z−γ(e2z−1)−g−(γe2z,t(z))+g−(γ,t(0))=Λ(z). (14)

Let us reparametrize this function in terms of . From (13)

 z=−12log(2γt(t−1)) , dzdt=−2t−12t(t−1), (15)

so that, if we denote we can write

 Λ(z)=ˆΛ(t)−ˆΛ(t0),

where

 ˆΛ(t):=−12log(1−1t)+12t.

It is clear that (resp. ) is a differentiable function of (resp. ) on its domain (resp. ). It is then enough to apply the Gärtner-Ellis theorem.

Moreover, since we see that

 Λ∗(x)=supz∈R(zx−Λ(z))=supt∈(1,∞)(−x2log(2γt(t−1))−ˆΛ(t)+ˆΛ(t0))

and the supremum is reached in so that

 Λ∗(x)=xlogx+1−x2log(1−x)−x2(1+log(2γ))+ˆΛ(t0).

This function is strictly convex and attains its minimum at and therefore, the proof follows. ∎

In the following proposition, we describe the precise asymptotic behavior of the three first cumulants of that are latter used to derive the mod-Gaussian convergence.

###### Proposition 2.2.

Let be the third cumulant of , then

 limn→∞E[Un]2n=1t0 , limn→∞Var(Un)2n=σ2(γ)=2(t0−1)t0(2t0−1) (16) limn→∞κ(3)n2n =κ(γ)=4(−2t20+5t0−1)(t0−1)t0(2t0−1)3. (17)
###### Proof.

The structure of the Laplace transform (11) and the representation (12) entail that

 limn→∞12nΛ′n(z)→Λ′(z) , limn→∞12nΛ′′n(z)→Λ′′(z) , limn→infty12nΛ(3)n(z)→Λ(3)z)

(we leave the details to the reader). Recall 1 and (15), which gives

 Λ′′(z)=2(t−1)t(2t−1) , Λ′′(0)=2(t0−1)t0(2t0−1). (18)

From (18) we have

 Λ(3)(z)=2(−2t2+4t−1)t2(2t−1)2×−2t(t−1)2t−1=−4(−2t2+4t−1)(t−1)t(2t−1)3,

and the limiting third cumulant is therefore given by taking . ∎

Set and, for , denote by

###### Theorem 2.3 (Mod-Gaussian convergence).

As , the sequence converges mod-Gaussian with parameters and the limiting function The convergence takes place on the whole complex plane.

###### Proof.

Since the support of the measure of orthogonality of the polynomials is , it is known (Theorem 6.73 in [18]) that the zeros of are positive and then

 Ln(x)=n∏k=1(x−ξk,n),

with for all . The latter, together with equation (2), implies that the probability generating function of can be written as

 Pn(t)=n∏k=1X2t2+ξk,nX2+ξk,n, (19)

that is, as a product of polynomials with positive coefficients of degree . To establish the mod-Gaussian convergence, now it suffices to apply Theorem 1.2 and Proposition 2.2. ∎

###### Remark 2.4.

It could be possible to apply Remark 1.3 to this case, but we did not push the computation to that step, not to enlarge this paper.

###### Corollary 2.5.

With and ,

1. the conclusions of Theorems 1.4 and 1.5 hold true,

2. the conclusions of Theorems 1.7 and 1.8 hold true.

###### Proof.

We have only to prove 2) and actually it suffices to show that the sequence converges mod-Gaussian with index of control and a zone of control , where . Using (19), can be written as

 Un\lx@stackrel(d)=2n∑k=1Xk,n,Xk,n∼B(X2X2+ξk,n),

where are independent and denotes the Bernoulli distribution with parameter Since are independent, by Theorems 9.1.6 and 9.1.7 in [6], we have the following bounds on the cumulants

 ∣∣κ(r)(¯¯¯¯Un)∣∣≤nrr−22r−12r,

for all . Then Lemma 4.2 in [7], implies that

 ∣∣ ∣∣E[eiξ¯¯¯Unn1/3]etnξ22−1∣∣ ∣∣≤K|ξ|3eK|ξ|3, (20)

for all with and . ∎

### 2.2. The fugacity parameter X=1

###### Theorem 2.6 (Ldp).

For , the sequence of random variables satisfies the LDP with rate and good rate function

 Λ∗(x)=xlogx−x+1 , (x>0).

This result was implicitly obtained in [16] p. 131, with a different proof.

###### Proof.

The Laplace transform of from Equation (3) is equal to

 E[ezUn]=L−1/2n(−e2z)L−1/2n(−1).

We use Perron’s formula (see Theorem 8.22.3 in [18] ) to characterize the asymptotic behavior of Laguerre polynomials

 L−1/2n(x)=12√πex/2n−1/2e2√−nx(1+O(n−1/2)),

which holds uniformly in any compact set of . We derive

 E[ezUn]=e12(1−e2z)+2√n(ez−1)(1+O(n−1/2)), (21)

uniformly in any compact set of . We conclude that

 12√nlog(E[ezUn])→ez−1=:Λ(z),

and we may apply the Gärtner-Ellis theorem, giving the LDP with rate function given by the Legendre dual of . ∎

###### Remark 2.7.

Since is the Cramér transform of the Poisson distribution, we could expect to have a mod-Poisson convergence (see [6] for the details on mod-Poisson convergence). We have indeed

 limn→∞E[ezUn]e2√n(ez−1)=e12(1−e2z)

but this convergence is valid only for Therefore, the assumptions required for the mod-Poisson convergence are not satisfied.

In contrast, the renormalized sequence converges mod-Gaussian. Indeed, from (21), the cumulant generating series of can be written as

 log(E[ezUnn1/6]) =12(1−e2zn1/6)+2√n(ezn1/6−1)+o(1) =2zn1/3+z2n1/6+z33+o(1),

for all , which readily implies the following theorem.

###### Theorem 2.8 (Mod-Gaussian convergence).

As , the sequence converges mod-Gaussian with parameters and the limiting function The convergence takes place on the whole complex plane.

###### Corollary 2.9.

With and , the conclusions of Theorems 1.4 and 1.5 hold true.

## 3. The limiting results for the two charge ensemble on the circle

Let us recall that for every .

###### Theorem 3.1 (Ldp).

The sequence of random variables satisfies the LDP in with speed and good rate function which is the Legendre dual of

 Λ(z):=2ρezarctan1ρez−2ρarctan1ρ+logρ2e2z+1ρ2+1.

The function is convex on , reaches its unique minimum at and satisfies

 Λ∗(0)=−2ρarctan(1ρ)−log(1+ρ2) , limx↓0(Λ∗)′(x)=−∞ , limx↑2Λ∗(x)=∞. (22)

It means that, when tends to infinity, tends to in probability and that

 limn→∞n−1logP(Vn=0)=−Λ∗(0),limn→∞n−1logP(Vn=2n)=−∞.
###### Proof.

We use the Gärtner-Ellis theorem. From (5), the moment generating function of the sequence can be written as

 E[ezVn]=n∏k=1ρ2e2z+(2k−12n)2ρ2+(2k−12n)2.

From the convergence of the Riemann sums we have

 limn→∞1nlogE[ezVn] =∫10log(ρ2e2z+t2ρ2+t2)dt (23) =2ρezarctan1ρez−2ρarctan1ρ+logρ2e2z+1ρ2+1=:Λ(z).

The first derivative is clearly

 Λ′(z) =∫102ρ2e2zρ2e2z+t2dt=2ρezarctan1ρez, (24)

and all the remaining assertions are straightforward. ∎

In the following proposition, we limiting results on the three first cumulants of .

###### Proposition 3.2.

If is the third cumulant of , then

 limn→∞E[Vn]n=2ρarctan1ρ , limn→∞Var(Vn)n=2ρarctan1ρ−2ρ21+ρ2=:σ2(ρ) (25) limn→∞κ(3)nn =κ(ρ)=2ρarctan1ρ−2ρ23+ρ2(1+ρ2)2 (26)
###### Proof.

The proof is the same as in Section 2, with given in (24), so that

 Λ′′(z) =2ρezarctan1ρez−2ρ2e2z1+ρ2e2z, Λ(3)(z) =2ρezarctan1ρez−2ρ2e2z3+ρ2e2z(1+ρ2e2z)2.

These limits coincide with the results of Theorem 2.2 in [17], up to a change of notation since with and
We set