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The Partition Function of Multicomponent Log-Gases Christopher D. Sinclair∗ January 4, 2012

Abstract We give an expression for the partition function of a one-dimensional log-gas comprised of particles of (possibly) different integer charge at inverse temperature β = 1 (restricted to the line in the presence of a neutralizing field) in terms of the Berezin integral of an associated nonhomogeneous alternating tensor. This is the analog of the de Bruijn integral identities [3] (for β = 1 and β = 4) ensembles extended to multicomponent ensembles.

Keywords: Partition function, Berezin integral, Pfaffian, Hyperpfaffian, Grand canonical ensemble MSC2010 Classification: 15B52, 82C22, 60G55

1 Introduction We imagine a finite number of charged particles interacting logarithmically on an infinite wire modelled by the real line. Different particles may have different charges (which we will assume are positive integers), but any two particles with the same charge, that is, of the same species, are assumed to be indistinguishable. A potential is placed on the wire to keep the particles from escaping to infinity. This system is placed in contact with a heat reservoir with inverse temperature β. We will consider two ensembles: 1. The Canonical Ensemble. We assume that the number of each species of particle is fixed. 2. The Grand Canonical Ensemble. We assume that the sum of the charges, that is the total charge of the system, is fixed but the number of each species is variable.1 Our goal is to provide a closed form of the partition function of these ensembles, for certain values of β, in terms of Berezin integrals. As is standard, we will find that the partition function for the Grand Canonical Ensemble is the generating function for the Canonical ensemble as a function of fugacities of the species of particles. After a minor modification, the partition function can also be seen as the generating function for the correlation functions of both the Canonical Ensemble and the Grand Canonical Ensemble. ∗ This

research was supported in part by the National Science Foundation (DMS-0801243) standard notion of the Grand Canonical Ensemble is that where the number of particles is not fixed. That is, in its traditional sense, the Grand Canonical Ensemble is the direct sum over all possible values of the sum of the charges. What we refer to as the Grand Canonical Ensemble might be better referred to as an isocharge or zero current Grand Canonical Ensemble. 1 The

1

The Partition Function of Multicomponent Log-Gases

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1.1 The Setup Let J > 0 be an integer and suppose q = (q1 , q2 , . . . , q J ) is a vector of positive integer (charges) with each of the q j distinct. We imagine a system of particles consisting of M1 indistinguishable particles of charge q1 , M2 indistinguishable particles of charge q2 and so on. We will refer to q as the charge vector and M = (M1 , M2 , . . . , M J ) as the population vector of the system. These particles are restricted to lie on an infinite wire, identified with the real axis2 , and interact logarithmically, so that the energy contributed to the system by a pair particles with charges q and q′ located at x and x′ is given by−qq′ log |x − x′ |. (Infinite energy is allowed in the situation where x = x′ ). We suppose that the particles of charge q1 are identified with the location vector x1 = (x11 , x12 , . . . , x1M1 ); the location vectors x2 , . . . , x J are similarly defined. If M j = 0 for some j then x j is taken to be the empty vector. The particles are placed in a neutralizing field with potential U so that the total potential energy of the system is given by EM (x1 , x2 , . . . , x J ) =

J X j=1

qj

Mj X m=1

U(xmj ) −

J X

q2j

j=1

X m

log |xnj − xmj | −

X

q j qk

j

Mj X Mk X m=1 n=1

log |xkn − xmj |.

We assume that the system is in contact with a heat reservoir at inverse temperature β, but energy is allowed to flow between the reservoir and the system of particles. In this situation the Boltzmann factor, which gives the relative density of states, is given by 1

J

ΩM (x1 , x2 , . . . , x J ) = e−βEM (x ,...,x ) Mj Mj Y Mk J Y J Y Y Y YY j j βq2j −βq j U(xmj ) = × × |xkn − xmj |βq j qk . |xn − xm | e j=1 m

j=1 m=1

(1.1)

j

The probability (density) of finding the system in a state determined by the location vectors x1 , x2 , . . ., x J is then given by ΩM (x1 , x2 , . . . , x J ) , pM (x1 , x2 , . . . , x J ) = ZM M1 !M2 ! · · · M J ! where the partition function of the system is given by Z Z 1 ZM = ΩM (x1 , x2 , . . . , x J ) dµ M1 (x1 ) dµ M2 (x2 ) · · · dµ MJ (x J ), ··· M1 !M2 ! · · · M J ! RM1 RMJ

(1.2)

and µ M is Lebesgue measure on R M .3 The factors of M1 !M2 ! · · · M J ! appear since a generic state of the system x1 , x2 , . . . , x J has this many different representatives. We will always assume that the external potential U is such that ZM is finite. For the grand canonical ensemble, We may view M as a random vector and the probability (density) of finding the system with a prescribed population vector M and state x1 , x2 , . . . , x J is 2 With a minor modification, much of what is presented here can be shown mutadis mutandis for multicomponent loggasses confined to the unit circle. See [5] for the circular case for β = 2, and [6] for some physical application. 3 If M = 0 for some j then we will use the convention that j Z ΩM (x1 , x2 , . . . , xJ ) dµ M j (x j ) = ΩM (x1 , x2 , . . . , xJ ); M R j

alternately, in this situation we may assume that the integral over RM j does not actually appear in our expression for ZM . Likewise we will assume that sums and products over empty sets are respectively taken to be 0 and 1.

Christopher D. Sinclair

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given by pM (x1 , x2 . . . , x J ) · prob(M). Classically, the probability of finding the system in a state with (allowed) population vector M is taken to be J prob(M) = z1M1 z2M2 . . . z M J

where ZN =

X

ZM , ZN

(1.3)

1 z1M1 z2M2 . . . z M J ZM

M M·q=N

and z = (z1 , . . . , z J ) is a vector of positive real numbers called the fugacity vector. It shall sometimes be convenient to view z as a vector of indeterminants and ZN = ZN (z) as a polynomial in these indeterminants. Our main result will be to show that, for certain values of β and q, ZN (z) can be expressed as a Berezin integral with respect to the volume form in RN of the exponential of an (explicitly given) alternating element (i.e. form) in the exterior algebra Λ(RN ). By J construction, ZM is the coefficient of z1M1 · · · z M J , and thus the integral formulation of ZN (z) is exactly the generating function we seek.

2 Wronskians, Berezin Integrals and Hyperpfaffians Here we collect the machinery necessary to state our main results. Given a non-negative integer L, let L = {1, 2, . . . L}, and, assuming K ≥ L is an integer, let t:LրK be a strictly increasing function, 0 < t(1) < t(2) < · · · < t(L) ≤ K. We will use such functions to keep track of minors of matrices, elements in exterior algebras and Wronskians of families of polynomials. Such indexing functions will always be written as fraktur minuscules.

2.1 Wronskians A complete family of monic polynomials is a sequence of polynomials P = (p1 , p2 , . . .) such that each pn is monic and deg pn = n − 1. We define the L-tuple Pt = (pt(1) , . . . , pt(L) ). And, given 0 ≤ ℓ < L we define the modified ℓth differentiation operator by D0 f (x) = f (x)

and

Dℓ f (x) =

1 dℓ f . ℓ! dxℓ

(2.1)

The Wronskian of Pt is then defined to be h iL Wr(Pt ; x) = det Dℓ−1 pt(k) (x)

k,ℓ=1

.

The Wronskian is often defined without the ℓ! in the denominator of (2.1); this combinatorial factor will prove convenient in the sequel. The reader has likely seen Wronskians in elementary differential equations, where they are used to test for linear dependence of solutions.

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2.2 The Berezin Integral If e1 , . . . , eK is a basis for RK , then ǫt = et(1) ∧ · · · ∧ et(L) is an element in ΛL (RK ), and {ǫt t : L ր K} is a basis for ΛL (RK ). In particular, we will denote ǫvol = e1 ∧ e2 ∧ · · · ∧ eK . Given 0 < k ≤ K we define the linear operator ∂/∂ek : ΛL (RK ) → ΛL−1 (RK ) by (−1)α et(1) ∧ · · · ∧ et(α−1) ∧ et(α+1) ∧ · · · ∧ et(L) if k = t−1 (α); ∂ ǫt = ∂ek 0 otherwise.

That is, if ek appears in ǫt then ∂ǫt /∂ek is formed by shuffling ek to the front of ǫt (taking into account the alternation of signs) and then dropping it. Given 0 < k1 , . . . , k M ≤ K we then define the Berezin integral as the linear operator on Λ(RK ) → Λ(RK ) specified by Z ∂ ∂ ∂ ǫt dek1 dek2 · · · dekM = ··· ǫt . ∂ekM ∂ek2 ∂ek1 Berezin integrals were introduced in [1] as a Fermionic analog to the Gaussian integrals which appear in Bosonic field theory. We will mostly be interested in Berezin integrals of the form Z Z ǫt dǫvol = ǫt de1 · · · deK . In this case, the Berezin integral is simply the projection operator Λ(RK ) → ΛK (RK ) R. Notice in particular that, if σ ∈ S K then Z eσ(1) ∧ eσ(2) ∧ · · · ∧ eσ(K) dǫvol = sgn σ.

2.3 Exponentials of Forms and Hyperpfaffians Given ω ∈ Λ(RK ) we define ω∧0 = 1 and for 0 < m ω∧m = ω ···∧ ω | ∧ {z }. m

Using this we define

eω =

∞ X ω∧m m=0

m!

.

If ω = ω0 + ω1 + · · · + ωK with ωk ∈ Λk (RK ) then it is easily verified that eω = eω0 ∧ eω1 ∧ · · · ∧ eωK . Moreover, eω0 is a real number equal to its traditional definition, and if k > 0 then the sum defining eωk is a finite sum. In the situation where k divides K, that is K = km, then we define the hyperpfaffian PF(ωk ) to be the real number defined by ω∧m k = PF(ωk ) ǫvol . m!

Christopher D. Sinclair

Alternately, PF(ωk ) =

Z

5

eωk ǫvol .

The hyperpfaffian is related to the Pfaffian of an antisymmetric K × K matrix by associating the matrix to a 2-form in the obvious manner. We see therefore that the Berezin integral formed with respect to ǫvol is a generalization of hyperpfaffians, which themselves are generalizations of Pfaffians.

3 Statement of Results Suppose b is a positive integer and β = b2 . Set K = bN and L j = bq j for j = 1, 2, . . . , J. For any complete family of monic polynomials P we define ω1 , ω2 , . . . , ω J ∈ Λ(RK ) as follows. 1. If L j is even, ωj =

X Z

t:L j րK

R

e−βq j U(x) Wr(Pt ; x) dx ǫt ;

(3.1)

2. If L j is odd, X 1 Z Z e−βq j U(x) e−βq j U(y) Wr(Pt ; x)Wr(Pu ; y) sgn(y − x) dxdy ǫt ∧ ǫu . ωj = 2 R R t,u:L րK

(3.2)

j

Notice that ω j is in ΛL j (RK ) when L j is even and is in Λ2L j (RK ) when L j is odd. Theorem 3.1. Suppose β = b2 and K = bN is even. Given a charge vector q let L j = bq j ;

j = 1, 2, . . . , J,

and, for any complete family of monic polynomials, define the form ω ∈ Λ(RbN ) by ω(z) =

J X

z jω j,

j=1

where ω j is defined as in (3.1) or (3.2). If the L j are positive integers, at most one of which is odd, then Z ZN (z) = eω(z) dǫvol . Remark. This is an algebraic identity which can be written more generally by replacing the integral over R with integrals over other sets (for instance, the partition functions for multicharge circular ensembles can be likewise expressed in terms of Berezin integrals). The only analytic prerequisite is the finiteness of the ZM which allows for the use of Fubini’s Theorem. This theorem covers certain situations which have appeared before. Certainly the Pfaffian partition functions of the classical one-species ensembles GOE and GSE (and their non-Gaussian variants) are a corollary. These cases follow from the ‘classical’ de Bruijn identities [3]. (see [7] and the references contained therein for their applications to random matrix theory). The classical ensembles can be viewed either as ensembles of charge 1 particles at respective inverse temperatures β = 1 and β = 4, or to ensembles of charge 1 and charge 2 particles (respectively) at inverse temperature

The Partition Function of Multicomponent Log-Gases

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β = 1. Recent work by the author, B. Rider and Y. Xu produced (among other things) a Pfaffian formulation of the partition function for the grand canonical ensemble for charge vector q = (1, 2) at inverse temperature β = 1 [9]. When the external field is Gaussian, and by tuning the fugacity, this latter ensemble provides an unusual interpolation between the classical ensembles GOE and GSE. Moreover, the skew-orthogonal polynomials necessary to solve the ensemble (that is explicitly derive the matrix kernel in which the correlation functions can be expressed and analyzed) were explicitly computed in terms of certain generalized Laguerre polynomials. Amongst other results, this allowed us to compute the distribution of the number of each type of particle for various fugacities. This analysis follows similar work for the two charge circular ensemble with charge vector q = (1, 2) initiated by P. Forrester and others (see [4, §7.10] and the references therein), and indeed the partition functions for those ensembles can be expressed as a Pfaffian, and hence in terms of (variants) of the above Berezin integrals. Recent work of the author [10] has lead to a hyperpfaffian expression for the partition functions of single-species ensembles of charge 1 particles when β = L2 is a perfect square, or β = L2 + 1 is even. In the former case, these ensembles can also be interpreted as systems of charge L particles at β = 1.

3.1 Correlation Functions Using a slight modification, the partition function gives a generating function for the correlation functions. For single species ensembles, the correlation functions are simply renormalized marginal densities. For multicomponent ensembles, however, the situation is more complicated (though the marginal probabilities are an important ingredient). For fixed population vector M and vector m = (m1 , m2 , . . . , m J ) with 0 ≤ m j ≤ M, we define j

j

ξ j = (ξ1 , . . . , ξm j )

and

j

j

y j = (y1 , . . . , y M j −m j ),

and set j ξ j ∨ y j = (ξ1j , . . . , ξmj j , y1j , . . . , y M ). j −m j

The mth marginal probability density of pM is then given by Z Z 1 2 J pM (ξ1 ∨ y1 , . . . , ξJ ∨ y J ) dµ M1 −m1 (y1 ) · · · dµ MJ −m j (y J ), ··· pM,m (ξ , ξ , . . . , ξ ) = R M1 −m1

R M J −m J

and by symmetry, the probability (density) that our system is in a state (x1 , . . . , x J ) which occupies the substate (ξ1 , . . . , ξJ ) (that is, viewed as sets, ξ j ⊆ x j for each j) is given by M1 ! MJ ! ··· pM,m (ξ1 , ξ2 , . . . , ξ J ) (M1 − m1 )! (M J − m J )! 1 = ZM (M1 − m1 )! · · · (M J − m J )! Z Z ΩM (ξ1 ∨ y1 , . . . , ξJ ∨ y J ) dµ M1 −m1 (y1 ) · · · dµ MJ −m j (y J ). ··· ×

RM,m (ξ1 , ξ2 , . . . , ξJ ) =

R M1 −m1

(3.3)

R M J −m J

This is the mth correlation function for the canonical ensemble with population vector M. To get the mth correlation function for the grand canonical ensemble we need to sum over the related correlation function for the canonical ensemble over all allowable population vectors M with m j ≤ M j for each j (a situation we will abbreviate by m ≤ M), taking into account the probability

Christopher D. Sinclair

7

of being in a state with prescribed population vector. That is, the probability (density) of the (grand canonical) system is in a state (x1 , . . . , x J ) which occupies the substate (ξ1 , . . . , ξ J ) is given by X prob (ξ1 , . . . , ξ J ) ⊆ (x1 , . . . , x J ) = prob(M) · RM,m (ξ1 , ξ2 , . . . , ξ J ). M.q=N M≥m

Denoting this density by RN,m , (1.3) and (3.3) yield RN,m (ξ1 , . . . , ξ J ) = ×

Z

J z1M1 · · · z M 1 X J ZN (z) M.q=N (M1 − m1 )! · · · (M J − m j )! M≥m Z ΩM (ξ1 ∨ y1 , . . . , ξ J ∨ y J ) dµ M1 −m1 (y1 ) · · · dµ MJ −m j (y J ). ···

R M1 −m1

R M J −m J

Notice that, by ignoring the prefactor ZN (z), and up to an easily recoverable constant, the coefficient J of z1M1 · · · z M J in RN,m is the mth correlation function for the corresponding canonical ensemble. We can in turn give a generating function for the correlation functions for the grand canonical j j j j j j ensemble as follows: Let c j = (c1 , c2 , . . . , cN ) and ζ j = (ζ1 , ζ2 , . . . , ζN ) and define the measures ν j and η j by N X dν j j j (x) = e−βq j U(x) and η j (x) = e−βq j U(x) cn δ(x − ζn ), dµ j n=1 where δ(x) is the probability measure with unit mass at x = 0. It is convenient at this point to index the forms ω j from (3.1) and (3.2) by ν j so that, for instance when L j is even, X Z νj ωj = Wr(Pt ) dν j ǫt . t:L j րK

R

ν Quantities which are dependent on ν = (ν1 , . . . , ν J ) will be denoted by, for instance, ZM , ZNν and ν ω (z). Theorem 3.1 is purely algebraic, and thus, we have, for instance that Z exp{ων (z)} dǫvol . ZNν (z) =

We can generalize these quantities by replacing ν with other vectors of measures. The following ν+η theorem gives particular relevance to ZN (z, c1 , . . . , c J ), where the notation indicates the additional dependence on the c j . Claim 3.2. The mth correlation function of the grand canonical ensemble is the coefficient of mj J Y Y

ν+η

cℓj

in

j=1 ℓ=1

ZN (z, c1 , . . . , c J ) . ZN (z)

That is, if ξ j = (ζ1j , . . . , ζmj j ), and we define ∂ ∂ ∂m j ··· j = ∂c j ∂c j ∂cm j 1 then, RN,m (ξ1 , . . . , ξ J ) =

" m1 1 ∂ ∂mJ ν+η 1 J · · · Z (z, c , . . . , c ) . 1 ZN (z) ∂c1 ∂c J N c =···=c J =0

The Partition Function of Multicomponent Log-Gases

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Moreover, the mth correlation function of the canonical ensemble with population vector M is given by " 1 ∂ M J ∂ m1 ∂ M1 ∂mJ ν+η 1 J RM,m (ξ1 , . . . , ξJ ) = · · · . Z (z, c , . . . , c ) · · · 1 ZM M1 ! · · · M J ! ∂z M1 ∂c J N ∂z MJ ∂c1 z=c =···=c J =0 1

J

The proof of this claim is standard (it is the multicomponent version of the ‘functional differentiation’ method), and follows mutatis mutandis that for Ginibre’s real ensemble [2, Prop. 6]. To write the correlation functions explicitly in terms of a Berezin integral (taking all of the L j to be even for convenience), we note that X Z ν η ν +η Wr(Pt ) d(ν j + η j ) ǫt = ω j j + ω j j , ωjj j = t:L j րK

R

and ων+η =

J X

ν +η j

ωjj

= ων + ωη ,

j=1

Hence, exp{ων+η } = exp{ων } ∧ exp{ωη }. This is useful, since the first term in the right hand side is independent of the c j . The following maneuvers are elementary ∂ m1 ∂ m1 ν+η ν exp{ω } = exp{ω } ∧ exp{ωη } ∂c1 ∂c1 m1 ^ X 1 ν η Wr(Pt ; ζℓ1 )ǫt ; = exp{ω } ∧ exp{ω } ∧ e−βq1 U(ζℓ ) ℓ=1

t:L1 րK

note that since all forms are even, we do not have to specify their order. It follows that m1 ^ X ∂ m1 ν η ν −βq1 U(ζℓ1 ) 1 exp{ω } ∧ exp{ω } = exp{ω } ∧ e , )ǫ Wr(P ; ζ t t ℓ c1 =0 ∂c1 ℓ=1 t:L րK 1

and that

mj J ^ ^ X ∂ m1 ∂m J j ν η ν −βq j U(ζℓj ) Wr(P ; ζ )ǫ exp{ω } ∧ exp{ω } = exp{ω } ∧ e · · · t ℓ t . c1 =···=cJ =0 ∂c J ∂c1 j=1 ℓ=1 t:L րK j

We therefore have the following corollary to Claim 3.2.

Corollary 3.3. If, for each 1 ≤ j ≤ J, L j is even, and ξ j = (ζ1j , . . . , ζmj j ) ∈ Rm j , then RN,m (ξ1 , . . . , ξJ ) =

1 ZN (z)

Z

exp{ων } ∧

mj J ^ ^ j=1 ℓ=1

j

e−βq j U(ζℓ )

X

t:L j րK

Wr(Pt ; ζℓj )ǫt dǫvol .

Note that we do not have to justify the exchange of the derivatives and the ‘integral’ in Claim 3.2, since the Berezin integral is not an integral in the traditional sense. That is, Claim 3.2 is an algebraic, not an analytic, identity. Notice also, that the quantity in braces is an (m · L)-form, and therefore only the projection of exp ων onto the space of (K − m · L)-forms will make a contribution to the mth correlation function. Finally, we note that a similar formula for the partial correlation function RM,m is available via functional differentiation with respect to the z variables.

Christopher D. Sinclair

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4 The Proof of Theorem 3.1 4.1 The Confluent Vandermonde Determinant Suppose 0 < L < K, x ∈ R and P = (p1 , p2 , . . .) is any complete family of monic polynomials. We define the K × L matrix h iK,L , VL (x) = Dℓ pn (x) n,ℓ=1

1

and given an admissible population vector M and x , . . . , x J with x j ∈ R M j , we define the K × K confluent Vandermonde matrix by ··· VLJ (x1J ) · · · VLJ (x JMJ ) . VM (x1 , . . . , x J ) = VL1 (x11 ) · · · VL1 (x1M1 ) | {z } | {z } M1

MJ

√ (Recall that L j = βq j ). In this case, the confluent Vandermonde determinant identity [8] has that det VM (x1 , . . . , x J ) =

Mj Y Mk J Y Y YY j j 2 j (xn − xm )L j × (xkn − xm )L j Lk j=1 m

=

(4.1)

j

Mj Y Mk J Y Y YY 2 2 2 (xkn − xmj )b q j qk . (xnj − xmj )b q j × j=1 m

j

When all of the L j are even, it follows from (1.1) that ΩM (x1 , x2 , . . . , x J ) =

Mj J Y Y j=1 m=1

j e−βq j U(xm ) det VM (x1 , . . . , x J )

(4.2)

We will deal with the situation where one of the L j is odd in Section 4.5.

4.2 The Laplace Expansion of the Determinant Each t : L ր K specifies a unique t′ : K − L ր N whose range is disjoint from t. Given a K × K matrix V = [vm,n ] and t, u : L ր K then we may create a L × L minor of V by selecting the rows and columns from the ranges of t and u. That is, we write L Vt,u = vt(k),u(ℓ) k,ℓ=1 .

Notice that the complementary minor to Vt,u is given by Vt′ ,u′ . We define sgn t by Z sgn t =

ǫt ∧ ǫt′ dǫvol .

More generally let tmj : L j → K

where

j = 1, 2, . . . J

and

and set ~t = (t1 , . . . , t1M , · · · , t J , . . . , t J ) M |1 {z }1 |1 {z }J M1

MJ

m = 1, 2, . . . , M j ,

10

The Partition Function of Multicomponent Log-Gases

We will use ~t to select minors of VM (x1 , . . . , x J ) each of which depends only on a single location variable. We denote the set of all such ~t by Im We define sgn~t by Z (4.3) sgn~t = ǫt11 ∧ · · · ∧ ǫt1M ∧ · · · ∧ ǫt1J ∧ · · · ∧ ǫtJM dǫvol . | {z }J | {z }1 MJ

M1

j

Clearly, sgn~t = 0 unless the ranges of the various tm are mutually disjoint, and otherwise sgn t is the signature of the permutation defined by concatenating the ranges of the various tmj in the appropriate order. We will reserve the symbol ~i for the vector whose coordinate functions are given by imj (ℓ) = ℓ + (m − 1)L j + M1 L1 + · · · + M j−1 L j−1 That is, for instance, if L = (2, 3) and M = (2, 2) then the ranges of i11 , i12 , i21 and i22 are given respectively by {1, 2}, {3, 4}, {5, 6, 7} and {8, 9, 10}. Clearly sgn~i = 1. This notation is convenient to represent the Laplace expansion of the determinant (which we will write in the form most useful for our ultimate goal). det V =

X

sgn~t

Mj J Y Y

det Vtmj ,imj .

(4.4)

j=1 m=1

~t∈IM

Applying (4.4) to VM (x1 , . . . , x J ) we find det VM (x1 , . . . , x J ) =

X

sgn~t

Mj J Y Y j=1 m=1

~t∈IM

j

det VM (xm ), t j ,i j m m

j j where the notation reflects the fact that VM t j ,i j (xm ) in independent of all location variables except xm .

From the definition of VM we see that det VMj j (xmj ) = Wr(Ptmj ; xmj ), and therefore tm ,im

M

1

J

det V (x , . . . , x ) =

X

~t∈IM

sgn~t

Mj J Y Y

Wr(Ptmj ; xmj ).

(4.5)

j=1 m=1

4.3 Fubini’s Theorem From (1.2), (4.2) and (4.5) we have that Z Y Z Z Mj J Y X j 1 ~ ZM = e−βq j U(xm ) Wr(Ptmj ; xmj ) ··· sgn t M M M M1 !M2 ! · · · M J ! R J j=1 m=1 R 1 R 2 ~t∈IM

× dµ M1 (x1 ) dµ M2 (x2 ) · · · dµ MJ (x J ). Fubini’s Theorem implies then that Mj Z J Y X Y 1 ~ ZM = e−βq j U(x) Wr(Ptmj ; x) dx. sgn t M1 !M2 ! · · · M J ! j=1 m=1 R ~t∈IM

(4.6)

Christopher D. Sinclair

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Thus, ZN (z) =

Mj Z J Y X z M1 z M2 · · · z MJ X Y J 1 2 e−βq j U(x) Wr(Ptmj ; x) dx sgn~t M !M ! · · · M ! 1 2 J R M j=1 m=1 ~t∈IM

M·q=N

=

X X

M M·q=N

sgn~t

~t∈IM

Z Mj J Y 1 Y e−βq j U(x) Wr(Ptmj ; x) dx. zj M ! j R m=1 j=1

4.4 Enter the Berezin Integral Using the definition of sgn~t (4.3) we find X X Z ZN (z) = ǫt11 ∧ · · · ∧ ǫt1M ∧ · · · ∧ ǫt1J ∧ · · · ∧ ǫtJM dǫvol M ~ | {z }1 | {z }J t∈IM M·q=N

M1

MJ

Z Mj J Y 1 Y × e−βq j U(x) Wr(Ptmj ; x) dx. zj M j ! m=1 R j=1

Exploiting the linearity of the Berezin integral, ZN (z) =

Z Z X X^ Mj J 1 ^ e−βq j U(x) Wr(Ptmj ; x) dx ǫtmj dǫvol , zj M j ! m=1 R M j=1 M·q=N

~t∈IM

where the wedge products are taken in the standard order. Next we may expand the sum over ~t ∈ IM as X X X ··· = ··· t11 ,...,t1M :L1 րK

~t∈IM

1

t21 ,...,t2M :L2 րK 2

X

t1J ,...,t JM :L J րK 1

··· ,

so that ZN (z) =

Z X ^ J 1 M j! M j=1 M·q=N

X

Mj Z ^ e−βq j U(x) Wr(Ptmj ; x) dx ǫtmj dǫvol . zj R

j t1j ,...,t M :L j րK m=1 j

We observe that X

Mj Z ^ e−βq j U(x) Wr(Ptmj ; x) dx ǫtmj zj

j t1j ,...,t M :L j րK m=1

R

j

∧M j X Z e−βq j U(x) Wr(Pt ; x) dx ǫt = zj R

t:L j րK

= (z j ω j ) and hence ZN (z) =

∧M j

,

Z X ^ J (z j ω j )∧M j dǫvol . M j! M j=1 M·q=N

The Partition Function of Multicomponent Log-Gases

12

Now, we can remove the restriction M · q = N from the sum in this expression, since the Berezin integral will be zero for any M not satisfying this condition. (If M does not satisfy this condition the form in the integrand will not be in ΛK (RK ) and hence its projection onto ΛK (RK ) R will be 0). Thus, Z X^ J (z j ω j )∧M j ZN (z) = dǫvol M j! M j=1 =

Z X ∞ X ∞

M1 =0 M2 =0

···

∞ ^ J X (z j ω j )∧M j dǫvol M j! M =0 j=1 J

Z ^ J X ∞ (z j ω j )∧M dǫvol = M! j=1 M=1 Z = ez1 ω1 ∧ ez2 ω2 ∧ · · · ∧ ezJ ω j dǫvol Z = eω(z) dǫvol , as desired.

4.5 When one of the L j is odd In the case where exactly one of the L j is odd, we will reorder the q j so that L1 is odd and L2 , . . . , L J are even. In this situation, (1.1) and (4.1) imply that ΩM (x1 , x2 , . . . , x J ) =

Mj J Y Y

j

e−βq j U(xm )

j=1 m=1

Y

1≤m

sgn(x1n − x1m ) det VM (x1 , x2 , . . . , x J ),

where the additional factors of the form sgn(x1n − x1m ) exist in order to make the expression nonnegative for all choices of x1 , x2 , . . . , x J . Defining the M1 × M1 antisymmetric matrix i M1 h , T(x1 ) = sgn(x1n − x1m ) m,n=1

When K is even, so is M1 , and in this situation

Y

Pf T(x1 ) =

1≤m

sgn(x1n − x1m ).

Thus, 1

2

J

ΩM (x , x , . . . , x ) =

Mj J Y Y j=1 m=1

j e−βq j U(xm ) Pf T(x1 ) det VM (x1 , x2 , . . . , x J ),

Following the analysis of the case where all L j even we find the analog of (4.6) in the current situation is Mj Z J Y Y X 1 ~ ZM = sgn t e−βq j U(x) Wr(Ptmj ; x) dx M1 !M2 ! · · · M J ! j=2 m=1 R ~t∈IM

×

Z

R M1

Pf T(x)

M1 Y m=1

e−βq1 U(xm ) Wr(Pt1m ; xm ) dµ M1 (x)

Christopher D. Sinclair

13

And, Mj Z J Y Y X z M1 z M2 · · · z MJ X J 1 2 ~ ZN (z) = e−βq j U(x) Wr(Ptmj ; x) dx sgn t M1 !M2 ! · · · M J ! M j=2 m=1 R ~t∈IM

M·q=N

×

Z

Pf T(x)

R M1

M1 Y m=1

e−βq1 U(xm ) Wr(Pt1m ; xm ) dµ M1 (x).

Using the same maneuvers as before, we can write Z X ^ J z M1 (z j ω j )∧M j ZM = ∧ 1 M j! M1 ! M j=2 M·q=N

Z

Pf T(x)

R M1

M1 Y m=1

X

t11 ,...,t1M :L1 րK 1

e−βq1 U(xm ) Wr(Pt1m ; xm ) dµ M1 (x) ǫt11 ∧ ǫt12 ∧ · · · ∧ ǫt1M dǫvol . 1

It is shown in [10, Section 4.2] that z1M1 M1 !

X

t11 ,...,t1M :L1 րK 1

Z

Pf T(x)

R M1

M1 Y m=1

e−βq1 U(xm ) Wr(Pt1m ; xm ) dµ M1 (x) ǫt11 ∧ ǫt12 ∧ · · · ∧ ǫt1M =

1

(z1 ω1 )∧M1 . M1 !

(The left hand side of this expression is the partition function of a system of M1 particles each of charge q1 when β is an odd square; showing partition functions of such systems is a hyperpfaffian was one of the goals of [10]). We therefore have that Z X^ Z J (z j ω j )∧M j ZN (z) = dǫvol = eω(z) dǫvol , M j! M j=1 as desired.

References [1] F. A. Berezin. The method of second quantization. Translated from the Russian by Nobumichi Mugibayashi and Alan Jeffrey. Pure and Applied Physics, Vol. 24. Academic Press, New York, 1966. [2] Alexei Borodin and Christopher D. Sinclair. The Ginibre ensemble of real random matrices and its scaling limits. Comm. Math. Phys., 291(1):177–224, 2009. [3] N. G. de Bruijn. On some multiple integrals involving determinants. J. Indian Math. Soc. (N.S.), 19:133–151 (1956), 1955. [4] Peter Forrester. Log-gases and Random Matrices. London Mathematical Society Monographs. Princeton University Press, 2010.

14

The Partition Function of Multicomponent Log-Gases

[5] Niko Jokela, Matti J¨arvinen, and Esko Keski-Vakkuri. The partition function of a multicomponent Coulomb gas on a circle. J. Phys. A, 41(14):145003, 12, 2008. [6] Niko Jokela, Matti J¨arvinen, Esko Keski-Vakkuri, and Jaydeep Majumder. Disk partition function and oscillatory rolling tachyons. J. Phys. A, 41(1):015402, 13, 2008. [7] Madan Lal Mehta. Random matrices, volume 142 of Pure and Applied Mathematics (Amsterdam). Elsevier/Academic Press, Amsterdam, third edition, 2004. [8] C. Meray. Sur un determinant dont celui de Vandermonde n’est qu’un particulier. Revue de Math´ematiques Sp´eciales, 9:217–219, 1899. [9] Brian Rider, Christopher D. Sinclair, and Yuan Xu. A solvable mixed charge ensemble on the line: global results. Accepted for publication, 2010. [10] Christopher D. Sinclair. Ensemble averages when β is a square integer. Submitted for publication, 2010. http://arxiv.org/abs/1008.4362.

Christopher D. Sinclair Department of Mathematics, University of Oregon, Eugene OR 97403

email: [email protected]