Painlevé III asymptotics of Hankel determinants for a perturbed Jacobi weight

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1 Painlevé III asymptotics of Hankel determinants for a perturbed Jacobi weight arxiv:4.8586v [math-ph] 9 May 05 Zhao-Yun Zeng a, Shuai-Xia Xu b, and Yu-Qiu Zhao a a Department of Mathematics, Sun Yat-sen University, GuangZhou 5075, China b Institut Franco-Chinois de l Energie Nucléaire, Sun Yat-sen University, GuangZhou 5075, China Abstract We study the Hankel determinants associated with the weight wx;t) = x ) β t x ) α hx), x,), where β >, α+β >, t >, hx) is analytic in a domain containing [,] and hx) > 0 for x [,]. In this paper, based on the Deift-Zhou nonlinear steepest descent analysis, we study the double scaling limit of the Hankel determinants as n and t. We obtain the asymptotic approximations of the Hankel determinants, evaluated in terms of the Jimbo-Miwa-Okamoto σ-function for the Painlevé III equation. The asymptotics of the leading coefficients and the recurrence coefficients for the perturbed Jacobi polynomials are also obtained. 00 Mathematics Subject Classification. Primary 33E7; 34M55; 4A60. Keywords and phrases: Hankel determinants; perturbed Jacobi weight; Painlevé III equation; orthogonal polynomials; Riemann-Hilbert approach. Introduction and statement of results Let wx;t) be the perturbed Jacobi weight wx;t) = x ) β t x ) α hx), x,),.) where β >, α+β >, t >, the function hx) is analytic in a domain containing [,] and hx) > 0 for x [,]. We study the Hankel determinants D n [wx;t)] = detµ j+k ) n j,k=0,.) Corresponding author Shuai-Xia Xu). address: xushx3@mail.sysu.edu.cn

2 where µ i is the i-th moment of wx;t), namely, µ i = x i wx;t)dx, i = 0,,. The Hankel determinants possess the well-known multiple integral representation [,..)], D n [wx;t)] = n! [,] n n i=wx i,t) i<j n x i x j dx i. Via the above integral representation, the Hankel determinants are closely related to various fundamental quantities in random matrix theory, such as the partition function, the gap probability of eigenvalues and the moment generating function of a certain random variable associated with the random matrix ensemble; see [0]. For example, in the Jacobi unitary ensemble corresponding to the weight w x) = x ) α, it is well-known that the probability distribution of the largest eigenvalues is P n λ max < s) = = n!d n [w x)] +s ) n +nα n!d n [w x)] [,s] n [,] n n i= i= x i )α i<j n n x i x j dx i i= i=+x i ) α s) x i ) α i<j n x i x j dx i, with s) = 3 s ; see [0]. Also, there is the remarkable Tracy-Widom formula for the +s large-n asymptotics of the distribution of the extreme eigenvalues near the hard edge, lim d n ds lnp nλ max < s n ) = σ JMs) ;.3) s see[], whereσ JM satisfiesthejimbo-miwa-okamotoσ-formofthepainlevéiiiequation [7, 3.3)]) sσ JM ) +σ JM σ JM sσ JM )4σ JM ) α σ JM = 0.4) and the boundary conditions σ JM s) 4 α+ Γ+α)Γ+α) s+α, s 0; σ JM s) s 4 α s, s..5) It is worth noting that the Tracy-Widom formula.3) holds for a large family of unitary ensembles, including the modified Jacobi unitary ensemble associated with the weight x) α +x) β hx), x,), α >, β >. The phenomenon is termed universality in random matrix theory. i=

3 In [4], Forrester and Witte apply the Okamoto τ-function theory to study the random matrix average for the Laguerre unitary ensemble Es,n) = n n x i s) β x α i e x i x i x j dx i, Z n [s, ) n i= where Z n is thenormalization constant. It is shown that the logarithmicderivative of the average Es, n) satisfies the Jimbo-Miwa-Okamoto σ-form of the Painlevé V equation, with parameters depending on n. By taking the scaling s = t and letting n, it 4n is found that the Painlevé V equation degenerates to a general Jimbo-Miwa-Okamoto σ-form of the Painlevé III equation. Thus the hard edge limiting average is obtained, generalizing the results of Tracy and Widom.3)-.5). The boundary conditions of the Painlevé III equation have also been studied in the follow-up paper [5] of the same authors. Now we mention several weights closely related to.). A decade ago, in [8, 9], Kuijlaars et al. considered the orthogonal polynomials associated with the weight +x) α x) β hx), x,), which, in the case α = β, is the weight.) with t =. The main focus of [8] is to obtain the asymptotics of the orthogonal polynomials, including those of the recurrence coefficients, the leading coefficients and the Hankel determinants. In [9], the results find applications in random matrix theory, the Bessel limit kernel is obtained at the hard edge, and the kernel is independent of the perturbed analytic function h. Later, in [3], Vanlessen studies the orthogonal polynomials associated with the further generalized Jacobi-type weight with several singularities, of the form +x) α x) β hx) i<j p x x ν λν, x,), ν= where p is a fixed integer, < x < x < < x p <, λ ν >, λ ν 0, α, β >, and h is real analytic and strictly positive on [,]. The asymptotics of the recurrence coefficients and the orthogonal polynomials associated are also obtained. Very recently, Basor, Chen and Haq [] study the Hankel determinants associated with the weight ŵx,k) = x ) β k x ) α, x,), β >, α R, which is a special case of the weight.) with t = /k, and h = k α. For n fixed, it is shown in [], via the ladder operator method, that the finite Hankel determinant D n [ŵx;k)] is the τ-function of the Painlevé VI equation; see also [3] for applications of the ladder operator method. Large-n asymptotics of the Hankel determinants are also obtained in [] for fixed k. In this paper, however, we focus on the asymptotics of the Hankel determinants, the leading coefficients, and the recurrence coefficients associated with the weight.), in 3 i=

4 the sense of a double scaling limit as n and t when the algebraic singularity x = t approaches the hard edge x =. Remarkable progress has been made in the study of the double scaling limit of Hankel determinants and Toeplitz determinants owning to the Riemann-Hilbert approach developed by Deift, Zhou et al. [8,, ]. A series of questions and conjectures arose in the analysis of the Ising model see [0]) and random matrices have been solved. For example, in [5, 9], Claeys, Deift and co-authors obtain Painlevé V asymptotics of Toeplitz determinants and Hankel determinants associated with the emergence of a Fisher-Hartwig singularity in the weight function. More recently, Claeys and Krasovsky [6] study a weight with merging Fisher-Hartwig singularities and again a Painlevé V function is involved to describe the transition between two different types of asymptotic behavior of the Toeplitz determinants. Other types of singularities have also been encountered. In [], Brightmore, Mezzadri and Mo consider the asymptotics of the partition function associated with the Gaussian weight perturbed by an essential singularity and they get Painlevé III type asymptotics. In [4, 5], Xu, Dai and Zhao also obtain Painlevé III type asymptotics of the Hankel determinants associated with the Laguerre weight with an essential singularity at the hard edge. In the double scaling limit of Hankel determinants, the appearance of Painlevé functions is of particular interests; cf., e.g., [4, 7, 6, 7]. The reader is referred to the comprehensive survey paper [0] for the historic background and updated results on the theory of Hankel determinants and Toeplitz determinants with applications in the Ising model. In the preceding papers [6, 8], the authors have studied the transition asymptotics of the eigenvalue correlation kernel for the perturbed Jacobi unitary ensemble defined by the perturbed Jacobi weight given in.), varying from the Bessel kernel J β to J α+β as the parameter t varies in,d] for a fixed d >. A new class of universal behavior at the edge of the spectrum for the modified Jacobi ensemble is obtained and described in terms of the generalized Painlevé V equation, which in this case is equivalent to the Painlevé III equation after a Möbius transformation. In the present paper, we focus on the asymptotic approximations of the Hankel determinants, the leading coefficients, and the recurrence coefficients of the polynomials orthogonal with respect to the weight.), in the sense of a double scaling limit as n and t. To simplify our discussion, we consider the even weight function.) by assuming h is even, then we have the recurrence relation zπ n z) = π n+ z)+b n π n z).6) for monic orthogonal polynomials with respect to the perturbed Jacobi weight. 4

5 . Statement of results To state the main results, we need the Jimbo-Miwa-Okamoto σ-form of the Painlevé III equation sσ JM ) +σ JM σ JM sσ JM )4σ JM ) c σ JM c σ JM c 0 = 0;.7) cf. [7, 3.3)], where c = α+β), c = βα+β) and c 0 = β 6. Our first result is on the Painlevé III asymptotic approximations of the Hankel determinants, in terms of the Jimbo-Miwa-Okamoto σ-notation. Theorem. Let β >, α + β >, t >, and D n t) be the Hankel determinants given in.) corresponding to the weight wx;t) in.). As n and t, we have the following asymptotic expansion lnd n t) = n+α+β)v 0 αlnht) βlnh)+ [ n +nα+β)+ ] ln+ α lnt+ 4nlnϕt) nlnϕt)) 4 0 k= [ n+α+β + σ JM s s 6 ) [ kvk + α+β) ] ln n 4 4 ] lnπ +ln G ) Gα+β +) ds+o),.8) where V k = π e kiθ lnhcosθ)dθ, k = 0,,, Gz) is the Barnes G-function defined π 0 in 5.40), ϕt) = t+ t, the Jimbo-Miwa-Okamoto σ-function σ JM is analytic on 0, + ) and solves the equation.7) with the boundary conditions and with σ JM s) = σ JM s) = 4 s α Cα,β) = β s ) +α+β 4α+β) s+cα,β) +Os +α+β )+Os ) as s 0.9) 4 s+ 4 α +αβ) αβ ) 4 4 +Os ) as s,.0) s αγ α β)γβ +) α+β)γ α)γα+β +)Γα+β +). Remark. For α+β = 0, the condition.9) is simplified to σ JM s) = s 4 +Os ) as s 0. 5

6 Remark. For β = 0, the theorem is reduced to the celebrated Tracy-Widom formula for the large-n asymptotic distribution of the largest eigenvalues near the hard edge; cf..3)-.5), see also []. Our second result is on the transition asymptotics of the leading coefficients of the corresponding orthonormal polynomials, where the Jimbo-Miwa-Okamoto σ-function is also involved. Theorem. Let n and t, then we have the asymptotic approximation of the leading coefficient of the orthonormal polynomial of degree n with respect to.) γ n = + α n πdt ) + q ) t +cn s )) ϕt) +Ot )+O α,.) s n where ϕt) = t+ t, D t ) = α+β) e V 0 ϕt) α, V 0 = lnhcosθ)dθ, qs) = π 0 4σ JM s s ) 6 6 α+β) +, s = 4nlnϕt), c 4 n is independent of t such that c n = On ), and the error term O ) s n is uniform for t,d]. The third result is on the transition asymptotics of the recurrence coefficients of the corresponding monic orthogonal polynomials; see.6). Theorem 3. Let n and t, then we have the asymptotic approximation of the recurrence coefficients b n = q ) ) ) ) t )+O t t 4 8 +O +O,.) s n n n 3 where qs) = 4σ JM s) s α β) +, s = 4nlnϕt), ϕt) = t + t 4, the derivative is taken with respect to s and the error term O ) n is uniform for t,d]. 3 Remark 3. The variable s = 4nlnϕt) 4 n t 0, ) describes the gap between the hard edge and the algebraic singularity of the weight function.). Theorems, and 3 describe the transition of asymptotics of the Hankel determinants, the leading coefficients and the recurrence coefficients associated with weights having two different hard edge singularities: On the one side it is of the form x ) α+β as s 0 +, on the other side with x ) β as s. For fixed s taken in the transition region 0, ), we obtain the Painlevé type transition asymptotics in the double scaling limit. Moreover, as s 0 and s, the limiting asymptotics of these quantities agree with the known asymptotics for the modified Jacobi weight with fixed singularities. Let s 0 or s, we obtain from Theorem the asymptotics of the leading coefficients corresponding to Jacobi polynomials of different orders. We state the results in the following corollary, which are in consistence with those obtained in [8]. Corollary. i) Let n and t such that s = 4nlnϕt) 0 +, we have the asymptotic approximation of the leading coefficients of the orthonormal polynomials )) γ n = 4α+β) +Os ) +O..3) n πd ) 8n n 6 π

7 ii) Let n and t such that s, we have the asymptotic approximation of the leading coefficients )) γ n = 4β +o..4) n πdt ) 8n n Similarly, as s 0 or s, we extract from Theorem 3 the asymptotics of the recurrence coefficients corresponding to Jacobi polynomials with different parameters. The results are also consistent with those obtained in [8]. Corollary. i) Let n and t such that s 0 +, we have the following asymptotic approximation of the recurrence coefficients b n = 4 4α+β) +Os) 6n +O n 3)..5) ii) Let n and t such that s and s 0, we have the asymptotic of the n recurrence coefficients b n = ) 4 4β +o..6) 6n n To prove the main results, first we derive several differential identities for the leading coefficients and the logarithmic derivative of the Hankel determinants, relating to the solution of the matrix Riemann-Hilbert RH) formulation for orthogonal polynomials. Then we make use of the results obtained in a preceding paper of the authors using the Deift-Zhou steepest descent method for the RH problems [8,, ]. The derivation is given in [6] with full details, and is briefly reviewed in Section 4 below. The rest of the paper is organized as follows. In Section, we formulate the model RH problem for the Painlevé III equation, which is introduced by the authors earlier in [6]. An equivalent σ-form of Painlevé III equation is then derived. In Section 3, we prove the differential identities for the Hankel determinants and the leading coefficients in terms of the RH problem for the orthogonal polynomials associated with the weight.). The identities are the starting points of our analysis in later sections. In Section 4, we outline the notations and formulas resulted from the RH analysis, obtained previously by the authors in [6]. The proofs of Theorems,, 3 are provided in the last section, Section 5. Painlevé III equation and σ-form of Painlevé III equation In [6], to construct the local parametrix in the nonlinear steepest descent analysis of the RH problems, Xu and Zhao introduce a modified version of the Painlevé V equation which is equivalent to the Painlevé III equation after a Möbius transformation. 7

8 Proposition. Xu and Zhao [6]) Assume that ys) solves d y ds y ) dy + dy y ds s ds + yy +) 4y ) + y s +αy s + β ) y + s = 0,.) where α and β are constants. The equation is converted to a generalized Painlevé V equation by putting ω = y, so that d ω ds ω + ω ) dω ds ) + dω s ds +α+)ω s + ωω+) ω ) ± β ) ω ω+) = 0, s.) which is reduced to the classical Painlevé V equation for β =. Applying the Möbius transformation vs) = ys)+ turns the equation.) into the Painlevé III equation ys) d v ds ) dv + dv v ds sds + α β v α β + + ) v3 s 6 + = 0..3) 6v Moreover, the equation.) is the compatibility condition for the Lax pair sσ3 Ψ λ λ,s) = + As) λ + Bs) λ+ + β )σ ) Ψλ,s), λ.4) ) λσ3 Ψ s λ,s) = +us)σ Ψλ,s), ) ).5) where σ = 0 0 and σ 3 = 0 0 are the Pauli matrices, As) = σ Bs)σ, and Bs) = bs) α bs) α)ys),.6) bs)/ys) bs)+ α with ys) being a particular solution of.), while bs) and us) are determined by the equations and s dy ds = sy + by ) αy )y β y )y ).7) us) = bs)/ys) bs) α)ys) s + β..8) s In view of the symmetry σ Ψ λ)σ = Ψλ), new Lax pair of differential equations are obtained by applying the transformation Ψ 0 ζ,s) = e πi 4 σ 3 ζ 4 σ I +iσ ) 3 Ψ ζ,s e πi 4 σ 3, argζ π,π)..9) 8

9 Then a model RH problem for Ψ 0 ζ,s) in the ζ-plane is formulated, and its unique solvability is proved for s > 0; see [6]. For later use, we recall the asymptotic behavior of the model RH problem for Ψ 0 ζ,s) as follows: Ψ 0 ζ) = ζ 4 σ 3 I iσ I + σs) s σ 3 +ius)σ ζ +O ) ) e s ζ σ 3.0) ζ as ζ for argζ π,π), where s 0, ), the function u is defined in.8) and σ is defined as also, σs) = bs) α/)s su) ;.) Ψ 0 ζ) = E 0 I + E k ζ /4) )ζ k /4) ασ 3.) k= as ζ /4 for argζ /4) π,π), where the coefficients can be determined by substituting.) into.4) and.9). For example, the leading coefficient is b α E 0 = α σ 3 +ys) ibs) +i ) ys)bs) α),.3) iys) ) bs) ys)bs) α) and the,) entry of E can be represented as E ) = σ su+α +β )..4) α 4 It is also noted that see [6, Sec..], with Θ = α) σ = b α, su) = ) b y +yb α)..5). Jimbo-Miwa-Okamoto σ-form of the Painlevé III equation We derive the Jimbo-Miwa-Okamoto σ-form of the Painlevé III equation. Let qs) = σs) us,.6) whereσs)andus)appearinthecoefficients oftheasymptoticbehaviorofψ 0 atinfinity, given in.0), then by the relation sσ σ = su) ;.7) cf..) and.5), we have q ) = u u, s.8) 9

10 and sq = σ +su) ssu)..9) Then taking derivative on both sides of.9), using the definition of q and the equation d su) = uσ + su β + ) ; ds 4 see [6, Sec..] with Θ = α and γ = β, and in view of.7), we obtain u = β 4 8 q q + s 4..0) Substituting.0) into.8), we get the following third order equation q q + s 8 q β )q + β + ) 4 8 q ) = 0..) q + s 8 ) s By the transformation qs) = 4σ JM s /6) s /6 c + 4,.) we get from.) the new third order equation III := s σ JM σ JM s σ JM +sσ JM σ JM 8sσ JM3 +4σ JM +s c )σ JM +c 0 = 0,.3) where c = α+β), c 0 = β 6. Let II denote the left-hand side of Jimbo-Miwa-Okamoto σ-form.7) of the Painlevé III equation, namely, IIs) := sσ JM ) +σ JM σ JM sσ JM )4σ JM ) c σ JM c σ JM c 0,.4) where c 0 and c are defined in.3), and c is a certain constant to be determined, then ) II σ JM 3 +II = III..5) σ JM σ JM From the above equality we see that the third order nonlinear equation.3) is equivalent to the Jimbo-Miwa-Okamoto σ-form.7) of the Painlevé III equation. Indeed, if σ JM satisfies the σ-form.7) for arbitrary coefficient c, in view of.4) we have IIs) = cσ JM for a constant c. Substituting it into.5) then yields.3). Conversely, if σ JM solves.3), then.5) is reduced to II II = σ JM σ JM. 0

11 Solving this equation gives IIs) = cσ JM, where c is a constant. Hence σ JM satisfies the Jimbo-Miwa-Okamoto σ-form.7) of the Painlevé III equation. The coefficient c = βα + β), as can be determined by substituting the behavior of σ JM at infinity into.7); see.30). We summarize the above derivation as follows: Proposition. Let qs) = σs) sus), where σs) and us) appear in.0) describing the asymptotic behavior of Ψ 0 at infinity, then qs) satisfies the third order nonlinear differential equation q q + s 8 By the transformation q β )q + β + ) 4 8 q ) = 0. q + s 8 ) s qs) = 4σ JM s /6) s /6 c + 4, the above third order equation is turned into the Jimbo-Miwa-Okamoto σ-form of the Painlevé III equation sσ JM) +σ JMσ JM sσ JM)4σ JM ) c σ JM c σ JM c 0 = 0,.6) where c = α+β), c = βα+β) and c 0 = β 6. Noting that for β = 0, the equation.6) is reduced to the special Jimbo-Miwa- Okamoto σ-form.4) of the Painlevé III equation, as appeared in []. In [6, Prop. ], it is proved that the RH problem for Ψ 0 ζ,s) has a unique solution for s 0, ). The nonlinear steepest descent analysis of the RH problem for Ψ 0 ζ,s) is also carried out as s 0 and s. As a by-product, the asymptotics of the specific Painlevé function are then obtained. We collect the results in the proposition that follows, obtaining directly from Proposition 3 in [6]. Proposition 3. The functions σs) and us) are analytic in s 0, ). For these and several other auxiliary functions, we have the asymptotic behavior as s : ) β ys) = ± +O, s s σs) = α s β bs) = 4αβ ) ) +O, s s 3 us) = β s ) 4αβ ) s ) +O, s + 4αβ ) s + 6α β ) s 3 +O s 4 )..7)

12 As s 0, they behave as ys) = + β α α+β) s+os )+Os +α+β) ), σs) = α+β) 4 + α s s α+β 3 +C 0α,β) 6 bs) = α+β) + α s + 3 α s s α+β 6 +4C 0α,β) 6 sus) = α s s α+β 3 +C 0α,β) 6 where C 0 α,β) = ) +α+β ) +Os )), ) +α+β ) +Os )), +α+β) Γ α β)γβ +) Γ α)γα+β +)Γα+β +). ) +α+β ) +Os )),.8) As a corollary of Proposition 3, we have the following asymptotic behavior of σ JM. Corollary 3. The Jimbo-Miwa-Okamoto σ-function σ JM s) is analytic for s 0, ) and satisfies the boundary conditions β s ) +α+β σ JM s) = 4α+β) s+cα,β) +Os +α+β )+Os ) as s 0 +,.9) 4 and with σ JM s) = 4 s α Cα,β) = s+ 4 α +αβ) αβ ) 4 4 +Os ) as s +,.30) s αγ α β)γβ +) α+β)γ α)γα+β +)Γα+β +). The asymptotic behavior of the σ-function σ JM s) has been considered in [5] and [7]. To compare the results with those obtained in [7], we take ζs) = σ JM 4s)+s, then by.6) we have sζ ) = 4ζ ζ )ζ sζ )+α+β)ζ α)..3) The related τ-function is defined as ζs) = s d lnτs) αβ +s ds in [7] in our notations. Then by.9), we get the asymptotic of the τ-function [ τs) = cs αβ βs α+β Cα,β) ] +α+β s+α+β +Os +α+β )+Os )+Os +α+β) ).3) with an arbitrary constant c. The result is in consistence with [7, Thm. 3.]. It is noted that, in [7], more general τ-function with three complex parameters is considered, thus more restrictions on the parameters are needed. Yet in [7], in our notation, a restriction < Reα+β) 0 is brought in, and the error estimate therein is given as O s +Reα+β))) ; see [7, Thm. 3.].

13 3 Riemann-Hilbert problem for orthogonal polynomials and differential identities Let π n z) be the monic polynomial of degree n with respect to the weight wx) = wx;t) in.), then π π n z) ns)ws) ds πi s z Yz) =, 3.) πiγn π n z) γn π n s)ws) ds s z is the unique matrix-valued function analytic in C\[, ], fulfilling the jump condition ) wx) Y + x) = Y x) for x,), 3.) 0 the asymptotic condition at infinity ) z n 0 Yz) = I +O/z)) 0 z n as z, 3.3) and certain behavior demonstrating weak singularities at z = ±; see [3] and [6]. To derive the asymptotic behavior of the Hankel determinants, the leading coefficients and the recurrence coefficients, we establish differential identities to represent several quantities in terms of the matrix-valued function Y in 3.). Lemma. Let h n = γ n = πn x)wx)dx, 3.4) then h n can be expressed in terms of Y±t) as d dt h n = πiαy t)y t) Y t)y t)), 3.5) where γ n is the leading coefficient of the orthonomal polynomial of degree n, and Y ij z) denotes the i,j) entry of Yz) given in 3.). Proof. Taking derivative with respect to t on both sides of 3.4) and making use of the orthogonality, we arrive at d dt h n = It follows from.) that π n x) t wx)dx. wx;t) = αwx;t) t t x + ). t+x Then the lemma is obtained by partial fraction decomposition of π n x)/x±t) and again using the orthogonality. For the Hankel determinants, we also have 3

14 Lemma. Let H n = d dt lnd nt), 3.6) then the following differential identity holds, H n = α Y Y z ) t) Y Y z ) t) ). 3.7) Proof. By the well-known relation between the Hankel determinants and the leading coefficients γ k = D k/d k+, D n = γ n γ n γ 0 ; 3.8) cf. Szegő [,..5)], and the orthogonal relation p n x) wx)dx =, 3.9) where p n z) = γ n π n z) is the orthonormal polynomial with respect to.), we get H n = n k=0 γkπ kx) wx) n dx = α t k=0 γkπ kx)wx) t x + ) dx. 3.0) t+x Now the Christoffel-Darboux formula [, 3..4)]) applies and we have n p kx) = γn k= ) d π n dx π d n π n dx π n 3.) Substituting 3.) into 3.0) and using the fraction decomposition techniques and the orthogonal relation, we obtain 3.7). 4 Nonlinear steepest descent analysis The nonlinear steepest descent analysis for the orthogonal polynomials has been provided by two of the present authors in [6, Sec. 3]. The central piece is the construction of the local parametrix in a domain containing singularities z = and z = t, in which a modified Painlevé V equation is involved. It is shown that the equation is equivalent to the Painlevé III equation after a Möbius transformation. In this section, we briefly review the results and collect several formulas to be used in the investigation of the Hankel determinants and the recurrence coefficients. In [6, Sec. 3.4], applying a certain normalization at infinity, Yz) in 3.) is approximated by N t z) = D t ) σ 3 M az) σ 3 M D t z) σ 3 4.) 4

15 for z kept away from [,], where M = I +iσ ), az) = z z+) /4 for z C\[,] with branches chosen such that argz ±) π,π) and thus ax) is positive for x > and a + x)/a x) = i for x,), and the Szegő function associated with wx) takes the form z D t z) = ϕz) ) β/ z exp π ln{t x ) α hx)} x dx z x ), z C\[,], 4.) in which ϕz) = z+ z is analytic in C\[,] and ϕz) z as z. In 4.), ) D t ) = lim D t z) = β ln[t x ) α hx)]dx exp. 4.3) z π x In a disc U,δ) centered at z = with fixed positive small radius δ, containing the hard edge z = and the algebraic singularity z = t of the weight function, the local parametrix is constructed as P ) z) = Ez)Ψ 0 f t z),n ρ t )ϕz) nσ 3 Wz) σ 3, 4.4) where Wz) = z ) β z t ) α hz), argz ± ) π,π), argz ± t) π,π), Ψ 0 ζ) = Ψ 0 ζ,s) is the solution to the model RH problem related to the Painlevé III equations; see.9). Also, in 4.4), Ez) = N t z)wz) σ 3 {Gf t z))}, 4.5) where Gζ) is a specific matrix function defined as { Gζ) = ζ 4 σ I iσ 3 α } ζ 4 dτ exp )σ 3, ζ C\,/4], 4.6) 0 τ τ ζ and the conformal mapping f t z) = lnϕz)) z ) = +Oz )), z U,δ), 4.7) ρ t ρ t with ρ t = 4lnϕt)) such that ρ t = 8t )+Ot ) ) as t. Accordingly, Yz) in 3.) is approximated by the local parametrix in U,δ). More precisely, we have Yz) = nσ 3 Rz)Ez)Ψ 0 f t z))wz) σ 3, t < z < +δ, 4.8) where E and f t is defined in 4.5) and 4.7), respectively, and Rz) = I +On ), 4.9) uniformly for z in the whole complex plane. 5

16 5 Proof of the theorems In the present section, we apply the differential identities in Lemma and Lemma and the results obtained via the nonlinear steepest descent analysis for the orthogonal polynomials summarized in Section 4, to derive the asymptotics of the Hankel determinants, the leading coefficients and the recurrence coefficients. Substituting 4.) into 4.5), we have Ez) = D σ 3 t )M az) σ 3 M D t z) σ 3 Wz)) σ 3 G f t z)), 5.) where az) = ) z 4 with argz ± ) π,π), the Szegő function D z+ t z), and the auxiliary functions G and f t are defined in 4.), 4.6) and 4.7), respectively. Making a change of variables s = τ in the integral in 4.6), we get a simpler representation for G, namely, Gζ) = ζ 4 σ I iσ )α 3 ζ σ 3. 5.) ζ + By the Cauchy theorem, the integral in 4.) for the Szegő function D t z) can be written as the summation of the following two integrals z lnt x ) α dx π x z x =α ln z t ϕz) z t α dx x x z z 5.3) α dx + x x z and z π t ln hx) dx x z x = lnhz) z ln hx) dx 4πi Γ x x z, 5.4) where argz ± ) π,π) and Γ is an anti-clockwise loop in the analytic domain of h encircling [, ]. Then the integrals on the right-hand side of 5.3) are expressed explicitly in terms of elementary functions as t dx x x z = zt+ z ln t ) t z t and t dx x x z = z ln zt t + ) t, z +t where argz ±) π, π) and the logarithmic functions take the principle branch. Thus we get an explicit expression of the Szegő function defined in 4.) Dtz) ) z ln hζ) dζ Wz) =ϕz) α+β) exp πi Γ ζ ζ z z +t zt+ z ) α 5.5) t z t zt z, t + 6

17 where in the fractional power we ) take the) principle branches, and argz ±) π,π). It is readily verified that = for x,). D t x) Wx) + D t x) Wx) In view of 4.3) and using a residue calculation argument, we have d dt lnd t ) = from which we obtain D t ) = D ) α t, 5.6) t+ t ) α. 5.7) Now a combination of 5.), 5.) and 5.5) gives Ez) = D σ 3 t )M az) σ 3 M Dz;t) σ 3 M ζ 4 σ 3, 5.8) where ) Dz;t) =ϕz) α+β z ln hζ) dζ exp 4πi Γ ζ ζ z ) α z +t)lnϕz) lnϕt)) zt+ z ) t α z t)lnϕz)+lnϕt)) zt z, t + and ζ = f t z) is defined in 4.7). Using Taylor expansions at z = t, we have lnϕz) lnϕt) z t = ϕ t) z=t ϕt) = t. Therefore, at z = t we have Dt;t) = ϕt) α+β t exp 4πi Γ ) ln hζ) dζ lnϕt) ζ ζ t t t By using the behaviors of ϕt) and lnϕt) as t, and noting that ζ t = ) n t for t ζ ζ ζ <, we obtain Dt;t) = + n=0 )α. 5.9) d k t ) k 5.0) k= for 0 < t < δ, where δ is a constant such that 0 < δ <, the coefficients d k are explicitly computable and the first two are d = α+β)+e 0) and d = 4 α+β)+e 0) 3 α 5.) 7

18 with e 0 = πi Γ ln hζ) dζ ζ ζ. 5.) For later use, we further compute the logarithmic derivative of Dz; t). Repeatedly using Taylor expansions at z = t, we have ) d lnϕz) lnϕt) dz ln = ) ϕ t) z t z=t ϕ t) ϕ t) t = ϕt) t ). Similarly, we get ) d lnϕz)+lnϕt) dz ln = z +t z=t t lnϕt) t = t ) 5 +Ot ), and d dz ln zt+ z t ) z=t = t t. Putting these formulas together, taking logarithm of 5.9) and differentiating both sides, and in view of 5.0), we have d dz lndz;t) = α+β + e ) 0 α z=t t ) 3 + d k t )k, 5.3) where e 0 is defined in 5.) and d k are computable constants. Substituting 5.0) into 5.8), we get the expansion Et) = D σ 3 t )M C k t ) )t )) k/ 4 σ 3, 5.4) k=0 where M = I +iσ ), the coefficients C k are explicitly computable matrices and the first few are ) ) 0 C 0 = 0 0 and C = id iα 0, 3 with d defined in 5.). By.3), we get b α σ 3 +ys) E 0 = α iys) ) ibs) +i ys)bs) α) bs) ys)bs) α) ) k=, 5.5) where s = 4nlnϕt). From the boundary conditions.7) and.8) for ys) and bs), we can derive the asymptotic behavior and E 0 = b α)e 0,0 I +Os)+Os α+β+) )) as s 0 +, 5.6) E 0 = E 0, k s k as s +, 5.7) k=0 where E 0,k are computable constant matrices and E 0,0 ) = )

19 5. Proof of Theorem Now we substitute the asymptotics we obtained for Y into the differential identity 3.7) for lnd n t). First, we consider the case for z close to t. From 4.8), we obtain Y Y z =Wz) σ 3 Ψ 0 ζ)e z)e z z)ψ 0ζ)Wz) σ 3 +Wz) σ 3 Ψ 0 ζ)ψ 0,z ζ)wz) σ 3 W z W σ 3 +Wz) σ 3 Ψ 0 ζ)e z)r z)r zz)ez)ψ 0 ζ)wz) σ 3, where ζ = f t z) is defined in 4.7). Noting that ζt) = ζ z=t =. By using the behavior.) of Ψ 0 ζ) at ζ = /4, and collecting 4.9), 5.4) and 5.6)-5.8) together, 4 eventually we have the,) entry of Y Y z at z = t: Y Y z ) t) = Ψ 0 ζt))e t)e z t)ψ 0ζt)) ) + Ψ ) s l W z W +O z=t n s +O, n) 0 ζ)ψ 0,z ζ)) z=t 5.9) where s = 4nlnϕt), l = min,α + β + )) and the error terms are uniform for t,d]. Using 5.8), we further obtain ) E t)e zt) I t) I t) =, 5.0) I 3 t) I t) where I t) = a t) D t,t)+d t,t) ) ζ z at) t), I t) = i a t) D t,t) D t,t) ) + d ) at) dz z=t lndz,t), a t) I 3 t) = i D t,t) D t,t) ) d ) at) dz z=t lndz,t) ; see 5.9) and 5.0) for the definition of D. It is readily seen that and dlnat) dt dζ dz = z=t = t ), 5.) t lnϕt). 5.) Then, from 5.0), 5.3), 5.) and 5.), we get the estimates as t +, namely, I t) = α+β)+e 0 ) ) +Ot ), 5.3) 4 3 I t) = i 3 α+ α+β)+e ) 0 +O t ), 5.4) t I 3 t) = O t ). 5.5) 9

20 From.), we have ) Ψ 0 ζt))e t)e z t)ψ 0ζt)) = E 0 I t) I t) I 3 t) I t) )E 0 ), 5.6) where ζt) = 4, and E 0 is defined in.3). Thus by.3) and 5.6), we obtain Ψ 0 ζt))e t)e z t)ψ 0ζt)) ) =su β + α iσ s)+su) ) I 3 t), α where the derivative is taken with respect to s, s = 4nlnϕt). By.), we get Ψ 0 ζ)ψ 0,zζ)) = ζ ze ) + αζ z ζ. 4 Recalling Wz) = z ) β z t ) α hz); cf. 4.4), we have W z W = βz z +α z t + ) + d z +t dz lnhz). I t)+ iσ s) su) ) I t) α Expanding the left-hand side at z = t, we obtain ζ z ζ ) t) = z t 4 t lnϕt) t ). t Collecting these formulas together, and using.4), we get Ψ 0 ζ)ψ 0,z ζ)) ) W z W z=t = σs) us+β + α 4 α t lnϕt) βt t α 4t d dz lnhz) Substituting 5.7) and 5.8) into 5.9), we obtain Y Y z ) t) = σs) us+β + α 4 α ) α t lnϕt) 4 +β t t α 4t α 4. z=t t t d dz lnhz) z=t + su β + I t)+ iσ s) su) ) I t) α ) α iσ s)+su) ) s l s I 3 t)+o +O. α n n) Applying a similar argument to the case z = t, we obtain Y Y z ) t) = σs) us+β + α 4 α ) α + t lnϕt) 4 +β t t + α 4t 5.7) 5.8) 5.9) d dz lnhz) z= t + su β + Ĩ t)+ iσ s) su) ) Ĩ t) α α ) iσ s)+su) ) s l s Ĩ 3 t)+o +O α n n) 0

21 5.30) with Ĩ t) = α+β)+e 0 ) ) +Ot ), 5.3) 4 3 Ĩ t) = i 3 α+ α+β)+e ) 0 +O t ), 5.3) t Ĩ 3 t) = O t ). 5.33) Here use has been made of the fact that hx) is an even function. Substituting 5.9) and 5.30) into 3.7) yields d dt lnd nt) = [ α+β)+e 0 ) 3 [ α+β)+e0 ]su β + ) σs) us+β 4 + α t lnϕt) α ] σ s) su) ) α +4αβ)t t 3 t ) α t α d dz lnhz) z=t d ) dz lnhz) z= t ) s l s +O +O +O n n) t ), 5.34) where ) = d. Integrating both sides of this identity from +ε to some t > gives ds lnd n t) =lnd n +ε) 4nlnϕt) ) t α +αβ 4nlnϕ+ε) +ε σs) us+β 4 + α s ds t t dt α lnt+ α ln+ε) α Vt) α V t)+ α V+ε)+ α V ε)+r nt)+o), 5.35) holding uniformly for arbitrary ε > 0, where Vz) = lnhz), and the remainder term R n t) = α+β)+e 0 ) ) t 3 t +ε su β + +ε )dt α+β)+e0 t 3 α )σ s) su) )dt. Since s = 4nlnϕt), then for small t, we may approximate the integral t t s t dt = ds+ot ). 5.36) 0 s From.7)and.8), namelytheboundaryconditionsforuandσ, wegettheestimates

22 for the integrals t t t su β + ) dt = Ot ), σ su) dt = o), t σ su) dt = o), which give the estimate for R n t) as ε 0 + R n t) = O t ) = o). 5.37) Letting ε 0 +, substituting 5.36) and 5.37) into 5.35) and making use of the fact that hx) is an even function, we obtain lnd n t) = lnd n ) αvt)+αv) α lnt s 0 σs) us+α+β) 4ds+o), 5.38) s where s = 4nlnϕt), ϕt) = t+ t. The convergence of the integral is guaranteed by the initial condition of σs) and us) in.8). The asymptotic approximation for lnd n ) has been given in [9, Thm..0] as lnd n ) = n +nα+β)+ ) ln+ α+β) ) ln n n+α+β + ) G lnπ +ln ) 5.39) Gα+β +) +n+α+β)v 0 α+β)v)+ k= kv k, where V k = π e kiθ lnhcosθ))dθ, k = 0,,. And the Barnes G-function is π 0 defined by the product G+z) = π) z e z+z +γe ) k= + z ) ) ke z k z, 5.40) k where γ E is the Euler constant. The Barnes G-function satisfies the well-known recurrence relation Gz +) = Gz)Γz), where G) =, and Γz) is the gamma function. Substituting 5.39) into 5.38) yields.8). Thus completing the proof of Theorem.

23 5. Proof of Theorem From.), 4.8) and 4.9), we obtain Yt) = nσ 3 I +O/n))Et)E 0 {lt)} σ 3, 5.4) where lt) = lim z t f t z) 4 )α Wz), E 0 and Et) are defined in.3) and 5.4), respectively. Thus, from 5.4) and the differential identity 3.5), we have d dt h n = πiα n )D t ) C E 0 ) E 0 ) t ) +C C E0 ) E 0 ) +E 0 ) E 0 ) ) +C E 0 ) E 0 ) t ) ), where C i,j stands for the i,j) entry of the matrix C = I +O )) n 5.4) M C k t ) k ; 5.43) k=0 see 5.4) for the constant matrices M, C 0 and C. In view of.3),.5) and Proposition, we may write and E 0 ) E 0 ) = i α σ su), E 0 ) E 0 ) = i α σ +su), 5.44) E 0 ) E 0 ) = α su β + ) α, E 0 ) E 0 ) = su β + ) α +α. 5.45) Now, substituting 5.43), 5.44) and 5.45) into 5.4), we obtain dh n dt = πd t ) n { σ su) t +O ) ) +Ot ) n +σ +su) O t ) 5.46) su β + ) ) +d +O +O )} t ), n where d is defined in 5.) and the error term O ) is uniform for t,d]. n From.7) and.8), we can derive the estimates of the integrals t su β + ) t ) t dt = O n ) +O t ) ) and t σ +su) ) t t dt = o n ) +O t ) ) 3

24 Now, in view of 5.7),.7) and.8), and integrating by parts once, we have t D t ) σ su) dt σ ) t +O )) =D t ) t ) t s u 3 α s ) t D 8n t ) σ su)ds+o 0 n ) D ) ) 4n 4 α+β) +O t ) ) Then, integrating both sides of 5.46) and using the estimates 5.47)-5.49), we obtain h n t) =h n ) πd ) 4α+β) n ) 4n where α Is) = 8D t ) +Is) n +O t n s 0 D t ) { +O By.7) and.8), we get the estimate πd t ) n { 4 σ s u ) t t ) 3 ) }, σ su)+ + d 4α 5.50) su β + ) } s ds. 5.5) Is) = Os )+Os). 5.5) To determined h n ), we use a result from [8, Thm..6], that is, h n ) = π ) ) nd ) + 4α+β) +c n, c n = O. 5.53) 4n n Substituting 5.5) and 5.53) into 5.50), and noting that the relation between D ) and D t ) 5.7), we have h n t) = π { [ nd t ) 4 σ ) s u +α ] t +O ) t ) 3 s ) +O +c n n t+ t ) } 5.54) α, where c n is independent of t and c n = O n ). It follows from Proposition that σ su = 4σ JM s 6 ) s 6 α+β) ) Since h n = γn, we obtain the asymptotic approximation of the leading coefficient as in.). Thus completing the proof of Theorem. 4

25 5.3 Proof of Theorem 3 To approximate the recurrence coefficients, we recall their relation with the leading coefficients b n = γ n. γn 5.56) Let Λ = α s+σ us, 5.57) then Λ is analytic for s 0, ) and Λ s) = 4 α+β) +Os l ) as s ) with l = min,+α+β)), and Λ s) = 4 β + k= l k s k as s +, 5.59) where l k are constant coefficients. Then it follows from 5.57), 5.58) and 5.59) that Λ s s/n) = Λ s) s ) s n Λ s)+o, 5.60) n where the error term is uniform for s 0. From 5.60), and using the fact that s 4 n t as t, we have Λ s s ) n s s = Λ ) s 4 Λ s ) t +O. 5.6) s n n Using the expression of Is) in 5.5), and in view of.7) and.8), we get Is s) n n ) = Is) ) t +O. 5.6) n n Then, a combination of 5.50), 5.53), 5.6) and 5.6) gives the following asymptotic formula for h n h n t) = 4πD { t ) 4 Λ ) s) Λ t +3 t )+ Is) n s s n +c n t+ ) ) α t ) )} t +O +O t ) 3 t +O. n n 5.63) From 5.50), 5.63) and 5.56), we obtain the asymptotic approximation of the recurrence coefficients stated in.). 5

26 Acknowledgements The authors are grateful to the referees for their valuable suggestions and comments. The work of Shuai-Xia Xu was supported in part by the National Natural Science Foundation of China under grant number 0493, GuangDong Natural Science Foundation under grant number S , and the Fundamental Research Funds for the Central Universities under grant number 3lgpy4. Yu-Qiu Zhao was supported in part by the National Natural Science Foundation of China under grant number 087. References [] E. Basor, Y. Chen and N. Haq, Asymptotics of determinants of Hankel matrices via non-linear difference equations, arxiv:40.073v. [] L. Brightmore, F. Mezzadri and M.Y. Mo, A matrix model with a singular weight and Painlevé III, Comm. Math. Phys., 04) doi:0.007/s z. [3] Y. Chen and L. Zhang, Painlev VI and the Unitary Jacobi Ensembles, Stud. Appl. Math., 5 00), 9. [4] T. Claeys, A.B.J. Kuijlaars and M. Vanlessen, Multi-critical unitary random matrix ensembles and the general Painlevé II equation, Ann. of Math., 68008), [5] T. Claeys, A. Its and I. Krasovsky, Emergence of a singularity for Toeplitz determinants and Painlevé V, Duke Math. J., 60 0), [6] T. Claeys and I. Krasovsky, Toeplitz determinants with merging singularities. arxiv preprint arxiv: [7] D. Dai and A.B.J. Kuijlaars, Painleve IV Asymptotics for Orthogonal Polynomials with Respect to a Modified Laguerre Weight, Stud. Appl. Math., 009), [8] P. Deift, Orthogonal polynomials and random matrices: a Riemann-Hilbert approach, Courant Lecture Notes 3, New York University, 999. [9] P. Deift, A. Its and I. Krasovsky, Asymptotics of Toeplitz, Hankel, and Toeplitz+Hankel determinants with Fisher-Hartwig singularities, Ann. of Math., 74 0), [0] P. Deift, A. Its and I. Krasovsky, Toeplitz matrices Toeplitz determinants under the impetus of the Ising model. Some history and some recent results, Comm. Pure Appl. Math., 66 03), [] P. Deift, T. Kriecherbauer, K.T.-R. McLaughlin, S. Venakides and X. Zhou, Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights 6

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