Correlations of RMT Characteristic Polynomials and Integrability: Hermitean Matrices
Integrable theory is formulated for correlation functions of characteristic polynomials associated with invariant non-Gaussian ensembles of Hermitean random matrices. By embedding the correlation functions of interest into a more general theory of ta…
Authors: Vladimir Al. Osipov, Eugene Kanzieper
Correlation functions of characteristic polynomials (CFCP) appear in various fields of mathematical and theoretical physics. (i) In quantum chaology, CFCP (i.a) provide a convenient way to describe the universal features of spectral statistics of a particle confined in a finite system exhibiting chaotic classical dynamics (Bohigas, Giannoni and Schmit 1984;Andreev, Agam, Simons and Altshuler 1996;Müller, Heusler, Braun, Haake and Altland 2004) and (i.b) facilitate calculations of a variety of important distribution functions whose generating functions may often be expressed in terms of CFCP (see, e.g., Andreev and Simons 1995). (ii) In the random matrix theory approach to quantum chromodynamics, CFCP allow to probe various QCD partition functions (see, e.g., Verbaarschot 2010). (iii) In the number theory, CFCP have been successfully used to model behaviour of the Riemann zeta function along the critical line (Keating andSnaith 2000a, 2000b;Hughes, Keating and O'Connell 2000). (iv) Recently, CFCP surfaced in the studies of random energy landscapes (Fyodorov 2004). (v) For the rôle played by CFCP in the algebraic geometry, the reader is referred to the paper by Brézin and Hikami (2008) and references therein.
In what follows, we adopt a formal setup which turns an n × n Hermitian matrix H = H † into a central object of our study. For a fixed matrix H, the characteristic polynomial det n (ς -H) contains complete information about the matrix spectrum. To study the statistics of spectral fluctuations in an ensemble of random matrices, it is convenient to introduce the correlation function Π n|p (ς; κ) of characteristic polynomials
Here, the vectors ς = (ς 1 , • • • , ς p ) and κ = (κ 1 , • • • , κ p ) accommodate the energy and the "replica" parameters, respectively. The angular brackets f (H) H stand for the ensemble average
with respect to a proper probability measure
dH jj n j 0, the vector c
, and ̺ = dim (c ′ ). The function f (λ) is, in turn, related to the confinement potential V n-s (λ) through the parameterisation
in which both g(λ) and f (λ) depend on n-s as do the coefficients b k and a k in the above expansions. The transformation Eq. (2.13) induces the Virasoro-like constraints ♯
where the differential operators
(2.17) act in the t-space whilst the differential operator
♯ The very notation LV q suggests that this operator originates from the confinement-potential-part e -Vn in Eqs. (2.5) and (2.4). On the contrary, the operator Ldet q is due to the determinant-like product α (ςα -λ) κα in Eq. (2.5). Indeed, setting κα = 0 nullifies the operator Ldet q . See Section 3.6 for a detailed derivation.
acts in the space of physical parameters {ς α } α∈Z+ . The notation s k (-p ̺ (c ′ )) stands for the Schur polynomial and p ̺ (c ′ ) is an infinite dimensional vector
(2.20)
Notice that the operator LV q (t) is expressed in terms of the Virasoro operators † † in terms of BV q (ς) Π n|p (ς; κ). This observation makes it tempting to project the hierarchical relations Eqs. (2.8) and (2.9) onto the hyperplane (s = 0, t = 0) in an attempt to generate their analogues in the space of physical parameters. In particular, such a projection of the first equation of the KP hierarchy, Having exhibited the general structure of the theory, let us turn to the detailed exposition of its main ingredients.
Integrability derives from the symmetry. In the context of τ functions Eq. (2.4), the symmetry is encoded into ∆ 2 n (λ), the squared Vandermonde determinant, as it appears in the integrand below :
(3.1) † † For q = -1, the second sum in Eq. (2.21) is interpreted as zero. § Whether or not the projected Virasoro constraints and the hierarchical equations always form a closed system is a separate question that lies beyond the scope of the present paper.
For the sake of brevity, the physical parameters ς and κ were dropped from the arguments of τ (s) n and Γ n-s .
In the random matrix theory language, the τ function Eq. (3.1) is said to posses the β = 2 symmetry. Using the identity
with P k (λ) being an arbitrary set of monic polynomials and the integration formula (Andréief 1883, de Bruijn 1955)
the τ function Eq. (3.1) can be written as the determinant
of the matrix of moments
Both the determinant representation and the scalar product
are dictated by the β = 2 symmetry § of the τ function.
In this subsection, the bilinear identity Eq. (2.7) will be proven.
The τ function and orthogonal polynomials.-The representation Eq. (3.4) reveals a special rôle played by the monic polynomials P (t; λ) orthogonal on D with respect to the measure Γ m (λ) e v(t;λ) dλ. Indeed, the orthogonality relation
shows that the choice P j (λ) → P (n-s) j
(t; λ) diagonalises the matrix of moments in Eq. (3.4) resulting in the fairly compact representation
(3.8) § The τ function Eq. (3.1) is a particular case of a more general τ function
In accordance with the Dyson "three-fold" way (Dyson 1962), the symmetry parameter β may also take the values β = 1 and β = 4. For these cases, the τ function Eq. (3.7) admits the Pfaffian rather than determinant representation (Adler and van Moerbeke 2001):
, where the matrix of moments µ
Γm e v is defined through the scalar product
Remarkably, the monic orthogonal polynomials P (m) k (t; λ), that were introduced as a most economic tool for the calculation of τ (s) n , can themselves be expressed in terms of τ functions:
.
(3.9)
Here, the notation t -[λ -1 ] stands for an infinite-dimensional vector with the components
(3.10)
The statement Eq. (3.9) readily follows from the definitions Eqs. (3.1) and (2.6), the formal relation
and the Heine formula (Heine 1878, Szegö 1939)
(3.12)
The τ function and Cauchy transform of orthogonal polynomials.-As will be seen later, the Cauchy transform of orthogonal polynomials is an important ingredient of our proof of the bilinear identity. Viewed as the scalar product,
it can also be expressed in terms of τ function:
To prove Eq. (3.14), we substitute Eq. (3.12) into Eq. (3.13) to derive:
Owing to the identity
in the integrand of Eq. (3.15) can be symmetrised
In view of Eq. (3.11), this is seen to coincide with
Comparison with the definition Eq. (3.1) completes the proof of Eq. (3.14).
Proof of the bilinear identity.-Now we are ready to prove the bilinear identity
where the integration contour C ∞ encompasses the point z = ∞, and a ∈ R is a free parameter.
We start with the needlessly fancy identity
whose structure is prompted by the scalar product Eq. (3.7) and which trivially holds due to a linearity of the t-deformation
The formulae relating the orthogonal polynomials and their Cauchy transforms to τ functions [Eqs. (3.9) and (3.14)] make it possible to express both sides of Eq. (3.19) in terms of τ functions with shifted arguments.
(i) Due to the Cauchy integral representation
the l.h.s. of Eq. (3.19) can be transformed as follows:
Taking into account Eqs. (3.9) and (3.14), this is further reduced to
To transform the r.h.s. of Eq. (3.19), we make use of the Cauchy theorem in the form
to get:
Taking into account Eqs. (3.9) and (3.14), this is further reduced to
The bilinear identity Eq. (3.18) follows from Eqs. (3.23) and (3.26) after setting n = m -s. End of proof.
The bilinear identity Eq. (3.18) can alternatively be written in the Hirota form:
where β = ±1 (not to be confused with the Dyson symmetry index!), p ≥ 1 and q ≥ -1. Let us remind that the vector D appearing in the scalar product
is defined in Appendix B. Also, s k (x) are the Schur polynomials defined in Eq. (2.11).
To prove Eq. (3.27), we proceed in two steps.
(i) First, we set the vectors t and t ′ in Eq. (3.18) to be (t, t ′ ) → (t + x, tx).
(3.28)
The parameterisation Eq. (3.28) allows us to rewrite the (t, t ′ ) dependent part of the integrand in the l.h.s. of Eq. (3.18)
as follows:
Here, we have used the linearity of the t-deformation, α v(t; z) = v(αt; z). Further, we spot the identity
.
The latter can be rewritten in terms of Hirota differential operators [see Eq. (B.
2)] with the result being
By the same token, the (t, t ′ ) dependent part of the integrand in the r.h.s. of Eq. (3.18),
(3.31) Thus, we end up with the alternative representation for the bilinear identity Eq. (3.18):
(3.32) (ii) Second, to facilitate the integration in Eq. (3.32), we rewrite the integrands therein in the form of Laurent series in z by employing the identity
Now, the integrals in Eq. (3.32) are easily performed to yield
It remains to verify that Eq. (3.34) is equivalent to the announced result Eq. (3.27). To this end we distinguish between two different cases. (i) If ℓ ≤ m, we set ℓ = p -1, m = p + q and s → s + q in Eq. (3.34) to find out that it reduces to Eq. (3.27) taken at β = +1; (ii) If ℓ > m, we set ℓ = p + q, m = p -1 and s → s -1 in Eq. (3.34) to find out that it reduces to Eq. (3.27) taken at β = -1. This ends the proof.
For an alternative derivation of Eq. (3.27) the reader is referred to Appendix A. ¶ Here,
The bilinear identity, in either form, encodes an infinite set of hierarchically structured nonlinear differential equations in the variables t. Two of these hierarchies -the KP and the TL hierarchies -were mentioned in Section 2. Below, we provide a complete list of integrable hierarchies associated with the τ function Eq. (3.1).
To identify them, we expand the bilinear identity in Hirota form [Eq. (3.27)] around x = 0 and a = 0, keeping only linear in x terms. Since s 0 (t) = 1 and
we obtain:
(
As soon as Eq. (3.27) holds for arbitrary a and x, Eq. (3.36) generates four identities.
(i) The first identity
holds for q ≥ 1 and k = 0, 1, . . . , q.
(ii) The second identity
holds for q ≥ 1 and k = 1, 2, . . . , q.
(iii) The third identity
holds for q ≥ -1 and k ≥ max(1, q + 1).
(iv) The last, fourth identity
holds for q ≥ 0 and k ≥ q + 1.
Equations (3.37) -(3.40) can further be classified to yield the following bilinear hierarchies:
• Toda Lattice (TL) hierarchy:
with k ≥ 1.
• q-modified Toda Lattice hierarchy:
with q ≥ 0 and k ≥ q + 1. (For q = 0, it reduces to the above Toda Lattice hierarchy.)
• Kadomtsev-Petviashvili (KP) hierarchy:
with ♯ k ≥ 3.
• q-modified Kadomtsev-Petviashvili hierarchy:
with q ≥ 0 and k ≥ q + 1. (For q = 0, it reduces to the above KP hierarchy.)
• Left q-modified Kadomtsev-Petviashvili hierarchy:
with q ≥ 1 and 1 ≤ k ≤ q.
• Right q-modified Kadomtsev-Petviashvili hierarchy:
with q ≥ 1 and 0 ≤ k ≤ q.
• (-1)-modified Kadomtsev-Petviashvili hierarchy:
Notice, that the modified hierarchies will play no rôle in further consideration.
As was pointed out in Section 2, the KP and Toda Lattice hierarchies are of primary importance for our formalism. In this subsection, we explicitly present a few first members of these hierarchies.
KP hierarchy.-Due to the properties of Hirota symbol reviewed in Appendix B, the first nontrivial equation of the KP hierarchy corresponds to k = 3 in Eq. (3.43). Consulting Table 1 and having in mind that [D] k = k -1 D k , we derive the first two members, KP 1 and KP 2 , of the KP hierarchy in Hirota form Making use of the Property 2b from Appendix B, the two equations can be written explicitly:
Only KP 1 will further be used.
Toda Lattice hierarchy.-The first nontrivial equations of the Toda Lattice hierarchy can be derived along the same lines from Eq. (3.41).
Explicitly, one has:
Higher order members of the KP and Toda Lattice hierarchies can readily be generated from Eqs. (3.43) and (3.41), respectively.
Virasoro constraints satisfied by the τ function Eq. (3.56) below is yet another important ingredient of the "deform-and-study" approach to the correlation functions of characteristic polynomials Π n|p (ς; κ). In accordance with the discussion in Section 2, Virasoro constraints are needed to translate nonlinear integrable hierarchies Eqs. (3.41) -(3.47), satisfied by the τ function
into nonlinear, hierarchically structured differential equations for the correlation function
obtained from Eq. (3.56) by setting t = 0 and s = 0. The normalisation constant N n is defined in Eq. (2.3). The Virasoro constraints reflect invariance of the τ function Eq. (3.56) under the change of integration variables
labeled by the integer q; here ǫ > 0 is an infinitesimally small parameter, and R(µ) is a suitable benign function (e.g., a polynomial). The function f (λ) is related to the confinement potential V n-s (λ) through the parameterisation (Adler, Shiota and van Moerbeke 1995)
in which both g(λ) and f (λ) depend on n -s as do the coefficients b k and a k in the above expansions. We also assume that
To derive the Virasoro constraints announced in Eqs. (2.15)-(2.18), we transform the integration variables in Eq. (3.56) as specified in Eq. (3.58) and further expand Eq. (3.56) in ǫ. Invariance of the integral under this transformation implies that the linear in ǫ terms must vanish: In the above formulae, we reinstated β > 0; it will be set to β = 2 when needed. The choice of R(µ) is dictated by problem in question and, hence, is not unique. If one is interested in studying matrix integrals as functions of the parameters
(3.64)
In this case, the differential operator (Adler, Shiota and van Moerbeke 1995)
becomes an essential part of the Virasoro constraints. In the context of characteristic polynomials, the integration domain D is normally fixed whilst the physical parameters {ς α } are allowed to vary. This prompts the choice
that nullifies the differential operator Eq. (3.65). Equivalently,
Here, the notation s k (-p ̺ (c ′ )) stands for the Schur polynomial and p ̺ (c ′ ) is an infinite dimensional vector
Remark. Equation (3.66) assumes that none of c ′ k 's are zeros of f (µ). If this is not the case, the set c ′ must be redefined:
where Z 0 is comprised of zeros of f (µ).
Substituting Eqs. (3.67), (3.59) and (2.6) into Eq. (3.61), we derive:
The ς-dependent part in Eq. (3.71),
originates from the term
in Eq. (3.61). Indeed, substituting Eqs. (3.67) and (3.59) into Eq. (3.73), the latter reduces to
The double sum in parentheses can conveniently be divided into two pieces, and
we conclude that Eq. (3.73) reduces to the sought Eq. (3.72). We found it more convenient to rewrite the ∂/∂ς α -term in Eq. (3.72) in a more compact way,
with the differential operator BV q (ς) being To complete the derivation of Virasoro constraints, we further notice that terms tr n (µ j ) in Eq. (3.71) can be generated by differentiating I (s) n over t j . Since tr n (µ 0 ) = n, the derivative ∂/∂t 0 should formally be understood as ∂/∂t 0 ≡ n. This observation yields Virasoro constraints for the τ function
in the form (q ≥ -1)
Here, the differential operators
act in the t-space whilst the differential operator BV q (ς) acts in the space of physical parameters {ς α } α∈Z+ . Notice that the operator LV q (t) is expressed in terms of the Virasoro operators
The Virasoro constraints derived in this section stay valid for arbitrary β > 0; for β = 2, they are reduced to Eqs. (2.15) -(2.22) announced in Sec. 2.
This concludes the derivation of three main ingredients of integrable theory of average characteristic polynomials -the bilinear identity, integrable hierarchies emanating from it, and the Virasoro constraints.
The general calculational scheme formulated in Section 2 and detailed in Section 3 applies to a variety of random matrix ensembles. In this Section we deal with CFCP for the Gaussian Unitary Ensemble (GUE) and Laguerre Unitary Ensemble (LUE). A detailed treatment of the GUE case is needed to lay the basis for further comparative analysis of the three variations of the replica approach that will be presented in Section 5. The study of the LUE relevant to the QCD physics (Verbaarschot 2010) is included for didactic purposes. A sketchy exposition of the theory for Jacobi Unitary Ensemble (JUE) and Cauchy Unitary Ensemble (CyUE) appearing in the context of universal aspects of quantum transport through chaotic cavities (Beenakker 1997) can be found in Appendices C and D.
The correlation function of characteristic polynomials in GUE is defined by the n-fold integral
where
is the normalisation constant. The associated τ function equals
see Section 2. (In the above definitions, the superscript G stands for GUE but it will further be omitted when notational confusion is unlikely to arise.)
In the notation of Section 3, the definition Eq. (4.1) implies that
This brings the Virasoro constraints Eqs. (2.15) -(2.21) for the τ function Eq. (4.3): In what follows, we need the three lowest Virasoro constraints labeled by q = -1, q = 0 and q = +1. Written for the logarithm of τ function, they read:
+n ϑ 1 (ς, κ) = B1 log τ n (ς, κ; t). (4.13)
Projection of the Toda Lattice hierarchy Eq. (2.9) for the t-dependent τ function Eq. ( 4.3) onto the hyperplane t = 0 generates the Toda Lattice hierarchy for the correlation function Π G n|p (ς; κ) [Eq. (4.1)] of the GUE characteristic polynomials,
Below, the first [Eq. (3.54)] and second [Eq. (3.55)] equation of the TL hierarchy will be considered:
with the help of Virasoro constraints. This is achieved in two steps. First, we differentiate Eq. (4.11) over t 1 and set t = 0 afterwards, to derive:
written in the space of physical parameters ς.
(ii) The second equation of the TL hierarchy for Π n|p (ς; κ) can be derived along the same lines. Equation (4.16) suggests that, in addition to the derivative ∂/∂t 1 log τ n at t = 0 given by Eq. ( 4.18), one needs to know the mixed derivative
It can be calculated by combining Eq. (4.11) differentiated over t 2 with Eqs. (4.12) and (4.18). The result reads:
Higher order equations of the TL hierarchy for the correlation functions Π n|p can be derived in a similar fashion.
Remark.-For p = 1, the equations TL
(4.23)
The technology used in the previous subsection can equally be employed to project the KP hirerachy Eq. (2.8) onto the hyperplane t = 0. Below, only the first KP equation
will be treated. Notice that no superscript (s) appears in Eq. (4.24) as the GUE confinement potential does not depend on the matrix size n. Proceeding along the lines of the previous subsection, we make use of the three Virasoro constraints Eqs. (4.11) -(4.12) to derive:
Notice that appearance of the single parameter κ instead of the entire set κ = (κ 1 , . . . , κ p ) in Eq. (4.28) indicates that correlation functions Π n|p (ς; κ) with different κ but with identical traces tr p κ satisfy the very same equation. It is the boundary conditions ‡ that pick up the right solution for the given set κ = (κ 1 , . . . , κ p ).
Remark.-For p = 1, the above equation reads:
This can be recognised as the Chazy I equation (see Appendix E)
see Appendix E for more details. The boundary condition to be imposed at infinity is
The correlation function of characteristic polynomials in LUE is defined by the formula
where
is the normalisation constant, and it is assumed that ν > -1. The associated τ function equals
dλ j e -λj +v(t;λj ) λ ν j p α=1
In the above definitions, the superscript L stands for LUE but it will be omitted from now on.
In the notation of Section 3, the definition Eq. (4.34) implies that
This brings the following Virasoro constraints Eqs. (2.15) -(2.21) for the τ function Eq. (4.36):
where Bq is defined by Eq. (4.8) and Lq (t) is the Virasoro operator given by Eq. (2.21).
In what follows, we need the three lowest Virasoro constraints for q = 0, q = 1 and q = 2. Written for log τ n (ς, κ; t), they read:
Notice that dim(c ′ ) = 0 follows from Eq. (3.70) in which Z 0 = {0}.
To generate the Toda Lattice hierarchy for the correlation function Π L n|p (ς; κ) [Eq. (4.34)] of characteristic polynomials we apply the projection formula
in which the τ function is defined by Eq. (4.36), to the first and second equation of the t-dependent TL hierarchy:
with the help of the Virasoro constraints. Differentiating Eq. (4.41) over t 1 and setting t = 0 afterwards, we obtain:
written in the space of physical parameters ς. Notice that the above equation becomes more symmetric if written for the correlation function
(4.51)
The corresponding TL equation reads:
The second equation of the TL hierarchy for Π n|p (ς; κ) can be derived along the same lines. Equation (4.46) suggests that, in addition to the derivative ∂/∂t 1 log τ n at t = 0 given by Eq. ( 4.48), one needs to know the mixed derivative
It can be calculated by combining Eq. (4.41) differentiated over t 2 with Eqs. (4.42) and (4.48). Straightforward calculations bring
The final result reads:
This equation takes a more compact form if written for the correlation function Πn|p defined by Eq. (4.51):
The same technology is at work for projecting the KP hirerachy Eq. (2.8) onto t = 0. Below, only the first KP equation
will be treated. Notice that no superscript (s) appears in Eq. (4.57) as the LUE confinement potential does not depend on the matrix size n. To make the forthcoming calculation more efficient, it is beneficial to introduce the notation
(4.60)
Hence,
and
(ii) Second, to determine T 13 , we differentiate the first Virasoro constraint Eq. (4.41) with respect to t 3 , and make use of the second and third constraints as they stand [Eqs. (4.42) and (4.43)] to obtain:
Although easy to derive, an explicit expression for T 13 is too cumbersome to be explicitly stated here.
(iii) To calculate T 22 , the last unknown ingredient of Eq. (4.59), we differentiate the first and second Virasoro constraints [Eqs. (4.41) and (4.42)] with respect to t 2 to realize that Finally, we substitute so determined T 1111 , T 11 , T 13 and T 22 into Eq. (4.59) to generate a nonlinear differential equation for log Π n|p (ς; κ) in the form
(4.65)
Remark.-For p = 1, the above equation reads:
(4.66)
It can further be simplified if written for the function
Straightforward calculations yield:
This can be recognized as the Chazy I form (see Appendix E) of the fifth Painlevé transcendent. Equivalently, ϕ satisfies the Painlevé V equation in the Jimbo-Miwa-Okamoto form (Forrester andWitte 2002, Tracy andWidom 1994):
This Section concludes the detailed exposition of integrable theory of correlation functions of RMT characteristic polynomials. Among the main results derived are: Other nonlinear multivariate relations between the correlation functions Π n|p (ς; κ) and Π n±q|p (ς; κ) can readily be obtained from the modified Toda Lattice and Kadomtsev-Petviashvili hierarchies listed in Section 3.4. Finally, let us stress that a similar calculational framework applies to other β = 2 matrix integrals depending on one (Osipov and Kanzieper 2007) or more (Osipov and Kanzieper 2009;Osipov, Sommers and Życzkowski 2010) parameters. The reader is referred to the above papers for further details.
In this Section, the integrable theory of CFCP will be utilised to present a tutorial exposition of the exact approach to zero dimensional replica field theories formulated in a series of publications (Kanzieper 2002, Splittorff and Verbaarschot 2003, Osipov and Kanzieper 2007). Focussing, for definiteness, on the calculation of the finite-N average eigenlevel density in the GUE (whose exact form (Mehta 2004)
has been known for decades), we shall put a special emphasis on a comparative analysis of three alternative formulations -fermionic, bosonic and supersymmetric -of the replica method. This will allow us to meticulously analyse the fermionic-bosonic factorisation phenomenon of RMT spectral correlation functions in the fermionic and bosonic variations of the replica method, where its existence is not self-evident, to say the least.
To determine the mean density of eigenlevels in the GUE, we define the average onepoint Green function
that can be restored from the replica partition function (n ∈ R + )
through the replica limit
(5.3) Equation (5.2) can routinely be mapped onto either fermionic or bosonic replica field theories, the result being (see, e.g., Kanzieper 2010)
and
(5.5)
Both integrals run over n×n Hermitean matrix Q; the normalisation constant c n equals
(5.6) By derivation, the replica parameter n in Eqs. (5.4) and (5.5) is restricted to integers, n ∈ Z + . Notably, Eqs. (5.4) and (5.5) are particular cases of the correlation function Π n|p (ς; κ) studied in previous sections.
Indeed, comparison of Eq. ( 5.4) with the definition Eq. (4.1) yields
(5.10)
Here and above, n ∈ Z + .
Equations (5.8) and ( 5.9) open the way for calculating the average Green function G(z; N ) via the fermionic replica limit
.
(5.11)
For the prescription Eq. ( 5.11) to be operational, the Painlevé representation of Z (+) n (z; N ) should hold * for n ∈ R + . Notice that for generic real n, the fermionic replica partition function Z (+) n (z; N ) is no longer an analytic function of z and exhibits a discontinuity across the real axis. For this reason, the Painlevé equation Eq. ( 5.9) should be solved separately for Re t < 0 (Im z > 0) and Re t > 0 (Im z < 0).
Replica limit and the Hamiltonian formalism.-To implement the replica limit, we employ the Hamiltonian formulation of the Painlevé IV (Noumi 2004, Forrester andWitte 2001) which associates ϕ(t; n, N ) with the polynomial Hamiltonian (Okamoto 1980a)
(5.12) of a dynamical system {Q, P, H f }, where Q = Q(t; n, N ) and P = P (t; n, N ) are canonical coordinate and momentum. For such a system, Hamilton's equations of motion read:
we need to develop a small-n expansion for the Hamiltonian H f {P, Q, t}. Restricting ourselves to the linear in n terms,
(5.16) and
(5.17)
(5.18)
+ Equation (5.10) follows from Eqs. (5.8), (5.7) and the footnote below Eq. (4.28). * Previous studies (Kanzieper 2002, Osipov andKanzieper 2007) suggest that this is indeed the case.
we conclude that q 0 (t; N ) = 0. This derives directly from the expansion Eq. (5.16) in which absence of the term of order O(n 0 ) is guaranteed by the normalisation condition
where
(5.21)
Here, p 0 = p 0 (t; N ) and q 1 = q 1 (t; N ) are solutions to the system of coupled first order equations:
(5.22)
Since the initial conditions are known for H 1 (t; N ) rather than for p 0 (t; N ) and q 1 (t; N ) separately, below we determine these two functions up to integration constants. The function p 0 (t; N ) satisfies the Riccati differential equation whose solution is (5.23) where
is, in turn, a solution to the equation of parabolic cylinder
(5.25)
Two remarks are in order. First, factoring out (-ι) N in the second term in Eq. ( 5.24) will simplify the formulae to follow. Second, the solution Eq. (5.23) for p 0 (t; N ) actually depends on a single constant (either c 1 /c 2 or c 2 /c 1 ) as it must be.
To determine q 1 (t; N ), we substitute Eq. (5.23) into the second formula of Eq. (5.22) to derive:
.
(5.26)
Making use of the integration formula (see Appendix F) .27) where two constants α 1 and α 2 are subject to the constraint
we further reduce Eq. (5.26) to
(5.29) Equations (5.21), (5.23), (5.29) and the identity
(5.30) (obtained from Eq. (5.24) with the help of Eqs. (F.7) and (F.8)) yield Ḣ1 (t; N ) in the form Ḣ1 (t;
(5.31)
Notice that appearance of four integration constants (c 1 , c 2 , α 1 and α 2 ) in Eq. (5.31) is somewhat illusive: a little thought shows that there is a pair of independent constants, either (c 1 /c 2 , α 2 c 2 ) or their derivatives.
To determine the unknown constants in Eq. ( 5.31), we make use of the asymptotic formulae for the functions of parabolic cylinder (collected in Appendix F) in an attempt to meet the boundary conditions ‡
Following the discussion next to Eq. ( 5.11), the two cases Re t < 0 and Re t > 0 will be treated separately.
• The case Re t < 0. Asymptotic analysis of Eq. ( 5.31) at t → -∞ yields
(5.33)
Here, we have used Eq. (F.5). To determine the remaining constant α 1 c 1 , we make use of the boundary condition Eq. ( 5.32) for t → ± ι∞ -0. Straightforward calculations bring α 1 c 1 = 0. We then conclude that Ḣ1 (t;
(5.34)
• The case Re t > 0. Asymptotic analysis of Eq. ( 5.31) at t → +∞ yields α 2 = 0 so that
(5.35)
To determine the remaining constant c 1 /c 2 , we make use of the boundary condition Eq. (5.32) for t → ± ι∞ + 0. Straightforward calculations bring c 1 /c 2 = 0. We then conclude that Ḣ1 (t;
(5.36)
The calculation of Ḣ1 (t; N ) can be summarised in a single formula (5.37) where σ ιt = sgn Im (ιt) = sgn Re t denotes the sign of Re t. In terms of canonical variables p 0 (t; N ) and q 1 (t; N ), this result translates to
(5.39)
Now H 1 (t; N ) can readily be restored by integrating Eq. (5.37). We proceed in three steps. (i) First, we make use of differential recurrence relations Eqs. (F.7) and (F.8) and the Wronskian Eq. (F.4) to prove the identity
(5.40) ‡ Equation (5.32) is straightforward to derive from Eqs. (5.16), (5.12) and (5.10).
The latter allows us to write down Ḣ1 (t; N ) as
(5.41) (ii) Second, it is beneficial to employ the differential equation Eq. (F.3) to derive
(5.43)
These two relations imply
(5.45) (iii) Third, we integrate the above equation to obtain
(5.46)
Here, the integration constant was set to zero in order to meet the boundary conditions Eq. ( 5.32) at infinities. The notation Ŵt stands for the Wronskian Ŵt [f, g] = f ∂g ∂t -∂f ∂t g.
(5.47)
Average Green function and eigenlevel density.-Now, the average one-point Green function readily follows from Eq. (5.19):
(5.48)
Here, σ z = sgn Im z denotes the sign of imaginary part of z = -ιt.
The average density of eigenlevels can be restored from Eq. (5.48) and the relation
(5.49) Indeed, noticing from Eqs. (F.2) and (5.48) that
we conclude, with the help of Eq. (F.1), that
(5.51)
Here, H N -1 (ǫ) is the Hermite polynomial appearing by virtue of the relation
(5.52)
Consequently,
(5.53)
Taken together with Eqs. (5.49) and (5.48), this equation yields the finite-N average density of eigenlevels in the GUE:
(5.54)
While this result, obtained via the fermionic replica limit, is seen to coincide with the celebrated finite-N formula (Mehta 2004)
originally derived within the orthogonal polynomial technique, the factorisation phenomenon (as defined in Section 1.2) has not been immediately detected throughout the calculation of either G(z; N ) or ̺ N (ǫ). We shall return to this point in Section 5.3.
Comparing Eq. ( 5.5) with the definition Eq. ( 4.1), we conclude that
subject to the boundary conditions
(5.59)
Here and above, n ∈ Z + .
The average Green function G(z; N ) we are aimed at is given by the bosonic replica limit
(5.60)
To implement it, we assume that the Painlevé representation of Z (-) n (z; N ) holds for n ∈ R + .
Replica limit and the Hamiltonian formalism.-Similarly to our treatment of the fermionic case, we employ the Hamiltonian formulation of the Painlevé IV (Noumi 2004, Forrester andWitte 2001) which associates ψ(t; n, N ) with the polynomial Hamiltonian (Okamoto 1980a)
(5.61) of a dynamical system {Q, P, H b }, where Q = Q(t; n, N ) and P = P (t; n, N ) are canonical coordinate and momentum. For such a system, Hamilton's equations of motion read:
(5.63)
Owing to Eq. ( 5.60), we need to develop a small-n expansion for the Hamiltonian H b {Q, P, t}:
(5.64)
Being consistent with yet another expansion
where
(5.69)
Here, p 0 = p 0 (t; N ) and q 1 = q 1 (t; N ) are solutions to the system of coupled first order equations: ṗ0 = -2p 2 0 -2p 0 t -N, q1 = 4p 0 q 1 + 2q 1 t + 2.
(5.70)
Since the initial conditions are known for H 1 (t; N ), rather than for p 0 (t; N ) and q 1 (t; N ) separately, below we determine these two functions up to integration constants. The function p 0 (t; N ) satisfies the Riccati differential equation whose solution is
where
(5.73)
Factoring out ιN in the second term in Eq. (5.72) will simplify the formulae to follow.
To determine q 1 (t; N ), we substitute Eq. (5.71) into the second formula of Eq. (5.70) to derive:
.
(5.74)
Making use of the integration formula (see Appendix F)
where two constants α 1 and α 2 are subject to the constraint
we further reduce Eq. (5.74) to
(5.77) Equations (5.69), (5.71), (5.77) and the identity
(obtained from Eq. (5.72) with the help of Eqs. (F.7) and (F.8)) yield Ḣ1 (t; N ) in the form Ḣ1 (t;
(5.79)
To determine the unknown constants in Eq. ( 5.79), we make use of the asymptotic formulae for the functions of parabolic cylinder (collected in Appendix F) to satisfy the boundary conditions [see Eq. (5.59)]
(5.80)
The two cases Im t < 0 and Im t > 0 should be treated separately.
• The case Im t < 0. Asymptotic analysis of Eq. ( 5.79) at t → -ι∞ yields c 1 = 0, so that Ḣ1 (t;
(5.81)
To determine the remaining constant α 2 c 2 , we make use of the boundary condition Eq. (5.80) for t → ±∞ -ι0. Straightforward calculations bring α 2 c 2 = 0. We then conclude that Ḣ1 (t;
(5.82)
• The case Im t > 0. Asymptotic analysis of Eq. ( 5.79) at t → + ι∞ yields
(5.84)
To determine the remaining constant α 1 c 1 , we make use of the boundary condition Eq. (5.80) for t → ±∞ + ι0. Straightforward calculations bring
(5.85)
We then conclude that Ḣ1 (t;
(5.86)
The calculation of Ḣ1 (t; N ) can be summarised in a single formula
where σ t = sgn Im t denotes the sign of Im t. In terms of canonical variables p 0 (t; N ) and q 1 (t; N ), this result translates to
(5.89)
In view of Eq. (5.67), the latter result is equivalent to the statement
(5.90)
This expression, obtained within the bosonic replicas, must be compared with its counterpart derived via the fermionic replicas [Eqs. (5.19) and (5.37)]:
As the two expressions coincide, we are led to conclude that the bosonic version of the replica limit reproduces correct finite-N results for the average Green function and the average density of eigenlevels as given by Eqs. (5.48) and (5.55), respectively. Again, as is the case of a fermionic calculation carried out in Section 5.2.1, the factorisation property did not show up explicitly in the above bosonic calculation. We defer discussing this point until Section 5.3.
The very same integrable theory of characteristic polynomials is at work for a "supersymmetric" variation of replicas invented by Splittorff and Verbaarschot (2003). These authors suggested that the fermionic and bosonic replica partition functions (satisfying the fermionic and bosonic Toda Lattice equations †, respectively) can be seen as two different branches of a single, graded Toda Lattice equation. Below we show that the above statement, considered in the context of GUE, is also valid beyond the first equation of the Toda Lattice hierarchy.
First (graded) TL equation.-Equations (5.7) and (4.22) imply that the fermionic replica partition function Z (+) n (z; N ) satisfies the first TL equation in the form:
Together with the initial conditions Z (+) 0 (z; N ) = 1 and (5.93) this equation uniquely determines fermionic replica partition functions of any order (n ≥ 2). Here, D N is the function of parabolic cylinder of positive order (see Appendix F).
The first TL equation for the bosonic replica partition function Z (-) n (z; N ) follows from Eqs. (5.56) and (4.22),
(5.94)
Together with the initial conditions Z (-) 0 (z; N ) = 1 and
where σ z = sgn Im z denotes the sign of Im z, this equation uniquely determines bosonic replica partition functions of any order (n ≥ 2). Here, D -N is the function of parabolic cylinder of negative order (see Appendix F). Further, defining the graded replica partition function as
(5.96)
we spot from Eqs. (5.92) and ( 5.94) that it satisfies the single (graded) TL equation
(5.97)
Here, the index n is an arbitrary integer, be it positive or negative. The first graded TL equation must be supplemented by two initial conditions given by Eqs. (5.93) and (5.95).
Second (graded) TL equation.-Equations (5.7), (5.56) and (4.23) imply that both fermionic and bosonic replica partition functions Z (±) n (z; N ) additionally satisfy the second TL equation
Consequently, the graded replica partition function Z n (z; N ) defined by Eq. (5.96) is determined by the second graded TL equation
supplemented by two initial conditions Eqs. (5.93) and (5.95).
Replica limit of graded TL equations.-To determine the one-point Green function G(z; N ) within the framework of supersymmetric replicas, one has to consider the replica limit
(5.100)
The first and second graded TL equations bring
and
respectively. Combining the two equations, we derive
where Ŵz is the Wronskian defined in Eq. (5.47); the prime ′ stands for the derivative ∂/∂z. Interestingly, the second graded TL equation has allowed us to integrate Eq. ( 5.102) at once! The resulting Eq. (5.103) is remarkable: it shows that the average Green function can solely be expressed in terms of bosonic Z -1 (z; N ) and fermionic Z 1 (z; N ) replica partition functions with only one flavor. This structural phenomenon known as the 'factorisation property' was first observed by Splittorff and Verbaarschot (2003) in the context of the GUE density-density correlation function. Striking at first sight, the factorisation property appears to be very natural after recognising that fermionic and bosonic replica partition functions are the members of a single graded TL hierarchy.
To make Eq. ( 5.103) explicit, we utilise Eqs. (5.93) and (5.95) to observe the identity
(5.104)
Consequently,
This expression for the average Green function, derived within the framework of supersymmetric replicas, coincides with the one obtained separately by means of fermionic and bosonic replicas (see, e.g., Eq. (5.48)). Hence, the result Eq. (5.55) for the finite-N average density of eigenlevels readily follows.
The factorisation property naturally appearing in the supersymmetric variation of the replica method suggests that a generic correlation function should contain both compact (fermionic) and non-compact (bosonic) contributions. Below, the fermionic-bosonic factorisation property will separately be discussed in the context of fermionic and bosonic replicas where its presence is by far less obvious even though the enlightened reader might have anticipated the factorisation property from Eqs. (5.19) and (5.21) for fermionic replicas and from Eqs. (5.67) and (5.69) for bosonic replicas. where
and
(5.108)
To derive the last two equations, we have used Eqs. (5.38) and (5.39) in conjunction with Eqs. (F.7) and (F.8).
With the help of Eq. (5.93), the canonical "momentum" p 0 can be related to the fermionic partition function for one flavor,
This is a compact contribution to the average Green function. A non-compact contribution is encoded in the canonical "coordinate" q 1 which can be related to the bosonic partition function via Eq. (5.95):
This is yet another factorised representation for G ′ (z; N ) [compare to Eq. (5.101)].
Bosonic replicas.-To identify both compact and non-compact contributions to the average Green function, we turn to Eqs. (5.67) and (5.69) to represent the derivative of the average Green function G(z; N ) in terms of canonical variables p 0 and q 1 as ∂ ∂z G(z; N ) = -2p 0 (z) q 1 (z), (5.112)
where
and
(5.114)
To derive the last two equations, we have used Eqs. (5.88) and (5.89) in conjunction with Eqs. (F.7) and (F.8).
With the help of Eq. (5.95), the canonical "momentum" p 0 can be related to the bosonic partition function for one flavor,
This is a non-compact contribution to the average Green function. A compact contribution comes from the canonical "coordinate" q 1 which can be related to the fermionic partition function via Eq. (5.93): (5.117) agreeing with the earlier result Eq. (5.111).
Brief summary.-The detailed analysis of fermionic and bosonic replica limits performed in the context of the GUE averaged one-point Green function G(z; N ) has convincingly demonstrated that the Hamiltonian formulation of the fourth Painlevé transcendent provides a natural and, perhaps, most adequate language to identify the factorisation phenomenon. In particular, we have managed to show that the derivative G ′ (z; N ) of the one-point Green function factorises into a product of canonical "momentum" p 0 (t; N ) = lim n→0 P (t; n, N ) (5.118) and canonical "coordinate"
(5.119)
As suggested by Eqs. (5.109) and (5.115), the momentum contribution p 0 to the average Green function is a regular one; it corresponds to a compact contribution in fermionic replicas and to a non-compact contribution in bosonic replicas:
On the contrary, the coordinate contribution q 1 is of a complementary nature: defined by a replica-like limit [Eq. (5.119)] it brings in a noncompact contribution in fermionic replicas [Eq. (5.110)] and a compact contribution in bosonic replicas [Eq. (5.116)]:
We close this section by noting that the very same calculational framework should be equally effective in performing the replica limit for other random-matrix ensembles and/or spectral observables.
In this paper, we have used the ideas of integrability to formulate a theory of the correlation function
of characteristic polynomials for invariant non-Gaussian ensembles of Hermitean random matrices characterised by the probability measure dµ n (H) which may well depend on the matrix dimensionality n. Contrary to other approaches based on various duality relations, our theory does not assume "integerness" of replica parameters κ = (κ 1 , • • • , κ p ) which are allowed to span κ ∈ R p . One of the consequences of lifting the restriction κ ∈ Z p is that we were unable to represent the CFCP explicitly in a closed determinant form; instead, we have shown that the correlation function Π n|p (ς; κ) satisfies an infinite set of nonlinear differential hierarchically structured relations. While such a description is, to a large extent, implicit, it does provide a useful nonperturbative characterisation of Π n|p (ς; κ) which turns out to be much beneficial for an in-depth analysis of the mathematical foundations of zero-dimensional replica field theories. With certainty, the replicas is not the only application of a nonperturbative approach to CFCP developed in this paper. With minor modifications, its potential readily extends to various problems of charge transfer through quantum chaotic structures (Osipov andKanzieper 2008, Osipov andKanzieper 2009), stochastic theory of density matrices (Osipov, Sommers and Życzkowski 2010), random matrix theory approach to QCD physics (Verbaarschot 2010), to name a few. An extension of the above formalism to the CFCP of unitary matrices may bring a useful calculational tool for generating conjectures on behaviour of the Riemann zeta function at the critical line (Keating andSnaith 2000a, 2000b).
Finally, we wish to mention that an integrable theory of CFCP for β = 1 and β = 4 Dyson symmetry classes is very much called for. Building on the insightful work by Adler and van Moerbeke (2001), a solution of this challenging problem seems to be within the reach.
A. Bilinear Identity in Hirota Form: An Alternative Derivation In Section 3.3, the bilinear identity in Hirota form [Eq. (3.27) or,equivalently,Eq. (3.34)] was derived from the one in the integral form [Eq. (3.18)]. As will be shown below, the latter is not actually necessary.
An alternative proof of the bilinear identity in Hirota form [Eq. (3.34)] starts with the very same Eq. (3.19),
to rewrite Eq. (A.1) in the form
Here, we have used the scalar product notation defined in Eq. (3.6). The bilinear identity is obtained from Eq. (A.4) after expressing all its ingredients in terms of the τ function Eq. (3.1).
(i) The coefficients A σ and B σ . To determine A σ , we employ two identities, .5) and
) is a consequence of Eq. (2.6) and the definition of Schur polynomials (see Table 1). Equation (A.6) follows from the Heine formula
and the identity
where p m (λ) is an infinite dimensional vector
Indeed, substituting Eq. (A.8) into Eq. (A.7), we derive:
In the last step, we have used the obvious formula
e v(t ′ ;λj ) . (A.11)
Having established Eq. (A.6), we substitute it and Eq. (A.5) into Eq. (A.2) to obtain
Similar in spirit calculation yields
(ii) Scalar products. Now we are ready to express the scalar products in Eq. (A.4) in terms of the τ function Eq. (3.1). Having in mind the Heine formula Eq. (A.7), we rewrite the scalar product in the l.h.s. of Eq. (A.4) as .16) Symmetrising the integrand, .17) and employing the remarkable relation (taken at n
we deduce:
For σ ≥ ℓ (which is the only nontrivial case), this scalar product reduces to
By the same token,
The bilinear identity. Now we are in position to derive the bilinear identity.
To this end we substitute Eqs. (A.12), (A.13), (A.20) and (A.21) into Eq. (A.4). Up to the prefactor 1
Owing to the identity
Interchange of the two series brings (1
Indeed, since the l.h.s. is the polynomial in x of the degree m, the Schur polynomials s k (-p m (λ)) in the r.h.s. must nullify for k > m.
we derive:
Finally, replacing n with n = m -s, we reproduce the l.h.s. of Eq. (3.34). The r.h.s. of Eq. (3.34) can be reproduced along the same lines starting with the r.h.s. of Eq. (A.4). This ends the alternative proof of the bilinear identity in Hirota form.
Definition
Property 1. Let P(D) be a multivariate function defined on an infinite set of differential operators D
Then it holds:
Taylor-expanded around x = 0 and the definition Eq. (2.11) of the Schur polynomials.
and
The exponential identity holds:
For a more exhaustive list of the properties of Hirota differential operators, the reader is referred to Hirota's book (Hirota 2004).
The correlation function of characteristic polynomials in JUE is defined by the formula
where
is the normalisation constant. It is assumed that both µ > -1 and ν > -1. The associated τ function equals
The superscript J standing for JUE will be omitted from now on. Although the very same technology is at work for a nonperturbative calculation of the JUE correlation function Eq. (C.1), its treatment becomes significantly more cumbersome. For this reason, only the final results of the calculations will be presented below.
In the notation of Section 3, the definition Eq. (C.1) implies that
Notice that dim(c ′ ) = 0 follows from Eq. (3.70) in which Z 0 = {-1, +1}.
where the operator Dq is defined through operators Eq. (4.8) as
and Lq (t) is the Virasoro operator given by Eq. (2.21).
The three lowest Virasoro constraints for q = -1, q = 0, and q = +1 read:
Projecting the first Toda Lattice equation Eq. (4.15) onto the hyperplane t = 0 with the help of the first [Eq. (C.9)] and second [Eq. (C.10)] Virasoro constraints, one derives:
Notice the structural similarity between TL
Here, α [j,k] , β j and γ are the short-hand notation for the following functions:
Remark. For p = 1, the equation KP J 1 simplifies. Indeed, introducing the function
where b j 's are given by b
one observes that Eq. (C.13) transforms to
with the parameters
Equation (C.16) can equivalently be written as
The boundary condition at infinity reads:
It is easy to convince yourself that Eq. (C.18) can be reduced to the sixth Painlevé transcendent. Introducing the new function (not to be confused with the operator Dq defined by Eq. (C.8) of the previous subsection) and Lq (t) is the Virasoro operator given by Eq. (2.21).
The three lowest Virasoro constraints for q = -1, q = 0, and q = +1 read: for which all movable singularities of y(t) are limited to poles, given F is a rational function in all its arguments. Note that this requirement, known as the Painlevé property, does not rule out the existence of immovable (i.e. fixed) essential singularities. We remind that a singularity is called movable if its location depends on one or more integration constants. Painlevé (1900Painlevé ( , 1902) ) and Gambier (1910) have shown that the requirement that all movable singularities are restricted to poles leads to 50 types of equations, six of which cannot further be reduced to either (i) linear second order differential equations or (ii) the differential equation for the Weierstrass P-function, (y ′ ) 2 = 4y 3 -g 2 y -g 3 , (E.2) (g 2 , g 3 are constants) or (iii) the Riccati equation Lemma. Let u 1 (t) and u 2 (t) be linearly independent functions whose Wronskian is a constant:
Then,
¶ See also their single variable reductions Eqs. (4.22) and (4.23) derived for the GUE.
† We will drop the superscript (f) wherever this does not cause a notational confusion.
† We will drop the superscript (b) wherever this does not cause a notational confusion.
† Notice thatSplittorff and Verbaarschot (2003) use the term "Toda Lattice equation" for the first equation of the TL hierarchy.
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