A Solvable Mixed Charge Ensemble on the Line: Global Results

A Solvable Mixed Charge Ensemble on the Line: Global Results

Abstract

We consider an ensemble of interacting charged particles on the line consisting of two species of particles with charge ratio in the presence of the harmonic oscillator potential. The system is assumed to be at temperature corresponding to and the sum of the charges is fixed. We investigate the distribution of the number as well as the spatial density of each species of particle in the limit as the total charge increases to . These results will follow from the fact that the system of particles forms a Pfaffian point process. We produce the skew-orthogonal polynomials necessary to simplify the related matrix kernels.

1 Introduction

Let and be non-negative integers so that , and consider 1-dimensional electrostatic system consisting of particles with unit charge and particles with charge 2. We will identify the state of the system by pairs of finite subsets of , and , where represent the locations of the charge 1 particles and represent the locations of the charge 2 particles.

The potential energy of state is given by

 ∑j

We assume that the system is in the presence of an external field, so that the interaction energy between the charges and the field is given by

 −L∑ℓ=1V(αℓ)−2M∑m=1V(βm)

for some potential . Eventually we will specify to the situation where is the harmonic oscillator potential, but for now we maintain generality. The total potential energy of the system is therefore

 E=∑j

Given a pair of vectors we will define to be the right hand side of (1.1), and call a state vector corresponding to the state . Generically, there are state vectors corresponding to a given state.

Assuming the system is placed in a heat bath corresponding to inverse temperature parameter , then the Boltzmann factor for the state vector is given by

 e−E(α,β)=L∏ℓ=1w(αℓ)M∏m=1w(βm)2∏j

where is the weight of the system. The partition function of the system is given by

 ZL,M=1L!M!∫RL∫RMe−E(α,β)dμL(α)dμM(β), (1.3)

where and are Lebesgue measure on and respectively. The multiplicative prefactor compensates for the multitude of state vectors associated to each state.

Here we will be interested in a form of the grand canonical ensemble conditioned so that the sum of the charges equals . That is, we consider the union of all two component ensembles with particles of charge 1 and particles of charge 2 over all pairs of non-negative integers and with . The partition function of this ensemble is given by

 Z(X)=∑(L,M)XLZL,M=∑(L,M)XLL!M!∫RL∫RMe−E(α,β)dμL(α)dμM(β).

Here is the fugacity of the system, a parameter which controls the probability that the system has a particular population vector . The sum over indicates that we are summing over all pairs of non-negative integers such that .

Note now that is itself a random vector, though we will continue to use this notation for the value of the population vector as well. For example, for each admissible pair , the joint density of particles given population vector is given by the normalized Boltzmann factor,

 XLZ(X)e−E(α,β). (1.4)

When the probability of seeing a particular pair , or , is the ratio , where .

Experts of random matrix theory will have already noticed that when the above reduces to a general orthogonal (or ) ensemble. Likewise, as , the above formally goes over to the corresponding symplectic (or ) ensemble. This provides then an unusual sort of interpolation between two classical and well studied point processes.

2 Statement of results

In this paper we will primarily be concerned with global statistics of the particles when the fugacity equals 1 and the potential is given by

 V(γ)=γ2/2,that is,w(γ)=e−γ2/2.

Many of the results presented here are valid for other potentials and other values of , however unless otherwise indicated we will restrict ourselves to these choices of and . We will also restrict ourselves to the situation where is an even integer.

Similar results for the two-charge ensemble constrained to the circle with uniform weight were obtained by P.J. Forrester (see 5.9 of [7] and the references therein).

The goal of this paper is to present global results about the distribution of and as well as the global spatial distribution of each of the species of particles. Along the way we will derive a Pfaffian point process for the particles (similar to that of another two-component ensemble, Ginibre’s real ensemble) as well as the skew-orthogonal polynomials which allow us to present a simplified matrix kernel for the process. The local analysis of this kernel (i.e. its scaling limits in the bulk and at the edge) as well an investigation of the right-most particle of each species will appear in a forthcoming publication.

2.1 Distribution of the population vectors

Sharp results on the law of the state vector are consequences of the following characterization.

Theorem 2.1.

For each non-negative integer , let be the th Laguerre polynomial with parameter . Then, is the coefficient of of the polynomial . That is,

 Z(X)Z=LN/2(−X2)LN/2(−1),

and so

Properties of the Laguerre polynomials now allow for nice expressions for the mean, variance, etc. of for all finite values of . For example, we have that

 E[L]=ddX[Z(X)Z]X=1=2N/2−1∑j=0Lj(−1)LN/2(−1)=2N/2−1∑i=0[Γ(N2−i)Γ(i+32)i!]−1N/2∑i=0[Γ(N2−i+1)Γ(i+12)i!]−1.

Asymptotic descriptions of the law of are just as readily obtained from Theorem 2.1.

Theorem 2.2.

As it holds:

1. and ,

2. converges in distribution to a standard Normal random variable,

3. with a numerical constant for any .

2.2 Spatial density of particles

We introduce the (mean) counting measures and for the charge 1 and charge 2 particles defined by

for Borel subsets (where, for instance, is the number of charge 1 particles in ). As we shall see in the sequel, these measures are absolutely continuous with respect to Lebesgue measure, and we will write and for their respective densities. (The cryptic notation will be resolved in Section 3.2, when we define the -correlation function of the ensemble to be ).

From Theorem 2.2 we see that, as ,

 ∫∞−∞R(N)1,0(x)dx=E[L]∼√2N, and ∫∞−∞R(N)0,1(x)dx=E[M]∼N2.

One then would ask, when suitably scaled and normalized as in

 s(N)1(x)=1√2R(N)1,0(√Nx) and s(N)2(x)=2√NR(N)0,1(√Nx),

whether and converge to proper probability measures. This is answered in the affirmative in Theorem 2.3 below.

The previous result shows that, with probability one, for all large the number of charge 1 particles is . This suggests that, in the thermodynamic limit, the statistics of the charge 2 particles should behave as though there are no charge 1 particles present, or like a copy of the Gaussian Symplectic Ensemble (again, arrived at from the present ensemble upon setting ). Indeed we find the scaled density of charge 2 particles approaches the semi-circle law.

On the other hand, though the charge 1 particles exhibit the same level repulsion amongst themselves as the eigenvalues in the Gaussian Orthogonal Ensemble (occurring here when ), the preponderance of charge 2 particles leads to a different limit distribution.

Ginibre’s real ensemble, the ensemble of eigenvalues of real asymmetric matrices with i.i.d. Gaussian entries, has superficial resemblance to the ensemble we are considering here. First, it is suggestive to think of the present ensemble as arising from real Ginibre by forcing the non-real eigenvalues, which occur in complex conjugate pairs, to be identified with one “charge two” particle on the line. A little more concretely, the (random) number of real eigenvalues in real Ginibre has both expectation and variance of , as does the number of charge 1 particles here. (See [6] for the mean, and [8] for the variance). It is perhaps not surprising, therefore, that the limiting scaled density of charge 1 particles is the same (up to a constant) as that of the real eigenvalues in Ginibre’s real ensemble [4].

Theorem 2.3.

As , converges weakly in the sense of measures to the uniform law on , and converges in the same manner to the semi-circular law with the same support. In particular, it is proved that

 ∫eitxs(N)1(x)dx→1√2tsin(√2t)

and

 ∫eitxs(N)2(x)dx→√2tJ1(√2t),

where the convergence is pointwise.

We give an elementary proof of the above, making use of the explicit skew-orthogonal polynomial system derived below. Given that the number of charge 1 particles is , one could undoubtedly make a large deviation proof along the lines of [2] or [3] of a stronger version of the second statement: that the random counting measure of charge 2 particles converges almost surely to the semi-circle law. However, it is not clear how to use such energy optimization ideas to access the charge 1 profile.

3 A Pfaffian point process for the particles

All of the results in this paper follow, in one way or another, from the fact that our interacting particles form a Pfaffian point process very much like that of Ginibre’s real ensemble and related to the Gaussian Orthogonal and Symplectic Ensembles.

The results in this section are valid for quite general weight functions and fugacities. Thus, for the time being, we will return to the general situation.

3.1 The joint density of particles

The joint density of particles for a particular choice of is given by

 XLZ(X)ΩL,M(α,β),whereΩL,M(α,β)=e−E(α,β).

More specifically,

 ΩL,M(α,β)=L∏ℓ=1w(αℓ)M∏m=1w(βm)2∏j

where, for now, the only assumptions we will make on are that it is positive and Lebesgue measurable with .

3.2 Correlation Functions

Given and , we define the -correlation function by

where, for instance, is the vector in formed by concatenating and . We will often write for in situations where is seen as being fixed.

The correlation functions encode statistical information about the configurations of the charged particles. To be more precise, given and with , we set

 ξ=ξ(α,β)=(ξ1,ξ2)=(ξ1(α),ξ2(β))=({α1,…,αℓ},{β1,…βm}).

Given an -tuple of mutually disjoint subsets of , , and an -tuple of mutually disjoint subsets of , , the probability that the system is in a state where there is exactly one charge 1 particle in each of the and exactly one charge 2 particle in each of the is given by

 Prob{|A1∩ξ1|=1,…,|AL∩ξ1|=1,|B1∩ξ2|=1,…,|BM∩ξ2|=1} =E[{L∏ℓ=1|Aℓ∩ξ1|}{M∏m=1|Bm∩ξ2|}].

This probability can also be represented by

 1L!M!∑σ∈SL∑τ∈SM∫Bτ(1)⋯∫Bτ(M)∫Aσ(1)⋯∫Aσ(L)ΩL,M(α,β)dμL(α)dμM(β).

Since the integrand is symmetric in the coordinates of and , we find

The correlation functions can be used to generalize this formula. If is a tuple of disjoint subsets of and another such tuple, then

3.3 Pfaffian point processes

Consider, for the moment, a simplified system of indistinguishable random points with correlation functions satisfying

 E[n∏j=1|Aj∩ζ|]=∫A1⋯∫AnRn(z)dμn(z)

for any -tuple of mutually disjoint sets.

If there exists a matrix valued function such that

 Rn(z)=Pf[KN(zj,zk)]nj,k=1,

then we say that our ensemble of random points forms a Pfaffian point process with matrix kernel . Much of the information about probabilities of locations of particles (e.g. gap probabilities) can be derived from properties of the matrix kernel. Moreover, in many instances, we are interested in statistical properties of the particles as their number (or some related parameter) tends toward . In these instances, it is sometimes possible to analyze in this limit (under, perhaps, some scaling of and dependent on ) so that the relevant limiting probabilities are attainable from this limiting kernel.

For the ensemble of charge 1 and charge 2 particles with total charge , we will demonstrate that the correlation functions have a Pfaffian formulation of the form,

where and are matrix kernels.

3.4 A Pfaffian form for the total partition function

In order to establish the existence of the matrix kernels we first need a Pfaffian formulation of the total partition function.

Given a measure on we define the operators and on by

 ϵν1f(x)=12∫Rf(y)sgn(y−x)dν(y)andϵν2f(y)=f′(y).

(Obviously does not depend on , but it is convenient to maintain symmetric notation). Using these inner products we define

 ⟨f|g⟩νb2=∫R[f(x)ϵbg(x)−g(x)ϵbf(x)]dν(x),b=1,2.

We specialize these operators and inner products for Lebesgue measure by setting and . We also write . It is easily seen that

 ⟨˜f|˜g⟩1=∫R[˜f(x)ϵ1˜g(x)−˜g(x)ϵ1˜f(x)]dμ(x)=⟨f|g⟩wμ1.

Similarly,

 ⟨˜f|˜g⟩4 =∫R[˜f(x)ddx˜g(x)−˜g(x)ddx˜f(x)]dμ(x) =∫Rw(x)2[f(x)g′(x)−g(x)f′(x)]dx=⟨f|g⟩w2μ4.

We call a family of polynomials, , a complete family of polynomials if . A complete family of monic polynomials is defined accordingly.

Theorem 3.1.

Suppose is even and is any complete family of monic polynomials. Then,

 Z(X)=Pf(X2Ap+Bp),

where

 Ap=[⟨˜pm|˜pn⟩1]N−1m,n=0andBp=[⟨˜pm|˜pn⟩4]N−1m,n=0.
Corollary 3.2.

With the same assumptions as Theorem 3.1, .

3.5 A Pfaffian formulation of the correlation functions

In order to describe the entries in the kernels and , we suppose is any complete family of polynomials and define

 Cp=Ap+Bp,

where and are as in Corollary 3.2. (Here we are setting , though similar maneuvers are valid for general ). Since we are assuming that is non-zero, is invertible and we set

 (Cp)−T=[ζj,k]N−1j,k=0.

The clearly depend on our choice of polynomials. We then define

 ϰN(x,y)=N−1∑j,k=0˜pj(x)ζj,k˜pk(y). (3.1)

The operators and operate on in the usual manner. For instance,

 ϵ2ϰN(x,y)=N−1∑j,k=0ϵ2˜pj(x)ζj,kϵ2˜pk(y)

and

 ϰNϵ1(x,y)=N−1∑j,k=0˜pj(x)ζj,kϵ1˜pk(y).

(That is, written on the left acts on the viewed as a function of , etc.).

Theorem 3.3.

Suppose is even, is any complete family of polynomials and is given as in (3.1). Then,

where

 K1,1N(x,y)=[ϰN(x,y)ϰNϵ1(x,y)ϵ1ϰN(x,y)ϵ1ϰNϵ1(x,y)+14sgn(y−x)],
Remark.

The factor can be moved inside the Pfaffian so that the entries in the various kernels where an appears are multiplied by 2. This maneuver is superficial, but has the effect of making these particular entries appear more like the entries in other ensembles (e.g. GOE). For instance, appears more natural to experts used to these other ensembles.

We can simplify the presentation of the matrix kernels with a bit of notation. First, let us write

 KN(x,y)=[ϰN(x,y)ϰN(x,y)ϰN(x,y)ϰN(x,y)],andEb=[100ϵb];b=1,2.

Then,

 K1,1N(x,y)=E1KN(x,y)E1+[00014sgn(y−x)],
 K2,2N(x,y)=E2KN(x,y)E2,K1,2N(x,y)=E1KN(x,y)E2,K2,1N(x,y)=E2KN(x,y)E1.

We notice in particular that the functions and given in Section 2.2 are given by

 R(N)1,0(x)=2N−1∑j,k=0˜pj(x)ζj,kϵ1˜pk(x)andR(N)0,1(x)=N−1∑j,k=0˜pj(x)ζj,kϵ2˜pk(x). (3.2)

3.6 Skew-orthogonal polynomials

The entries in the kernel themselves can be simplified (or at least presented in a simplified form) by a judicious choice of . If we define

 ⟨f|g⟩=⟨f|g⟩1+⟨f|g⟩4,

then

 Cp=[⟨˜pm|˜pn⟩]N−1m,n=0.

Since (and by extension all other entries of the various kernels) depend on the inverse transpose of , it is desirable to find a complete family of polynomials for which can be easily inverted.

We say is a family of skew-orthogonal polynomials for the skew-inner product with weight if there exists real numbers (called normalizations) such that

 ⟨˜p2j|˜p2k⟩=⟨˜p2j+1|˜p2k+1⟩=0and⟨˜p2j|˜p2k+1⟩=−⟨˜p2k+1|˜p2j⟩=δj,krj.

Using these polynomials, the entries in the matrix kernels presented in Section 3.5 have a particularly simple form. For instance,

 ϰN(x,y) =J−1∑j=0˜p2j(x)˜p2j+1(y)−˜p2j+1(x)˜p2j(y)rj,

and the entries of the kernels are computed by applying the appropriate operators to this expression.

3.7 Specification to the Harmonic Oscillator Potential

We now return to the case where the weight function is .

Theorem 3.4.

Let

 ⟨⋅|⋅⟩(X)=X2⟨⋅|⋅⟩1+⟨⋅|⋅⟩4.

A complete family of skew-orthogonal polynomials for the weight with respect to is given by

 P(X)2j(x)=j∑k=0(−1)kLk(−X2)Lk(0)Lk(x2), (3.3)

and

 P(X)2j+1(x) =2xP(X)2j(x)−2ddxP(X)2j(x) =4X2xm−1∑k=0(−1)kL12k(−X2)L12k(0)L12k(x2)+2x(−1)mL−12m(−X2)L−12m(0)L12m(x2). (3.4)

where is the generalized th Laguerre polynomial. The normalization of this family of polynomials is given by

 ⟨˜P(X)2m|˜P(X)2m+1⟩(X)=4π(m+1)!Γ(m+12)Lm(−X2)Lm+1(−X2). (3.5)

We can recover a family of monic skew-orthogonal polynomials by dividing by the leading coefficient. Specifically,

Corollary 3.5.

A complete family of monic skew-orthogonal polynomials for the weight with respect to is given by

 p(X)2j(x)=Lj(0)j!Lj(−X2)j∑k=0(−1)kLk(−X2)Lk(0)Lk(x2),

and

 p(X)2j+1(x)=xp(X)2j(x)−ddxp(X)2j(x).

The normalization for this family of monic skew-orthogonal polynomials is given by

 r(X)j=⟨˜p(X)2j|˜p(X)2j+1⟩(X)=4(j+1)!Γ(j+12)j!Lj+1(−X2)Lj(−X2).

Setting , we recover a family of skew-orthogonal polynomials for the harmonic oscillator two charge ensemble with fugacity equal to one, and we will write for and for .

4 Proofs

4.1 Proof of Theorem 2.1

We set . To prove 1, we use Theorem 3.1 and the skew-orthogonal polynomials from Corollary 3.5 to write

 Z(X)=Pf⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣0r(X)0−r(X)00⋱0r(X)J−1−r(X)J−10⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦=J−1∏j=0r(X)j.

Hence,

 Z(X)Z=J−1∏j=0r(X)jrj=LJ(−X2)L0(−1)LJ(−1)L0(−X2)=LJ(−X2)LJ(−1),

where again . Note .

The remaining claims follow from the above by definition and the properties of Laguerre polynomials.

4.2 Proof of Theorem 2.2

Point 3 of Theorem 2.1 specified to the first two moments produces

 E[L]=ddX[Z(X)Z]X=1,  Var(L)=⎡⎣ddX(XddXZ(X)Z)−(ddXZ(X)Z)2⎤⎦X=1.

Now, since and , we have that

 E(L)=2L1/2J−1(−1)LJ(−1)=2L1/2J(−1)LJ(−1)−2.

Further, using the differential equation , we also have that

 ddx(xddxLJ(−x2)) =−4xL′J(−x2)+4x3L′′J(−x2) =(−2x+4x3)L′J(−x2)+4xJLJ(−x2).

This yields

 Var(L)=4J−E(L)−E(L)2,

and so asymptotics of the variance follow from those for the mean.

Next introduce a version of Perron’s formula (see [5]),

 Lαn(−1)=12√πemα/2−1/4e2√m(1+C1(α)m−1/2+C2(α)m−1+O(m−3/2)),

where and are known explicitly. In particular, , , , and . Substituting into the above we then obtain

 E(L)=2√J+1−1−23√J+1+O(J−1)=2√J−1+13√J+O(J−1),

and which completes the proof of point 1 (recall ).

Moving to the limit law for , we introduce a little new notation. Set

 pN(k)=CNΓ(N2−k2+1)Γ(k2+12)Γ(k2+1)=CNqN(k)−1

with . For even, is the probability of particles of charge 1, otherwise this probability is zero, compare point 1 of Theorem 2.1. In the continuum limit this distinction is unimportant; we will show that, as

 (2N)1/4pN((2N)1/2+(2N)1/4c)=e−c2/2√2π(1+O(N−1/4)) (4.1)

uniformly for on compact sets.

First note that by Stirling’s approximation (in the form ) and again Perron’s formula (now in the simpler form ),