DYSON S CONSTANT FOR THE HYPERGEOMETRIC KERNEL

Transcription

1 DYSON S CONSTANT FOR THE HYPERGEOMETRIC KERNEL OLEG LISOVYY Abstract We study a Fredholm determinant of the hypergeometric kernel arising in the representation theory of the infinite-dimensional unitary group It is shown that this determinant coincides with the Palmer-Beatty-Tracy tau function of a Dirac operator on the hyperbolic disk Solution of the connection problem for Painlevé VI equation allows to determine its asymptotic behavior up to a constant factor for which a conjectural expression is given in terms of Barnes functions We also present analogous asymptotic results for the Whittaker and Macdonald kernel Introduction Connections between Painlevé equations and Fredholm determinants have long been a subject of great interest mainly because of their applications in random matrix theory and integrable systems see eg One of the most famous examples is concerned with the Fredholm determinant F t = det K sine where K sine is the integral operator with the sine kernel sinx y πx y on the interval 0 t It is well-known that F t is equal to the gap probability for the Gaussian Unitary Ensemble GUE in the bulk scaling limit As shown in 7 the function σt = t d ln F t dt satisfies the σ-form of a Painlevé V equation tσ + tσ σ tσ σ + σ = 0 Equation and the obvious leading behavior F t 0 = t + O t provide an efficient method of numerical computation of F t for all t Further as t one has N F t = f 0 t e t / + f k t k + O t N The coefficients f f in this expansion can in principle be determined from It was conjectured by Dyson that the value of the remaining unknown constant is f 0 = e 3ζ where ζz is the Riemann ζ-function Dyson s conjecture was rigorously proved only recently in Similar results were also obtained in 3 8 for the Airy-kernel determinant describing the largest eigenvalue distribution for GUE in the edge scaling limit 6 The present paper is devoted to the asymptotic analysis of the Fredholm determinant of the hypergeometric kernel on L 0 t with t 0 This determinant to be denoted by Dt arises in the representation theory of the infinite-dimensional unitary group 5 and provides a -parameter class of solutions to Painlevé VI PVI equation 6 Rather surprisingly it turns out to coincide with the Palmer-Beatty-Tracy PBT τ-function of a Dirac operator on the hyperbolic disk 0 3 under suitable identification of parameters Relation to PVI allows to give a complete description of the behavior of Dt as t up to a constant factor analogous to Dyson s constant f 0 in the sine-kernel asymptotics Relation to the PBT τ-function on the other hand suggests a conjectural expression for this constant in terms of Barnes functions The paper is planned as follows In Section we recall basic facts on Painlevé VI and the associated linear system The F kernel determinant Dt and the PBT τ-function are introduced in Sections 3 and Section 5 gives a simple proof of a result of 6 relating Dt to Painlevé VI In Section 6 we discuss Jimbo s asymptotic formula for PVI and determine the monodromy corresponding to the F kernel solution Section 7 contains the main results of the paper: the k=

2 OLEG LISOVYY asymptotics of Dt as t obtained from the solution of PVI connection problem Proposition 7 and a conjecture for the unknown constant Conjecture 8 Numerical and analytic tests of the conjecture are discussed in Sections 8 and 9 Similar asymptotic results for the Whittaker and Macdonald kernel are presented in Section 0 Appendix A contains a brief summary of formulas for the Barnes function Consider the linear system Painlevé VI and JMU τ-function dφ dλ = A0 λ + A λ + A t Φ λ t where A ν sl C ν = 0 t are independent of λ with eigenvalues ±θ ν / and θ / 0 A 0 + A + A t = θ 0 θ / 0 The fundamental matrix solution Φλ is a multivalued function on P \{0 t } Fix the basis of loops as shown in Fig and denote by M 0 M t M M SL C the corresponding monodromy matrices Clearly one has M M M t M 0 = Fig : Generators of π P \{0 t } Since the monodromy is defined up to overall conjugation it is convenient to introduce following 6 a 7-tuple of invariant quantities 3 p ν = Tr M ν = cos πθ ν ν = 0 t p µν = Tr M µ M ν = cos πσ µν µ ν = 0 t These data uniquely fix the conjugacy class of the triple M 0 M M t unless the monodromy is reducible The traces 3 satisfy Jimbo-Fricke relation p 0t p t p 0 + p 0t + p t + p 0 p 0 p t + p p p 0t p p t + p 0 p p t p 0 p + p t p p 0 = As a consequence for fixed {p ν } p 0t p t there are at most two possible values for p 0 It is well-known that the monodromy preserving deformations of the system are described by the so-called Schlesinger equations da 0 = A t A 0 da dt t dt which are equivalent to the sixth Painlevé equation: d q dt = q + q + dq 5 q t dt qq q t + t t θ θ 0t q + θ t Relation between A 0t t and qt is given by A0 λ + = A t A t t + t + dq q t dt + q + θ t tt q t A λ + At = λ t ktλ qt λλ λ t

3 DYSON S CONSTANT FOR THE HYPERGEOMETRIC KERNEL 3 Jimbo-Miwa-Ueno JMU τ-function 8 of Painlevé VI is defined as follows: d 6 dt ln τ JMUt; θ = tr A 0A t + tr A A t t t where θ = θ 0 θ θ t θ Introducing a logarithmic derivative 7 σt = tt d dt ln τ JMUt; θ + t θt θ θ t + θ0 θ θ 8 it can be deduced from the Schlesinger system that σt satisfies the following nd order ODE σ-form of Painlevé VI: σ tt σ + σ tσ σ σ θ t θ θ 0 θ 8 = 6 = σ + θ t + θ σ + θ t θ σ + θ 0 + θ σ + θ 0 θ In terms of qt the definition of σt reads 9 σt = q t t qq q t θ q qq tt θ t t + θ t + θ0 θ θ 8 θ 0t q + θ t q θ t tt q t 3 Hypergeometric kernel determinant It was shown in 5 that the spectral measure associated to the decomposition of a remarkable -parameter family of characters of the infinite-dimensional unitary group U gives rise to a determinantal point process with correlation kernel where λ = Kx y = λ sin πz sin πz π AxBy BxAy x y 0 y x + z + w + z + w Γ + z + w + z + w + z + z + w + w + z + z + w + w Ax = x z+z +w+w x z+z +w z + w F z + w z + z + w + w x x Bx = x z+z +w+w + x z+z +w + z + w F + z + w + z + z + w + w + x x Note that our notation slightly differs from the standard one 5 6; to shorten some formulas from Painlevé theory the interval of 5 6 is mapped to 0 by x / + x The kernel Kx y has a number of symmetries: S It is invariant under transformations z z and w w ; the latter symmetry follows from a b F z = z c a bf c a c b c z c S It is also straightforward to check that Kx y is invariant under transformation z z z z w w + z + z w w + z + z S3 We can simultaneously shift z z ± z z ± w w w w ; together with S this allows to assume without loss of generality that 0 Re z + z We are interested in the Fredholm determinant 3 Dt = det K 0t t 0 Assume that the parameters z z w w C satisfy the conditions: C z = z C\Z or k < z z < k + for some k Z

4 OLEG LISOVYY C w = w C\Z or l < w w < l + for some l Z C3 z + z + w + w > 0 z + z < w + w < Then as was shown by Borodin and Deift in 6 the determinant 3 is well-defined and Dt = τ JMU t; θ for the following choice of PVI parameters: 35 θ = z + z + w + w z z 0 w w The original proof in 6 that Dt satisfies Painlevé VI is rather involved In Section 5 we give an alternative simple derivation of this result in the spirit of 7 Lemma Assume C C3 Then the asymptotic expansion of Dt as t 0 has the form 36 Dt = κ t +z+z +w+w + O t +z+z +w+w where 37 κ = Proof sin πz sin πz π + z + w + z + w Γ + z + w + z + w + z + z + w + w + z + z + w + w As t 0 one has Dt t Kx x dx The result then follows from 0 Ax x z+z +w+w Bx x z+z +w+w + as x 0 Note that in the expression for κ given in Remark 7 in 6 the gamma product is missing which seems to be a typesetting error The asymptotics 36 and σpvi equation 8 uniquely fix Dt by a result of 7 Gamma sin πz sin πz product in 37 is a function of θ 0 θ θ only but depends on an additional parameter eg z + z ; hence we are dealing with a -parameter family of initial conditions The results of 6 can be extended to a larger set of parameters This follows already from the observation that the subset of C defined by C C3 is not stable under the transformations S S3 However instead of trying to identify all admissible values of z z w w in the remainder of this paper we simply replace C C3 by a much weaker invariant condition C z + w z + w z + w z + w / Z <0 and Re z + z + w + w > 0 and define Dt as the JMU τ-function of Painlevé VI with parameters 35 whose leading behavior as t 0 is specified by Our aim in the next sections is to determine the asymptotics of Dt as t π PBT τ-function Palmer-Beatty-Tracy τ-function 0 3 is a regularized determinant of the quantum hamiltonian of a massive Dirac particle moving on the hyperbolic disk in the superposition of a uniform magnetic field B and the field of two non-integer Aharonov-Bohm fluxes πν < ν < 0 located at the points a Denote by m and E the particle mass and energy by /R the disk curvature and write b = BR µ = m E R +b s = tanh daa R where da a denotes the geodesic distance between a and a Then τ P BT s can be expressed 3 in terms of a solution us of the sixth Painlevé equation 5: d ds ln τ s s du P BT s = u uu s ds u s u θ θ 0 + θ t + s s u u s where the corresponding PVI parameters are given by θ = + ν + ν b 0 µ + ν ν

5 DYSON S CONSTANT FOR THE HYPERGEOMETRIC KERNEL 5 The initial conditions are specified by the asymptotics of τ P BT s as s computed in 0: τ P BT s = κ P BT s +µ + O s +µ κ P BT = sin πν sin πν + µ + ν b µ ν π Γ + b + µ + ν b µ ν + b + µ + µ Some resemblance between 9 and suggests that τ P BT s is a special case of the JMU τ-function Indeed consider the following transformation: s t u t q In the notation of Table of this corresponds to Bäcklund transformation r x P xy for Painlevé VI If us is a solution with parameters θ = θ 0 θ θ t θ then qt solves PVI with parameters θ = θ t θ θ θ 0 + Straightforward calculation then shows that τ P BT t = τ JMU t; θ provided θ = 0 Lemma Under the following identification of parameters 3 z + z + w + w = µ z z = ν ν w w = + ν + ν b we have Dt = τ P BT t cos πz + z = cos πν + ν Proof It was shown above that if 3 holds then both Dt and τ P BT t are JMU τ- functions with the same θ To show the equality it suffices to verify that implies κ = κ P BT Symmetries of Dt imply that τ P BT s is invariant under transformations S µ µ ν ν b + ν + ν b; S µ µ ν ν b b These symmetries of τ P BT s are by no means manifest although they can also be deduced from the Fredholm determinant representation in 0 Theorem 5 Painlevé VI from Tracy-Widom equations 5 Basic notation Tracy and Widom 7 have developed a systematic approach for deriving differential equations satisfied by Fredholm determinants of the form 5 D I = det K I where K I is an integral operator with the kernel 5 K I x y = ϕxψy ψxϕy x y on L J with J = M a j a j The kernels of the form 5 are called integrable and possess j= rather special properties: eg it was observed in 5 that the kernel of the resolvent K I K I is also integrable The method of 7 requires that ϕ ψ in 5 obey a system of linear ODEs of the form 53 mx d dx ϕ ψ N = A k x k ϕ ψ where mx is a polynomial and A k sl C k = 0 N Note that a linear transformation ϕ ϕ G leaves K ψ ψ I x y invariant provided det G = and therefore {A k } can be conjugated by an arbitrary SL C-matrix k=0

6 6 OLEG LISOVYY Our aim is to show that in the special case mx = x x N = J = 0 t the determinant 5 i coincides with the F kernel determinant Dt and ii considered as a function of t is a Painlevé VI τ-function Let us temporarily switch to the notation of 7 and introduce the quantities qt = K I ϕ t pt = K I ψ t ut = ϕ K I ϕ vt = ϕ K I ψ wt = ψ K I ψ where the inner products are taken over J Then D I D I = qp pq with primes denoting derivatives with respect to t Tracy-Widom approach gives a system of nonlinear first order ODEs for q p u v w which we are about to examine 5 Derivation Let A be diagonalizable so that one can set α0 β A 0 = 0 α 0 A γ 0 α = 0 0 α The Tracy-Widom equations then read q α β q 5 t t p = γ α p 55 u = q v = pq w = p where α = α 0 + α t + v β = β 0 + α u γ = γ 0 α + w The system 5 55 has two first integrals I = αpq + βp + γq α v I = v + α 0 βγ α t tpq + α tv I t Consider the logarithmic derivative ζt = tt D I D I It can be easily checked that 50 ζ = αpq + βp + γq = α v + I ζ = α pq t tζ = α βp γq Note that v α are expressible in terms of ζ and pq in terms of ζ Using 57 and 58 one may also write βγ and βp + γq in terms of ζ and ζ Now squaring 50 we find a second order equation for ζ: 5 t tζ + ζ α tζ ζ ζ tζ ζ ζ + α 0 α I = I + I ζ If we parameterize the integrals I I as and define I = k k + α α 0 + α I = k + k α α 0 + α 5 σt = ζt αt + α + k k then 5 transforms into σpvi equation 8 with parameters θ = k k k + k 0 α Moreover 5 and the definition of ζt imply that D I t coincides with the corresponding JMU τ-function

7 DYSON S CONSTANT FOR THE HYPERGEOMETRIC KERNEL 7 The system 53 has two linearly independent solutions only one of which can be chosen to be regular as x 0 This is the only solution of interest here as if ϕ ψ have an irregular part the operator K I fails to be trace-class The regularity further implies that q p u v w vanish as t 0 and therefore the integrals I I are given by I = 0 I = α 0 β 0 γ 0 Choosing A 0 A as above one can still conjugate them by a diagonal matrix Use this freedom to parameterize α 0 β 0 γ 0 α as follows: α 0 = c ab c a b c ac b β 0 = c a b ab γ 0 = c a b α = c a b so that I = c and therefore k + k = a b k k = c Now if Re c > 0 the regular solution of 53 is given by 53 ϕ c ac b c+c x = const ψ ab c+c x c x a+b x + c a+b x F a b c x x F + a + b + c x x Setting a = z + w b = z + w c = z + z + w + w and comparing 53 with 3 33 we see that K I t coincides up to an adjustable constant factor with the F kernel of Section 3 Remark 3 A system similar to 5 55 has already appeared in the Tracy-Widom analysis of the Jacobi kernel see Section VC of 7 As the integral I was not noticed there the final result of 7 was a third order ODE as one may well guess it represents the first derivative of 5 in a disguised form Later Haine and Semengue have derived another third order equation for the Jacobi kernel determinant using the Virasoro approach of and obtained Painlevé VI as the compatibility condition of the two equations Our calculation gives among other things an elementary proof of this result Remark For non-diagonalizable A it can be assumed that A = 55 remain unchanged whereas instead of 5 we get q α β q t t p = γ α p where As before we have two first integrals α = α 0 + v w β = β0 + s u + v γ = γ 0 w I = αpq + βp + γq + I = α β γ t tp + t γ I t The equations The rest of the computation is completely analogous to the diagonalizable case As a final result one finds that the determinant Dt with w = w is a τ-function of Painlevé VI with parameters θ = z + z + w z z 0 0

8 8 OLEG LISOVYY 6 Jimbo s asymptotic formula A remarkable result of Jimbo 6 relates the asymptotic behavior of the JMU τ-function 6 near the singular points t = 0 to the monodromy of the associated linear system Theorem 5 Theorem in 6 Assume that J J J3 θ 0 θ θ t θ / Z 0 Re σ 0t < θ 0 ± θ t ± σ 0t θ ± θ ± σ 0t / Z Then τ JMU t has the following asymptotic expansion as t 0: τ JMU t = const t σ 0t θ 0 θ t θ 0 θ t σ 0t θ θ σ 0t 6σ0t + σ 0t ŝ t +σ0t θ 0 θ t + σ 0t θ θ + σ 0t 6 6σ0t σ 0t ŝ t σ0t + θ 0 θ t σ 0tθ θ σ 0t 8σ 0t t + O t Re σ0t where σ 0t 0 and σ0t σ ŝ = Γ 0t + θ0+θt+σ0t θ0 θt σ0t + θ +θ+σ0t θ θ σ0t + σ 0t + σ 0t + θ0+θt σ0t θ0 θt+σ0t + θ +θ σ0t θ θ+σ0t If σ 0t = 0 then s ± cos πθ t σ 0t cos πθ 0 cos πθ σ 0t cos πθ = = ± i sin πσ 0t cos πσ t cos πθ t cos πθ cos πθ 0 cos πθ e ±iπσ0t ± i sin πσ 0t cos πσ 0 + cos πθ t cos πθ + cos πθ cos πθ 0 τ JMU t = const t θ 0 +θ t θ θ t t θ θθ 0 θt tω + Ω where Ω = ŝ ln t and ŝ = s + ψ + θ 0 + θ t + ψ + θ tθ θ + θ θ 0 θ t + θt θ 0 Here ψx denotes the digamma function and + ψ + θ + θ + ψ tω + + o t + θ θ s = iπ cos πσ t + cos πσ 0 cos πθ 0 e iπθ cos πθ e iπθt + i sin πθ + θ t cos πθt cos πθ 0 cos πθ cos πθ s ψ When one tries to determine from Theorem 5 the monodromy associated to the F kernel solution Dt of σpvi it turns out that all three assumptions J J3 are not satisfied: Firstly J does not hold since in our case θ t = 0 This requirement can nevertheless be relaxed as the appropriate non-resonancy condition for is θ 0 θ θ t θ / Z\{0} The proof of asymptotic formulas when some θ s are equal to zero differs from that in 6 only in technical details; see eg If we blindly accept 6 then from Dt 0 follows σ 0t = θ 0 = z + z + w + w Thus J is violated unless Re θ 0 < and J3 does not hold in any case Note however that 6 admits a well-defined limit as θ t = 0 σ 0t θ 0 In this limit the coefficients of t and

9 6 t σ0t DYSON S CONSTANT FOR THE HYPERGEOMETRIC KERNEL 9 vanish; we also have cos πσ 0 cos πθ + cos πθ cos πσ t e iπθ0 sθ 0 σ 0t π sin πθ 0 cos πθ cos πσ t sin π θ θ 0 + θ sin π θ + θ 0 θ and hence the coefficient of t +σ0t becomes cos πθ cos πσ t + θ 0+θ +θ π Γ + θ0+θ θ + θ0 θ+θ + θ0 θ θ + θ 0 + θ 0 Suppose that in our case the error estimate in 6 can be improved to O t +θ0 or at least to o t +θ0 Then assuming that 0 Re z + z and comparing 6 with one would conclude that σ t = z + z The above steps can indeed be justified after some tedious analysis going into the depths of Jimbo s proof Alternatively the monodromy can be extracted from Sections 3 of 6 where σpvi equation for Dt has itself been derived from a Riemann-Hilbert problem 7 Asymptotics of Dt as t Once the monodromy is known the asymptotics of τ JMU t as t can be determined from Jimbo s formula after substitutions t t θ 0 θ σ 0t σ t σ 0 σ 0 where 7 cos π σ 0 = Tr M 0 M t M M t = p0 p + p t p p 0t p t p 0 Remark 6 The transformation σ 0 σ 0 is missing in 6 due to an incorrectly stated symmetry: the relation τ JMU t; M 0 M t M = const τ JMU t; M M t M 0 on p of 6 should be replaced by τ JMU t; M 0 M t M = const τ JMU t; Mt M 0 M M t M 0 M 0 M t M 0 M 0 which can be understood from Fig Fig : Homotopy basis after transformation λ λ t t Proposition 7 Assume that 0 Re z + z < and z z w w z + z + w z + z + w / Z If z + z 0 then the following asymptotics is valid as t : Dt = C t zz + zz z + z + wz + z + w + ww z + z t a + t +z+z a t z z + O t Rez+z 7 where C is a constant and a ± = Γ z z z z ± z ± z + w + z+z ± z+z + w + z+z ± z+z ± z ± z ± z ± z z z w + z+z z+z w + z+z z+z

10 0 OLEG LISOVYY Similarly if z + z = 0 then 73 Dt = C t z + z ww t Ω + Ω z w + w t Ω + + o t where Ω = a ln t and a = ψ + z + ψ z + ψ + w + ψ + w ψ Proof Take into account that in our case θ t = 0 σ 0t = θ 0 and replace θ 0 θ σ 0t σ t σ 0 σ 0 Different quantities in Theorem 5 then transform into s s t = s s t = 0 σt σ ŝ ŝ t = Γ t + θ+σt θ σt + θ0+θ +σt + θ0 θ +σt + σ t + σ t + θ σt θ+σt + θ0+θ σt + θ0 θ σt ŝ ŝ t = ψ + ψ + θ θ The statement now follows from σ t = z + z and 35 + ψ + θ 0 + θ + ψ + θ 0 θ ψ The constant C in 7 73 remains as yet undetermined We will find an expression for it using Lemma and earlier results of Doyon 0 who conjectured that for vanishing magnetic field τ P BT s coincides with a correlation function of twist fields in the theory of free massive Dirac fermions on the hyperbolic disk The asymptotics of τ P BT s as s 0 and s is then fixed respectively by conformal behavior of the correlator and its form factor expansion The basic statement of 0 supported by numerics is that there indeed exists a solution of the appropriate σpvi equation which interpolates between the two asymptotics Although the proof that the correlator of twist fields satisfies σpvi has not yet been found there are further confirmations of Doyon s hypothesis: long-distance asymptotics with b = 0 and the exponent zz in the short-distance power law 7 reproduce the conjectures of 0 The QFT analogy also implies that for real z z 0 such that 0 < z + z < and w = w z z this corresponds to b = 0 the constant C in 7 can be expressed in terms of vacuum expectation values of twist fields which have been computed in 0 see also The resulting conjectural evaluation is: 7 C z + z z w = G + z + w + w + z + z + w z z + w =w z z a a where G m = b b n z z + z + z + z + w z + w + z + w z + w m k= G a k n k= G b and Gx denotes the Barnes function: k { } Gx + = π x/ ψx exp xx + n= + x n n exp { x + x In spite of what one might expect extension of the above approach to the case b 0 turns out to be rather complicated However the simple structure of 7 and the symmetries of the F kernel suggest the following: Conjecture 8 Under assumptions of Proposition 7 the constant C in the asymptotic expansions 7 73 is given by z + z z 75 C = G + z + w + w + z + z + w + z + z + w z z + z + z + z + w + z + w + z + w + z + w The formula 75 is clearly compatible with 7 and S S It has been checked both numerically and analytically as described below n }

11 DYSON S CONSTANT FOR THE HYPERGEOMETRIC KERNEL 8 Numerics To verify Conjecture 8 one can proceed in the following way: The solution of PVI associated to the F kernel solution Dt of σpvi uniquely determined by has the following asymptotic behavior as t 0: 8 qt = t λ 0 t +z+z +w+w + Ot +z+z +w+w 8 In fact one can show that in this case λ 0 = + z + z + w + w z + wz + w qt = t λ 0 t +z+z +w+w t +z z F + Ot +z+z +w+w κ z + w + z + w + z + z + w + w t 3 Use this asymptotics as initial condition and integrate the corresponding PVI equation numerically for some admissible choice of θ It is then instructive to check Proposition 7 by verifying that for 0 < Rez + z < the asymptotic expansion of qt as t is given by qt = λ t z z + o t Rez+z where λ = Γ z + z z + z z z w + w z z z z z z z + z + w + z + z + w = = z z w z + z + w a Similarly for z + z = 0 one has a logarithmic behavior Ω qt = + t z + w + O t ln t Finally use qt and the initial condition Dt as t 0 to compute Dt from 7 9 Looking at the asymptotics of Dt as t one can numerically check the formula 75 for C 9 Special solutions check For special choices of parameters and initial conditions Painlevé VI equation can be solved explicitly All explicit solutions found so far are either algebraic or of Picard or Riccati type Algebraic solutions have been classified in ; up to parameter equivalence their list consists of 3 continuous families and 5 exceptional solutions It turns out that the parameters of exceptional algebraic solutions cannot be transformed to satisfy F kernel constraints p 0 = p 0t p t = Continuous families however do contain representatives verifying these conditions Explicit computation of the corresponding τ-functions provides a number of analytic tests of Conjecture 8 some of which are presented below Our notation for PVI Bäcklund transformations follows Table in Example 9 Painlevé VI equation with parameters θ = θ 0 θ is satisfied by qt = θ θ + t θ 3 θ t t This two-branch solution is obtained by applying Bäcklund transformation s δ s x s y s z s δ s z s δ P xy to Solution II in set θ a = θ b = θ An explicit formula for the corresponding JMU τ-function

12 OLEG LISOVYY can be found from 7 9: τ JMU t = t / + t θ Note that τ JMU t 0 = θ 8 t + O t 3 and therefore τ JMU t coincides with the hypergeometric kernel determinant Dt if we set z = w = +θ z = w = θ The asymptotics of τ JMU t as t has the form τ JMU t = θ t θ 6 + O t which implies that C = θ To verify that this coincides with the expression 3+θ 3 θ 5+θ 5+θ 5 θ 5 θ 7+θ 7 θ C = G θ 3 θ given by Conjecture 8 one can use the recursion relation Gz + = ΓzGz the duplication formulas A A3 for Barnes and gamma functions and the value of G from Appendix A Example 0 Consider the rational curve q = s + s 5s + ss 5s t = s + s s s + It defines a three-branch solution of PVI with parameters θ = 0 0 /3 which can be obtained from Solution III in with θ = 0 by the transformation t x = s x s δ s y s z s s δ The associated τ-function is given by τ JMU ts = s s s s 7 7 where the normalization constant is introduced for convenience The map ts bijectively maps the interval onto 0 Choosing the corresponding solution branch one finds that τ JMU t 0 = t3 + O t τ JMU t t First asymptotics implies that τ JMU t coincides with Dt provided z = z = 6 w = 7 6 w = From the second asymptotics we obtain C = whereas Conjecture 8 gives 3 C = G Equality of both expressions can be shown using the known evaluations of G k 6 k = 5 see or Appendix A Example Applying the transformation s δ s x s y 3 s z s s δ r x to Solution IV in and setting θ = 0 one obtains a four-branch solution of PVI with θ = / 0 parameterized by The corresponding τ-function has the form 36 q = s s5s 5s + s s3 t = 3 s3 s 3 s τ JMU ts = s 5 6 s 5 s 5 8

13 DYSON S CONSTANT FOR THE HYPERGEOMETRIC KERNEL 3 Choose the solution branch with s 0 From the asymptotics τ JMU t 0 = t + O t 3 follows that τ JMU t coincides with Dt provided z = 5 z = w = 5 6 w = 6 Leading term in the asymptotic behavior of τ JMU t as t is τ JMU t t 5 so that we have C = On the other hand Conjecture 8 implies that 5 6 C = G To prove that these expressions are equivalent i use the multiplication formula A with n = and z = 5 to compute G G 5 G 7 G and ii combine the resulting expression with the evaluations of G k G k 6 0 Limiting kernels 0 Flat space limit: PVI PV The interpretation of Dt as a determinant of a Dirac operator Section suggests to consider the flat space limit R This corresponds to the following scaling limit of the F kernel: w + t s s 0 w In this limit Dt transforms into the Fredholm determinant D L s = det K s L with the kernel K L x y = lim w + w K x w y A L xb L y B L xa L y w = λ L x y A L x = x W z+z +w λ L = x B z z L x = x W z+z +w sin πz sin πz π Γ + z + w + z + w z z where W αβ x denotes the Whittaker s function of the nd kind K L x y is the so-called Whittaker kernel see eg which plays the same role in the harmonic analysis on the infinite symmetric group as the F kernel does for U The function σ L s = s d ds ln D Ls satisfies a Painlevé V equation written in σ-form: 0 s σ L = σ L z + z + w + sσ L + σ L σ L σ L z w σ L z w This can be shown by considering the appropriate limit of the σpvi equation for Dt An initial condition for 0 is provided by the asymptotics D L s = λ L e s s z z w + O s To link our notation with the one used in the PV part of Jimbo s paper 6 we should set θ 0 θ t θ V Jimbo = z + w z w z z which gives D L s = e z+ws τ V Jimbo s This in turn allows to obtain from Theorem 3 in 6 the asymptotics of D L s as s 0: Proposition Assume that 0 Re z + z < and z z w z + z + w / Z If z + z 0 then D L s = C L s zz + zz z + z + w z + z s a + L s+z+z a L s z z + O s Rez+z with a ± L = Γ z z z z ± z ± z + w + z+z ± z+z ± z ± z ± z ± z z z w + z+z z+z x

14 OLEG LISOVYY If z + z = 0 then D L s = C L s z + z ws Ω L + Ω L z s ΩL + + os where Ω L = a L ln s and a L = ψ + z + ψ z + ψ + w ψ Note that the same result is obtained by considering the formal limit of the leading terms in the asymptotics of Dt This further suggests an expression for constant C L : Conjecture 3 Under assumptions of Proposition we have C L = lim w w zz z + z z + z + w + z + z + w C = G z z + z + z + z + w + z + w 0 Zero field limit: PV PIII Next we consider the limit of vanishing magnetic field B 0 In terms of the parameters of the Whittaker kernel this translates into w + s ξ ξ 0 w The scaled kernel is given by x K M x y = lim w + w K L w y sin πz sin πz = w π AM xb M y B M xa M y x y A M x = x K z z+ x BM x = K z z x where K α x is the Macdonald function Denote D M ξ = det K s M and introduce σ M ξ = ξ d dξ ln D M ξ Then σ M ξ solves the σ-version of a particular Painlevé III equation: 0 ξσ M = σ M σ M σ M ξσ M + z z σ M To match the notation in 6 we have to set θ 0 θ III Jimbo = z z z z which gives D L s = e ξ τ III Jimbo ξ The appropriate initial condition for this σpiii is given by sin πz sin ξ πz 03 D M ξ = e + z z 3 π ξ 8 + O ξ ξ The asymptotics of D M ξ as ξ 0 can now be obtained from Theorem 3 in 6: Proposition Assume that 0 Re z + z < and z z / Z If z + z 0 then D M ξ 0 = C M ξ zz + zz z + z ξ a+ M ξ+z+z a M ξ z z + O ξ Rez+z with a ± z z M = Γ z z ± z ± z ± z ± z ± z ± z z z If z + z = 0 then D M ξ 0 = C M ξ z + z ξ Ω M + Ω M oξ where Ω M = a M ln ξ and a M = ψ + z + ψ z ψ Analogously to the above we suggest a conjectural expression for C M : Conjecture 5 Under assumptions of Proposition we have z + z z C M = lim w w zz C L = G + z z z + z + z

15 DYSON S CONSTANT FOR THE HYPERGEOMETRIC KERNEL 5 Partial proof This formula can in fact be proved for real z = z 0 though in an indirect way Consider the solution ψr of the radial sinh-gordon equation d ψ dr + dψ r dr = sinh ψ satisfying the boundary condition ψr ν νk 0 r as r + Define the function { } dψ τr ν = exp u sinh ψu ν du du and consider the logarithmic derivative σξ = ξ d dξ ln τ ξ ν τr ν = πν e r r r It is straightforward to show that σξ satisfies σpiii equation 0 with z = z Further a little calculation shows that as r + 3 r + O r Comparing this asymptotics with 03 we conclude that D M ξ = τ sin πz ξ ± z=z π On the other hand τr ν = τ B r ν where τ Br ν is a special case of the bosonic -point tau function of Sato Miwa and Jimbo which can be represented as an infinite series of integrals formulas in with l = l By direct asymptotic analysis of this series Tracy 5 has proved that for ν 0 π it has the following behavior as r 0: with τ B r ν = e βν r αν + o αν = σ ν σν = π βν = 3αν ln + ln π ν ln cos πσν sin πz From ν = ± π arcsin πν ln G +σν σν one readily obtains σ = α = z Thus in order to show that βν reproduces the conjectured expression for C M with z = z it is sufficient to prove the identity + z + z z z G = z cos πz G + z z + z + z z z This however is a simple consequence of the duplication formula for Barnes function and the known evaluation of G Appendix A Multiplication formula for Barnes function 8: n x n nx ln Gnx = nx ln ln π + 5 ln n n A + + n n n ln A + ln G x + j + k n j=0 k=0 where A = exp ζ denotes Glaisher s constant Asymptotic expansion as z arg z π: z A ln G + z = ln z 3z + z ln π ln A + + O z

16 6 OLEG LISOVYY Special values see eg : ln G = ln ln π 3 ln A + 8 ln G = ln π ln Γ 3 3 ln A π 3 ψ ln G = ln π ln Γ 3 3 ln A π 3 ψ ln ln G = 6 + π ln Γ ln A 0π 3 ψ ln ln G = 6 + π ln Γ ln A 5 0π 3 ψ ln G = 3 ln Γ 9 8 ln A K π 3 ln G = 3 ln Γ 9 8 ln A K π where K is Catalan s constant When checking Conjecture 8 with explicit examples one also needs the relations n A3 Γnx = π n n nx Γ x + k ψ x + ψ x = π n sin πx k=0 References V S Adamchik On the Barnes function Proc 00 Int Symp Symbolic and Algebraic Computation Academic Press M Adler T Shiota P van Moerbeke Random matrices vertex operators and the Virasoro algebra Phys Lett A J Baik R Buckingham J DiFranco Asymptotics of Tracy-Widom distributions and the total integral of a Painlevé II function Comm Math Phys ; preprint arxiv: mathfa A Borodin Harmonic analysis on the infinite symmetric group and the Whittaker kernel St Petersburg Math J A Borodin G Olshanski Harmonic analysis on the infinite-dimensional unitary group and determinantal point processes Ann Math ; preprint math/0099 mathrt 6 A Borodin P Deift Fredholm determinants Jimbo-Miwa-Ueno tau-functions and representation theory Comm Pure Appl Math ; preprint math-ph/ O Costin R D Costin Asymptotic properties of a family of solutions of the Painlevé equation P V I Int Math Res Notices ; preprint math/0035 mathca 8 P Deift A Its I Krasovsky Asymptotics of the Airy-kernel determinant Comm Math Phys ; preprint math/06095 mathfa 9 P Deift A Its I Krasovsky X Zhou The Widom-Dyson constant for the gap probability in random matrix theory J Comput Appl Math ; preprint math/ mathfa 0 B Doyon Two-point correlation functions of scaling fields in the Dirac theory on the Poincaré disk Nucl Phys B ; preprint hep-th/03090 B Dubrovin M Mazzocco Monodromy of certain Painlevé VI transcendents and reflection groups Inv Math ; preprint mathag/ F J Dyson Fredholm determinants and inverse scattering problems Comm Math Phys T Ehrhardt Dyson s constant in the asymptotics of the Fredholm determinant of the sine kernel Comm Math Phys ; preprint math/0005 mathfa L Haine J-P Semengue The Jacobi polynomial ensemble and the Painlevé VI equation J Math Phys A R Its A G Izergin V E Korepin N A Slavnov Differential equations for quantum correlation functions Int J Mod Phys B M Jimbo Monodromy problem and the boundary condition for some Painlevé equations Publ RIMS Kyoto Univ

17 DYSON S CONSTANT FOR THE HYPERGEOMETRIC KERNEL 7 7 M Jimbo T Miwa Y Môri M Sato Density matrix of an impenetrable Bose gas and the fifth Painlevé transcendent Physica D M Jimbo T Miwa K Ueno Monodromy preserving deformations of linear ordinary differential equations with rational coefficients I Physica D I V Krasovsky Gap probability in the spectrum of random matrices and asymptotics of polynomials orthogonal on an arc of the unit circle Int Math Res Not ; preprint math/0058 mathfa 0 O Lisovyy On Painlevé VI transcendents related to the Dirac operator on the hyperbolic disk J Math Phys ; preprint arxiv:07057 math-ph O Lisovyy Yu Tykhyy Algebraic solutions of the sixth Painlevé equation preprint arxiv: mathca S Lukyanov A B Zamolodchikov Exact expectation values of local fields in quantum sine-gordon model Nucl Phys B ; preprint hep-th/ J Palmer M Beatty C A Tracy Tau functions for the Dirac operator on the Poincaré disk Comm Math Phys ; preprint hep-th/ M Sato T Miwa M Jimbo Holonomic quantum fields IV Publ RIMS Kyoto Univ C A Tracy Asymptotics of a τ-function arising in the two-dimensional Ising model Comm Math Phys C A Tracy H Widom Level-spacing distributions and the Airy kernel Comm Math Phys ; preprint hep-th/9 7 C A Tracy H Widom Fredholm determinants differential equations and matrix models Comm Math Phys ; preprint hep-th/ I Vardi Determinants of Laplacians and multiple gamma functions SIAM J Math Anal T T Wu B M McCoy C A Tracy E Barouch Spin-spin correlation functions for the two-dimensional Ising model: exact theory in the scaling region Phys Rev B Laboratoire de Mathématiques et Physique Théorique CNRS/UMR 6083 Université de Tours Parc de Grandmont 3700 Tours France address: lisovyi@lmptuniv-toursfr Bogolyubov Institute for Theoretical Physics Kyiv Ukraine