Matrix integral solutions to the discrete KP hierarchy and its Pfaffianized version

0.17 MB·

by  Stéphane Lafortune

Matrix integral solutions to the discrete KP hierarchy and its
Pfaffianized version
St´ephane Lafortune1 and Chun-Xia Li2,1
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1 1 Department of Mathematics, College of Charleston, Charleston, SC 29401, USA
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2 School of Mathematical Sciences, Capital Normal University, Beijing 100048, CHINA
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Abstract

0 Matrix integrals used in random matrix theory for the study of eigenvalues of Hermitian ensembles
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have been shown to provide τ-functions for several hierarchies of integrable equations. In this article,

] we extend this relation by showing that such integrals can also provide τ-functions for the discrete KP
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S hierarchy and a coupled version of the same hierarchy obtained through the process of Pfaffianization.
. To do so, we consider the first equation of the discrete KP hierarchy, the Hirota-Miwa equation. We
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i write the Wronskian determinant solutions to theHirota-Miwa equation and consider a particular form
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n of matrix integrals, which we show is an example of those Wronskian solutions. The argument is then
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generalizedtothewholehierarchy. AsimilarstrategyisusedforthePfaffianizedversionofthehierarchy

2 except that in that case, thesolutions are written in terms of Pfaffians rather than determinants.
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Key words. Matrix integrals, Hirota-Miwa equation, Wronskian determinant solutions, Pfaffianized
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0 Hirota-Miwa system, Pfaffian solutions.
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1 Introduction
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: Inrandommatrixtheory,thanks to a trick introducedby Mehta [1,2], the matrix integralsused to compute
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the probability that the eigenvalues of the matrices in an Hermitian ensemble belong to a certain interval
i
X can be written as integrals over the eigenvalues. These integrals take the form
r
a N
··· |x −x |βexp f(x ) dx dx ... dx , (1.1)
j k j 1 2 N
 
Z Z 1≤j<k≤N j=1
Y X
 
where β = 1,2, or 4 correspond, respectively, to the orthogonal, unitary, and symplectic ensembles and
exp(f(x)) is a weight function. Note that the terminology orthogonal, unitary, and symplectic for the
different values of β refers to the symmetry groups of the underlying measures [2,3].
Integrals of the form (1.1) have been studied in the context of integrable systems and matrix models in
string theory [4–15]. In this context, a dependence on the variables t , i = 1,2,3,... is introduced in the
i
following way
1 ∞ ∞ N ∞
Z(β)(t)= ··· |x −x |βexp c η(x ,t) dx dx ... dx , η(x,t)= xit +f(x),
N N! j k  j  1 2 N i
Z−∞ Z−∞1≤j<k≤N j=1 i=1
Y X X
  (1.2)
where the parameter c is chosen such that c = 1 for β = 1,2 and c = 2 for β = 4. In the case β = 2, the
integral(1.2)givesthepartitionfunctionfortheHermitianone-matrixmodel[4–7]anditwasindependently
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showntoprovideaτ-functionfortheToda[4]andtheKP[8,9]hierarchies. Inthecasesβ =1,4theintegrals
(1.2)provideτ-functions forthecoupledKPhierarchy[12,16]andthe PfaffLatticehierarchy[14,17]. While
related through their bilinear forms, the KP and Pfaff hierarchies were discovered in different ways as the
Pfafflatticewasfirstderivedin[14,17]throughLiealgebraictechniques,whilethecoupledKPequationwas
obtained using the procedure now knownas Pfaffianizationapplied to the KP equation [16]. The procedure
of Pfaffianization allows to generalize soliton equations with solutions represented by determinants to their
coupled systems with solutions represented by Pfaffians.
This connection of (1.1) with integrable systems was deepened by the authors of [18] who introduced
dependence on the continuous variables t ,t , and s and the discrete variables n and n in the following
1 2 1 2
way
1 ∞ ∞ N N
Z(β)(t ,t ,n )= ··· (1+x )cn1 |x −x |βexp cη(x ,t) dx ···dx ,
N 1 2 1 N! i j k  j  1 N
Z−1 Z−1 i=1 1≤j<k≤N j=1
Y Y X
 
1 ∞ ∞ N N
Z(β)(t ,s,n )= ··· xcn1 |x −x |βexp cη(x ,t ,s) dx ···dx ,
N 1 1 N! i j k  j 1  1 N
Z0 Z0 i=1 1≤j<k≤N j=1
Y Y X
1 ∞ ∞ N  b N 
Z(β)(t ,n ,n )= ··· xcn1(1+x )cn2 |x −x |βexp cη(x ,t ) dx ···dx ,
N 1 1 2 N! i i j k  j 1  1 N
Z0 Z0 i=1 1≤j<k≤N j=1
Y Y X
  (1.3)
where η(x,t) = xt +x2t +f(x), η(x,t ,s) = xt −x−1s+f(x), η(x,t ) = xt +f(x), and where the
1 2 1 1 1 1
parameter c is chosen such that c = 1 for β = 1,2 and c = 2 for β = 4. The authors of [18] show that the
matrix integrals in (1.3) with β = 2bsatisfy the bilinear versionbs of the differential-difference KP equation,
the two-dimensionalTodalattice, andthe semi-discreteTodalattice,respectively. Furthermore,inthe cases
β =1 and β =4, the integrals in (1.3) are shown to satisfy the Pfaffianized versions of those equations.
Themostdirectwaythatintegralsoftheform(1.2)solvebilinearequationsistousetheidentitiesbelow,
which were introduced in [19,20]:
1
··· det[φ (x )] ·det[ψ (x )] dx dx ... dx
N! i j 1≤i,j≤N i j 1≤i,j≤N 1 2 N
Z Z
=det φ (x)ψ (x)dx , (1.4)
i j
(cid:20)Z (cid:21)i,j=1,...,N
1
··· det[φ (x )] dx dx ... dx
i j i,j=1,...,N 1 2 N
N!
Z Z
=Pf (φ (x)φ (y)−φ (y)φ (x))dxdy , (1.5)
i j i j
 
ZZ
x<y i,j=1,···,N
 
1
... det[φ (x ),ψ (x )] dx ...dx
i j i j i=1,...,2N,j=1,...,N 1 N
N!
Z Z
=Pf (φ (x)ψ (x)−φ (x)ψ (x))dx . (1.6)
i j j i
(cid:20)Z (cid:21)i,j=1,...,2N
For the case β = 2, the integrals in (1.2) and (1.3) are written as determinants using (1.4), while in the
case β = 1,4, they are written as Pfaffians using (1.5) and (1.6). As pointed out by Kakei in [12], once
written in that form, the integrals in (1.2) in the cases β = 1,2 can be interpreted as the continuum limits
ofN-solitonsolutions. Itcanalsobe verifiedthatthe sameobservationappliesto the integrals(1.3)studied
in[18]. ItisinterestingtonotethatTracyandWidom[3]usedtheidentities(1.4),(1.5),and(1.6),toobtain
Fredholmdeterminant expressionsfor expressions involvedin the study of the spacings between consecutive
eigenvalues.
In this article, we will extend the relation between integrals of the form (1.1) and integrable systems by
obtaining matrix integral solutions to the discrete KP hierarchy and its Pfaffianized version. This will be
done by introducing a dependence of (1.1) on discrete variables and on an auxiliary continuous variable.
We will be using the identities (1.4), (1.5), and (1.6) in order to interpret our integrals as determinants and
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Pfaffians to prove our results. Once more, the case β = 2 will give rise to solutions to the initial equations
(in our case the discrete KP hierarchy),while the casesβ =1,4 will give rise to solutions of the Pfaffianized
versions.
Specifically, our strategy will be to consider the following integrals
1 1 N
Z(β)(n ,n ,n ,s)= 1 a3 ··· a3 (1−a x )−cn1/a1(1−a x )−cn2/a2
N 1 2 3 N! 1 i 2 i
Z−∞ Z−∞i=1
Y (1.7)
N
(1−a x )−cn3/a3 |x −x |βexp c η(x ,s) dx ···dx ,
3 i i j i 1 N
" #
1≤i<j≤N i=1
Y X
where n , i = 1,2,3 are discrete variables, s is a continuous variable, the lattice parameters satisfy 0 <
i
a ,a ≤ a , η(x,s) = xs+η (x), c = 1 for β = 1,2 and c = 2 for β = 4. The integrals (1.7) will provide
1 2 3 0
solutionstotheHirota-MiwaanditsPfaffianizedversion. Thoseintegralswillthenbegeneralized(see(2.9))
to obtained τ-functions for the whole discrete KP hierarchy and its Pfaffianized version.
The rest of this paper is organized as follows. In Section 2, we present the Wronskian determinant
solutions to the Hirota-Miwa equation and prove that the matrix integral (1.7) with β = 2 is a particular
exampleofsucha solution. We thengeneralizethe argumentto the whole discreteKP hierarchy. InSection
3, we show that the integrals (1.7) with β = 1 and β = 4 give solutions to the Pfaffianized version of
the Hirota-Miwa equation. Matrix integral solutions for the whole Pfaffianized version of the discrete KP
hierarchy are also obtained. Conclusions are given in Section 4.
2 Matrix integral solutions to the discrete KP hierarchy
2.1 The Hirota-Miwa equation and its determinant solutions
The Hirota-Miwa equation is the first bilinear equation of the discrete KP hierarchy. It is given by [21,22]
a (a −a )τ(n +a ,n ,n )τ(n ,n +a ,n +a )
1 2 3 1 1 2 3 1 2 2 3 3
+a (a −a )τ(n ,n +a ,n )τ(n +a ,n ,n +a )
2 3 1 1 2 2 3 1 1 2 3 3
+a (a −a )τ(n ,n ,n +a )τ(n +a ,n +a ,n )=0, (2.1)
3 1 2 1 2 3 3 1 1 2 2 3
where a ,a ,a are lattice parameters and τ is a function of the discrete variables n , n and n .
1 3 3 1 2 3
In [23], the following Casoratideterminant solution was obtained
φ (n ,n ,n ,0) φ (n ,n ,n ,1) ··· φ (n ,n ,n ,N −1)
1 1 2 3 1 1 2 3 1 1 2 3
φ (n ,n ,n ,0) φ (n ,n ,n ,1) ··· φ (n ,n ,n ,N −1)
2 1 2 3 2 1 2 3 2 1 2 3
τ(n1,n2,n3)=(cid:12)(cid:12)(cid:12) ... ... ... ... (cid:12)(cid:12)(cid:12),
(cid:12) (cid:12)
(cid:12)φ (n ,n ,n ,0) φ (n ,n ,n ,1) ··· φ (n ,n ,n ,N −1)(cid:12)
(cid:12) N 1 2 3 N 1 2 3 N 1 2 3 (cid:12)
(cid:12) (cid:12)
where φ ,i=1,··· ,N satisfy(cid:12) the dispersion relations (cid:12)
i (cid:12) (cid:12)
φ (n )−φ (n −a )
i j i j j
∆ φ (n ,n ,n ,s)= =φ (n ,n ,n ,s+1), j =1,2,3. (2.2)
nj i 1 2 3 a i 1 2 3
j
Here s is an auxiliary discrete variable.
For our purpose of obtaining matrix integral solutions, it will be more convenient to work with a Wron-
skian determinant solution. To find such a solution, we interpret s as a continuous variable and write a
determinant of a matrix whose columns are derivatives in the variable s. More precisely, the expression for
the Wronskian solution has the form
(0) (1) (N−1)
φ (n ,n ,n ,s) φ (n ,n ,n ,s) ··· φ (n ,n ,n ,s)
1 1 2 3 1 1 2 3 1 1 2 3
(cid:12)φ(20)(n1,n2,n3,s) φ(21)(n1,n2,n3,s) ··· φ(2N−1)(n1,n2,n3,s)(cid:12)
τ(n1,n2,n3)=(cid:12)(cid:12)(cid:12) ... ... ... ... (cid:12)(cid:12)(cid:12), (2.3)
(cid:12)(cid:12)φ(0)(n ,n ,n ,s) φ(1)(n ,n ,n ,s) ··· φ(N−1)(n ,n ,n ,s)(cid:12)(cid:12)
(cid:12) N 1 2 3 N 1 2 3 N 1 2 3 (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
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where φ(k)(n ,n ,n ,s)= ∂kφi(n ,n ,n ,s) and φ for i=1,··· ,N satisfy the dispersion relations
i 1 2 3 ∂sk 1 2 3 i
φ (n )−φ (n −a )
∆ φ (n ,n ,n ,s)= i j i j j =φ(1)(n ,n ,n ,s), j =1,2,3. (2.4)
nj i 1 2 3 a i 1 2 3
j
Forcompleteness,wepresenttheproofthattheexpressiongivenin(2.3)solvestheHirota-MiwaEquation.
Here we adopt the compact notation introduced by Freeman and Nimmo [24,25]. We denote the expression
given in (2.3) as
τ(n ,n ,n )=|0,1,··· ,N −1|, (2.5)
1 2 3
where each index j for j =0,...,N −1 stands symbolically for a column vector given by
φ(j)(n ,n ,n ,s)
1 1 2 3
φ(j)(n ,n ,n ,s)
 2 1 2 3 
. .
.
 . 
φ(j)(n ,n ,n ,s)
 N 1 2 3 
 
Based on the dispersion relations (2.4), we have
τ(n +a )=|0,1,··· ,N −2,N −1 |,
i i ni+ai
a τ(n +a )=|0,1,··· ,N −2,N −2 |,
i i i ni+ai
1
τ(n +a ,n +a )= |0,1,··· ,N −3,N −2 ,N −2 |,
i i j j a −a ni+ai nj+aj
j i
where the notation j is the column vector obtained by replacing n with n +a in the column vector
ni+ai i i i
corresponding to the index j. By substituting the expressions (2.5) into (2.1), the Hirota-Miwa equation is
reduced to nothing but the Plu¨cker relation [26–28]
|0,1,··· ,N −3,N −2,N −2 ||0,1,··· ,N −3,N −2 ,N −2 |
n1+a1 n2+a2 n3+a3
−|0,1,··· ,N −3,N −2,N −2 ||0,1,··· ,N −3,N −2 ,N −2 |
n2+a2 n1+a1 n3+a3
+|0,1,··· ,N −3,N −2,N −2 ||0,1,··· ,N −3,N −2 ,N −2 |=0.
n3+a3 n1+a1 n2+a2
The N-soliton solution for equation (2.1) obtained in [23] can be written as a Wronskian solution of the
form (2.3) with
3 3
φ (n ,n ,n ,s)=α (1−p a )−nν/aνexp(p s)+β (1−q a )−nν/aν exp(q s), (2.6)
i 1 2 3 i i ν i i i ν i
ν=1 ν=1
Y Y
where p ,q ,α ,β are arbitrary constants, which correspond to the wave numbers and phase parameters of
i i i i
the i-th soliton, respectively.
2.2 Matrix integral solutions of the Hirota-Miwa equation
(β) (β=2)
Considerthecaseofβ =2fortheintegralZ (n ,n ,n ,s)givenby(1.7),whichwedenoteasZ (n ,n ,n ,s).
N 1 2 3 N 1 2 3
Inwhat follows,we show thatZ(β=2)(n ,n ,n ,s) solvesthe Hirota-Miwaequation(2.1) by showing thatit
N 1 2 3
is a particular case of the Wronskian solution (2.3).
Using the Vandermonde determinant
(x −x )=det xj−1 ,
j k k
j,k=1···,N
1≤jY<k≤N h i
we can express the integrand of Z(β=2)(n ,n ,n ,s) as a product of two determinants
1 2 3
det xj−1 ·det xj−1(1−a x )−n1/a1(1−a x )−n2/a2(1−a x )−n3/a3exp(η(x ,s)) .
k k 1 k 2 k 3 k k
j,k=1···,N j,k=1···,N
h i h i
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Then, using the identity (1.4), the matrix integral Z(β=2)(n ,n ,n ,s) can be written as
N 1 2 3
1
Z(β=2)(n ,n ,n ,s)=det a3 xi+j−2(1−a x)−n1/a1(1−a x)−n2/a2(1−a x)−n3/a3exp(xs+η (x))dx
N 1 2 3 1 2 3 0
"Z−∞ #1≤i,j≤n
(0) (1) (N−1)
g (n ,n ,n ,s) g (n ,n ,n ,s) ··· g (n ,n ,n ,s)
1 1 2 3 1 1 2 3 1 1 2 3
(cid:12)g2(0)(n1,n2,n3,s) g2(1)(n1,n2,n3,s) ··· g2(N−1)(n1,n2,n3,s)(cid:12)
=(cid:12)(cid:12)(cid:12) ... ... ... ... (cid:12)(cid:12)(cid:12),
(cid:12)(cid:12)g(0)(n ,n ,n ,s) g(1)(n ,n ,n ,s) ··· g(N−1)(n ,n ,n ,s)(cid:12)(cid:12)
(cid:12) N 1 2 3 N 1 2 3 N 1 2 3 (cid:12)
(cid:12) (cid:12)
with (cid:12) (cid:12)
(cid:12) (cid:12)
1
g (n ,n ,n ,s)= a3 xi−1(1−a x)−n1/a1(1−a x)−n2/a2(1−a x)−n3/a3exp(xs+η (x))dx,
i 1 2 3 1 2 3 0
Z−∞ (2.7)
∂jg
g(j)(n ,n ,n ,s)= i(n ,n ,n ,s).
i 1 2 3 ∂sj 1 2 3
It is then a simple exercise to prove that the g satisfy the dispersion relation (2.4). The integral
i
Z(β=2)(n ,n ,n ,s)thusisaWronskiansolutiontotheHirota-Miwaequation(2.1). Moreover,thissolution
N 1 2 3
can be seen as a continuum limit of the N-soliton solutions given by the Wronskian solution (2.3) with φ
i
as in (2.6). Note that a similar interpretation applies to the whole discrete KP hierarchy treated below.
2.3 Matrix integral solution of the discrete KP hierarchy
The Hirota-Miwa equation is the first member of the discrete KP hierarchy [23,29,30]. The Casorati
determinant solutions of this hierarchyis presented in [23]. Just like we did in Section 2.1, we canrewrite it
as a Wronskian solution, which takes the form
φ(0)(n,s) φ(1)(n,s) ··· φ(N−1)(n,s)
1 1 1
(cid:12)φ(20)(n,s) φ(21)(n,s) ··· φ(2N−1)(n,s)(cid:12)
τ(n)=(cid:12)(cid:12)(cid:12) ... ... ... ... (cid:12)(cid:12)(cid:12), (2.8)
(cid:12)(cid:12)φ(0)(n,s) φ(1)(n,s) ··· φ(N−1)(n,s)(cid:12)(cid:12)
(cid:12) N N N (cid:12)
(cid:12) (cid:12)
where n=(n1,n2,...,nm), φ(ik)(n,s(cid:12)(cid:12))= ∂∂ksφki(n,s) and φi for i=1,··· ,N sa(cid:12)(cid:12)tisfy the dispersion relations
φ (n )−φ (n −a )
∆ φ (n,s)= i j i j j =φ(1)(n,s), j =1,2,3,...,m,
nj i a i
j
where m ≥ 3. The case m = 3 corresponds to the Hirota-Miwa equation treated in Section 2.2. We now
generalize the integral (1.7) so that it now depends on the discrete variables n = (n ,n ,...,n ) in the
1 2 m
following way
1 1 N m N
Z(β)(n,s)= 1 am ··· am (1−a x )−cnk/ak |x −x |βexp cη(x ,s) dx ···dx ,
N N! k i i j i 1 N
Z−∞ Z−∞ i=1k=1 1≤i<j≤N "i=1 #
YY Y X
(2.9)
whereweassumethata ≥a forj =1,2,...m. TheargumentusedinSection2.2istheneasilygeneralized
m j
to show that the integral above in the case β = 2 (with c =1) provides a solution of the form (2.8) for the
discrete KP hierarchy.
3 Matrix integral solutions to the Pfaffianized version of the dis-
crete KP hierarchy
In this section, we first recall some facts about Pfaffians and then present Pfaffian solutions and matrix
integral solutions to the Pfaffianized Hirota-Miwa equation and the Pfaffianized version of the discrete KP
hierarchy.
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As is known [26–28,32], a Pfaffian (1,2,...,2N) is defined recursively by
2N
(1,2,...,2N), (−1)j(1,j)(2,3,...,ˆj,...,2N), (3.1)
j=2
X
where(i,j)=−(j,i)andˆj meansthattheindexj isomitted. Foranygiven2N×2N antisymmetricmatrix
A =[a ] , the Pfaffian associated with A is defined as
2N ij 1≤i,j≤2N 2N
Pf[A ],(1,2,...,2N) (3.2)
2N
with (i,j) = −(j,i) = a . On the other hand, for any given Pfaffian (1,2,...,2N), we can construct an
ij
antisymmetric matrix
0 (1,2) (1,3) ··· (1,2N −1) (1,2N)
−(1,2) 0 (2,3) ··· (2,2N −1) (2,2N)
(cid:12) (cid:12)
A2N =(cid:12)(cid:12) ... ... ... ... ... ... (cid:12)(cid:12)
(cid:12) (cid:12)
(cid:12)−(1,2N −1) −(2,2N −1) −(3,2N −1) ··· 0 −(2N −1,2N)(cid:12)
(cid:12) (cid:12)
(cid:12) −(1,2N) −(2,2N) −(3,2N) ··· −(2N −1,2N) 0 (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
whose Pfaffian(cid:12)is Pf[A2N]=(1,2,...,2N) by the definition (3.1). (cid:12)
3.1 Pfaffianized version of the Hirota-Miwa
The Pfaffianized version of the Hirota-Miwa equation was obtained in [31] using the procedure of Pfaffi-
anization proposed by Hirota and Ohta [16]. It takes the form of the following system of three coupled
equations
a a a [a (a −a )τ(n −a ,n ,n )τ(n ,n −a ,n −a )
1 2 3 1 2 3 1 1 2 3 1 2 2 3 3
+a (a −a )τ(n ,n −a ,n )τ(n −a ,n ,n −a )
2 3 1 1 2 2 3 1 1 2 3 3
+a (a −a )τ(n ,n ,n −a )τ(n −a ,n −a ,n )]
3 1 2 1 2 3 3 1 1 2 2 3
+(a −a )(a −a )(a −a )σ(n ,n ,n )ρ(n −a ,n −a ,n −a )=0,
1 2 2 3 3 1 1 2 3 1 1 2 2 3 3
a a a [a (a −a )σ(n −a ,n ,n )τ(n ,n −a ,n −a )
1 2 3 1 2 3 1 1 2 3 1 2 2 3 3
+a (a −a )σ(n ,n −a ,n )τ(n −a ,n ,n −a )
2 3 1 1 2 2 3 1 1 2 3 3
(3.3)
+a (a −a )σ(n ,n ,n −a )τ(n −a ,n −a ,n )]
3 1 2 1 2 3 3 1 1 2 2 3
+(a −a )(a −a )(a −a )σ(n ,n ,n )τ(n −a ,n −a ,n −a )=0,
1 2 2 3 3 1 1 2 3 1 1 2 2 3 3
a a a [a (a −a )τ(n −a ,n ,n )ρ(n ,n −a ,n −a )
1 2 3 1 2 3 1 1 2 3 1 2 2 3 3
+a (a −a )τ(n ,n −a ,n )ρ(n −a ,n ,n −a )
2 3 1 1 2 2 3 1 1 2 3 3
+a (a −a )τ(n ,n ,n −a )ρ(n −a ,n −a ,n )]
3 1 2 1 2 3 3 1 1 2 2 3
+(a −a )(a −a )(a −a )τ(n ,n ,n )ρ(n −a ,n −a ,n −a )=0.
1 2 2 3 3 1 1 2 3 1 1 2 2 3 3
The system above has the Pfaffian solutions
τ =τ(n ,n ,n )=(1,2,··· ,N), N even,
1 2 3
σ =σ(n ,n ,n )=(1,2,··· ,N,N +1,N +2), (3.4)
1 2 3
ρ=ρ(n ,n ,n )=(1,2,··· ,N −2),
1 2 3
wherethe Pfaffianentries(i,j)inthe above-mentionedthree Pfaffiansarefunctions ofthe discretevariables
n , n and n which satisfy the following dispersion relations
1 2 3
(i,j) =(i,j)−a (i+1,j)−a (i,j+1)+a2(i+1,j+1), k =1,2,3. (3.5)
nk−ak k k k
In (3.5), the notation (i,j) represents the operation on (i,j) which replaces the discrete variable n
nk−ak k
in (i,j) by n −a . The terms on the right hand side of (3.5) are various Pfaffian entries. Based on the
k k
definitionofaPfaffian(3.1),onceweknowthedispersionrelations(3.5)thatthePfaffianentries(i,j)satisfy
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for i,j = 1,2,...,N +2, it is enough to prove that τ,σ and ρ given by (3.4) are Pfaffian solutions to the
Pfaffianized versionof the Hirota-Miwa equation (3.3). As for the antisymmetric matrices associated to the
three Pfaffians in (3.4), they are related but not the same. For example, if we assume that A , A and
N−2 N
A arethreeantisymmetricmatricessuchthatρ=Pf[A ], τ =Pf[A ]andσ =Pf[A ],respectively,
N+2 N−2 N N+2
then it is obvious that A is an enlarged matrix of A and A is an enlarged matrix of A .
N+2 N N N−2
We would like to point out that an example of the Pfaffian entry (i,j) which satisfies (3.5) is
N 3
(i,j)= pi−1qj−1−qi−1pj−1 [(1−a p )(1−a q )]−nν/avexp[η(p ,s)+η(q ,s)], (3.6)
k k k k ν k ν k k k
Xk=1(cid:16) (cid:17)νY=1
where p , andq arearbitraryconstants. Under this choice,the Pfaffiansolutionτ =(1,2,...,N)gives the
k k
(N/2)-soliton solutions to the Pfaffianized version of the Hirota-Miwa equation.
Remark. Byreferingto[16,26,27],weknowthatthecoupledKPequationhasWronski-typepfaffiansolutions
τ =(0,1,··· ,2N−1), σ =(0,1,...,2N−3)and σˆ =(0,1,...,2N+1)with (l,m)satisfying the dispersion
relation ∂ (l,m) = (l + n,m) + (l,m + n) where x = x,x = y,x = t. One such an example is
∂xn 1 2 3
2N
(l,m)= Φ(l−1)Ψ(m−1)−Φ(m−1)Ψ(l−1) , where Φ(l) and Ψ(l) stand for the l-th derivatives with respect
k k k k k k
k=1
to x ofPfunhctions Φ and Ψ satisfying i∂ Φ = Φ(n) and ∂ Ψ = Ψ(n). Further on, if we choose
1 k k ∂xn k k ∂xn k k
2N
Φ =exp[p x+p2y+p3t]andΨ =exp[q x+q2+q3t], then (l,m)= pl−1qm−1−ql−1pm−1 exp[(p +
k k k k k k k k k k k k k
k=1
q )x+(p2 +q2)y+(p3 +q3)t] coincides with the expression a givenPin(cid:2)[12] and can be used t(cid:3)o generate
k k k k k l,m
multi-solitionsolutionstothecoupledKPequationgivenin[34]. Alongthisline,wecanderivemulti-soliton
solutions to the coupled Hirota-Miwa equation with the pfaffian entry (i,j) given by (3.6) in the same way.
Matrix integral solutions (I). Consider the case β = 1. By virtue of the identity (1.5), the matrix
integral (1.7) turns into
Z(β=1)(n ,n ,n ,s)
N 1 2 3
1 1
= 1 a3 ··· a3 det xi−1(1−a x )−n1/a1(1−a x )−n2/a2(1−a x )−n3/a3exp(η(x ,s))
N!Z−∞ Z−∞ h j 1 j 2 j 3 j j ii,j=1···,N
=Pf (xi−1yj−1−yi−1xj−1)[(1−a x)(1−a y)]−n1/a1
1 1

ZZ
x<y

[(1−a x)(1−a y)]−n2/a2[(1−a x)(1−a y)]−n3/a3expη(x,s)+η(y,s)dxdy ,
2 2 3 3
i,j=1,···,N
i
whose Pfaffian entry (i,j) is given by
(i,j)= (xi−1yj−1−yi−1xj−1)[(1−a x)(1−a y)]−n1/a1
1 1
ZZ (3.7)
x<y
[(1−a x)(1−a y)]−n2/a2(1−a x)(1−a y)]−n3/a3expη(x,s)+η(y,s)dxdy.
2 2 3 3
It is easy to check that the Pfaffian entry (3.7) satisfies the dispersion relation (3.5). Therefore, the matrix
integrals τ = Z(β=1)(n ,n ,n ,s), σ = Z(β=1)(n ,n ,n ,s) together with ρ = Z(β=1)(n ,n ,n ,s) provide
N 1 2 3 N+2 1 2 3 N−2 1 2 3
solutions to the coupledHirota-Miwasystem(3.3). Moreover,the Pfaffiansolutions obtainedwith (3.7) can
be interpreted as a continuum of the multisoliton Pfaffian solutions defined by (3.6).
Matrix integral solutions (II). With regard to the twofold Vandermonde determinant, the following
identity was presented in [3]
(x −x )4 =det xj,(j−1)xj . (3.8)
j k k k
j=1,···,2N,k=1,···,N
1≤jY<k≤N h i
7
Consider the case β =4. By resorting to the identities (1.6) and (3.8), the matrix integral (1.7) becomes
Z(β=4)(n ,n ,n ,s)
N 1 2 3
1 1
= 1 a3 ··· a3 det xi(1−a x )−n1/a1(1−a x )−n2/a2(1−a x )−n3/a3exp(η(x ,s)),
N! j 1 j 2 j 3 j j
Z−∞ Z−∞ h
(i−1)xi(1−a x )−n1/a1(1−a x )−n2/a2(1−a x )−n3/a3exp(η(x ,s)) dx ···dx
j 1 j 2 j 3 j j 1 N
1 i
=Pf a3(j−i)xi+j−3(1−a x)−2n1/a1(1−a x)−2n2/a2(1−a x)−2n3/a3exp(2η(x,s))dx ,
1 2 3
"Z−∞ #i,j=1,···,2N
whose Pfaffian entry (i,j) entry is given by
1
(i,j)= a3(j−i)xi+j−3(1−a x)−2n1/a1(1−a x)−2n2/a2(1−a x)−2n3/a3exp(2η(x,s))dx. (3.9)
1 2 3
Z−∞
Similarly,one canprovethat the Pfaffianentry (3.9)satisfies the dispersionrelation(3.5) aswell. Thus, the
matrix integrals τ = Z(β=1)(n ,n ,n ,s), σ = Z(β=1)(n ,n ,n ,s) as well as ρ = Z(β=1)(n ,n ,n ,s) give
N 1 2 3 N+1 1 2 3 N−1 1 2 3
solutions to the coupled Hirota-Miwa system (3.3).
3.2 Pfaffianized version of the discrete KP hierarchy
The Pfaffianized version of the whole discrete KP hierarchy and its Pfaffian solutions were derived in [33].
The Pfaffian solutions take the form
τ =τ(n)=(1,2,··· ,N), N even,
σ =σ(n)=(1,2,··· ,N,N +1,N +2), (3.10)
ρ=ρ(n)=(1,2,··· ,N −2),
where n=(n ,n ,...,n ) and where the Pfaffian entry (i,j) satisfies
1 2 m
(i,j) =(i,j)−a (i+1,j)−a (i,j+1)+a2(i+1,j+1), k =1,2,...m.
nk−ak k k k
If we now consider the general integral (2.9), the argument used in Section 3.1 is then easily generalized to
show that the integral above in the cases β = 1 (with c = 1) and β = 4 (with c = 2) provide a solution of
the form (3.10) for the Pfaffianized discrete KP hierarchy.
4 Conclusion
WehaveconsideredmatrixintegralsusedincertainprobabilitycomputationsforHermitianensembles. Such
matriceshavebeenshownbeforetobeusefulinthecontextofintegrablesystems,stringtheory,andrandom
matrix theory by providing partition functions and solutions to integrable equations. More precisely, by
inserting an appropriate dependence on various variables in the integrals (1.1), one can obtain τ-functions
for a number of integrable equations, together with their Pfaffianized versions [12,14,16–18]. In all the
examples studied in the literature, the case β = 2 relates to a given integrable equation, while the cases
β =1,4relatesto the Pfaffianizedversionofthe sameequation. Furthermore,the solutionsobtainedin this
way can often be interpreted as a continuous limit of multi-soliton solutions.
Inthis article,wehaveshownthatintegralsofthe form(1.1)canprovideτ-functions forthe discreteKP
hierarchy and its Pfaffianized version by introducing an appropriate dependence on the discrete variables.
More precisely, we have considered the integrals (1.7) and (2.9) and showed that in the case β = 2, they
provide solutions to the discrete KP hierarchy (starting with the Hirota-Miwa equation (2.1)), while the
cases β = 1,4 provide solutions for the Pfaffianized version of the discrete KP hierarchy (starting with the
Pfaffianized Hirota-Miwa Equation given in (3.3)).
8
Acknowledgement
This work was supported by the National Natural Science Foundation of China (Grant No. 11271266 and
11371323), Beijing Natural Science Foundation (Grant No. 1162003) and the China Scholarship Council.
The authors would like to thank Professor Xing-Biao Hu, Alex Kasman, and Virgil Pierce for valuable
discussions. Dr. Li would also like to thank for the hospitality of the Department of Mathematics during
her visit to the College of Charleston.
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