Asymptotics for Hankel Determinants Associated to a Hermite Weight with a Varying Discontinuity

We study n×n Hankel determinants constructed with moments of a Hermite weight with a Fisher-Hartwig singularity on the real line. We consider the case when the singularity is in the bulk and is both of root-type and jump-type. We obtain large n asymptotics for these Hankel determinants, and we obser...

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Hauptverfasser:	Charlier, C., Deaño, A.
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spelling	nasplib_isofts_kiev_ua-123456789-2094462025-11-22T01:07:13Z Asymptotics for Hankel Determinants Associated to a Hermite Weight with a Varying Discontinuity Charlier, C. Deaño, A. We study n×n Hankel determinants constructed with moments of a Hermite weight with a Fisher-Hartwig singularity on the real line. We consider the case when the singularity is in the bulk and is both of root-type and jump-type. We obtain large n asymptotics for these Hankel determinants, and we observe a critical transition when the size of the jumps varies with n. These determinants arise in the thinning of the generalised Gaussian unitary ensembles and in the construction of special function solutions of the Painlevé IV equation. C. Charlier was supported by the European Research Council under the European Union’s Seventh Framework Programme (FP/2007/2013)/ ERC Grant Agreement n. 307074. A. Deaño acknowledges financial support from projects MTM2012-36732-C03-01 and MTM2015-65888-C4-2-P from the Spanish Ministry of Economy and Competitiveness. The authors are grateful to A.B.J. Kuijlaars for sharing a simplified proof for the first part of [11, Proposition A.1]. This inspired us to simplify the proof of Lemma 7.4. We also thank T. Claeys for a careful reading of the introduction and for useful remarks. The authors acknowledge the referees for their careful reading and useful remarks. 2018 Article Asymptotics for Hankel Determinants Associated to a Hermite Weight with a Varying Discontinuity / C. Charlier, A. Deaño // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 41 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 30E15; 35Q15; 15B52; 33E17 arXiv: 1708.02519 https://nasplib.isofts.kiev.ua/handle/123456789/209446 https://doi.org/10.3842/SIGMA.2018.018 en Symmetry, Integrability and Geometry: Methods and Applications application/pdf Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	We study n×n Hankel determinants constructed with moments of a Hermite weight with a Fisher-Hartwig singularity on the real line. We consider the case when the singularity is in the bulk and is both of root-type and jump-type. We obtain large n asymptotics for these Hankel determinants, and we observe a critical transition when the size of the jumps varies with n. These determinants arise in the thinning of the generalised Gaussian unitary ensembles and in the construction of special function solutions of the Painlevé IV equation.
format	Article
author	Charlier, C. Deaño, A.
spellingShingle	Charlier, C. Deaño, A. Asymptotics for Hankel Determinants Associated to a Hermite Weight with a Varying Discontinuity Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Charlier, C. Deaño, A.
author_sort	Charlier, C.
title	Asymptotics for Hankel Determinants Associated to a Hermite Weight with a Varying Discontinuity
title_short	Asymptotics for Hankel Determinants Associated to a Hermite Weight with a Varying Discontinuity
title_full	Asymptotics for Hankel Determinants Associated to a Hermite Weight with a Varying Discontinuity
title_fullStr	Asymptotics for Hankel Determinants Associated to a Hermite Weight with a Varying Discontinuity
title_full_unstemmed	Asymptotics for Hankel Determinants Associated to a Hermite Weight with a Varying Discontinuity
title_sort	asymptotics for hankel determinants associated to a hermite weight with a varying discontinuity
publisher	Інститут математики НАН України
publishDate	2018
url	https://nasplib.isofts.kiev.ua/handle/123456789/209446
citation_txt	Asymptotics for Hankel Determinants Associated to a Hermite Weight with a Varying Discontinuity / C. Charlier, A. Deaño // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 41 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT charlierc asymptoticsforhankeldeterminantsassociatedtoahermiteweightwithavaryingdiscontinuity AT deanoa asymptoticsforhankeldeterminantsassociatedtoahermiteweightwithavaryingdiscontinuity
first_indexed	2025-11-30T21:24:59Z
last_indexed	2025-11-30T21:24:59Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 14 (2018), 018, 43 pages Asymptotics for Hankel Determinants Associated to a Hermite Weight with a Varying Discontinuity Christophe CHARLIER † and Alfredo DEAÑO ‡ † Department of Mathematics, KTH Royal Institute of Technology, Lindstedtsvägen 25, SE-114 28 Stockholm, Sweden E-mail: cchar@kth.se ‡ School of Mathematics, Statistics and Actuarial Science, University of Kent, Canterbury CT2 7FS, UK E-mail: A.Deano-Cabrera@kent.ac.uk Received November 02, 2017, in final form February 27, 2018; Published online March 07, 2018 https://doi.org/10.3842/SIGMA.2018.018 Abstract. We study n × n Hankel determinants constructed with moments of a Hermite weight with a Fisher–Hartwig singularity on the real line. We consider the case when the singularity is in the bulk and is both of root-type and jump-type. We obtain large n asymptotics for these Hankel determinants, and we observe a critical transition when the size of the jumps varies with n. These determinants arise in the thinning of the generalised Gaussian unitary ensembles and in the construction of special function solutions of the Painlevé IV equation. Key words: asymptotic analysis; Riemann–Hilbert problems; Hankel determinants; random matrix theory; Painlevé equations 2010 Mathematics Subject Classification: 30E15; 35Q15; 15B52; 33E17 1 Introduction and motivation We consider the Hankel determinant Hn(v, s, α) = det (∫ R xj+kw(x; v, s, α)dx )n−1 j,k=0 , n ∈ N, (1.1) with a Gaussian weight on the real line of the form w(x; v, s, α) = e−x 2 \|x− v\|α { s, if x < v, 1, if x > v, (1.2) where v ∈ R, s ∈ [0, 1] and α ∈ (−1,∞). There is a root-type Fisher–Hartwig (FH) singularity if α 6= 0. The piecewise constant factor in (1.2) is a jump-type FH singularity only if s 6= 0 and s 6= 1. If s = 1 there is no jump, and if s = 0 the weight is supported on the interval [v,∞). By Heine’s formula, Hn(v, s, α) admits the following n-fold integral representation: Hn(v, s, α) = 1 n! ∫ Rn ∆(x)2 n∏ i=1 w(xi; v, s, α)dxi, ∆(x) = ∏ 1≤i<j≤n (xj − xi). (1.3) In this paper we are interested in large n asymptotics for Hn(v, s, α), uniformly in s ∈ [0, 1]. An analogous case was analysed in [6] for Fredholm determinant associated with the sine kernel This paper is a contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applica- tions (OPSFA14). The full collection is available at https://www.emis.de/journals/SIGMA/OPSFA2017.html mailto:cchar@kth.se mailto:A.Deano-Cabrera@kent.ac.uk https://doi.org/10.3842/SIGMA.2018.018 https://www.emis.de/journals/SIGMA/OPSFA2017.html 2 C. Charlier and A. Deaño and in [10, 11] for Toeplitz determinants with a weight defined on the unit circle. We briefly summarize here some known results for particular values of the parameters and we present some applications. The Hankel determinant Hn(v, s, α) arises in random matrix theory. Consider the set of n×n Hermitian matrices M endowed with the probability measure 1 Ẑn(v, α) \|det(M)− vI\|αe−TrM2 dM, dM = n∏ i=1 dMii ∏ 1≤i<j≤n d<Mijd=Mij , where Ẑn(v, α) is the normalisation constant. We will refer to this random matrix ensemble as the generalised Gaussian unitary ensemble, which we denote by GUE(v, α). Such a measure of matrices M induces a probability measure on the eigenvalues x1, . . . , xn of M which is of the form 1 n!Zn(v, α) ∆(x)2 n∏ i=1 w(xi; v, 1, α)dxi, (1.4) where Zn(v, α) is called the partition function of GUE(v, α). From (1.4) and the integral repre- sentation for Hankel determinants given by (1.3), we have the relation Zn(v, α) = Hn(v, 1, α). The special case of GUE(0, 0) (note that if α = 0, the parameter v is irrelevant) is called the Gaussian unitary ensemble (GUE), and has already been widely studied (see, e.g., [32]). The partition function of the GUE is a Selberg integral and is explicitly known (see, e.g., [32]). Its exact expression and its large n asymptotics are given by logHn(0, 1, 0) = −n 2 2 log 2 + n 2 log(2π) + n−1∑ j=1 log(j!) (1.5) = n2 2 log (n 2 ) − 3 4 n2 + n log(2π)− log n 12 + ζ ′(−1) +O ( n−1 ) , as n→∞, where ζ is Riemann’s zeta-function. When α 6= 0 but v = 0, the partition function of GUE(0, α) is again a Selberg integral and is also known exactly for finite n, see [33] or [15, equation (A.8)]. This is not true for v 6= 0. In [29], Krasovsky obtained via the Riemann–Hilbert method that large n asymptotics of Hn( √ 2nt, 1, α), when t is in a compact subset of (−1, 1), are given by log Hn (√ 2nt, 1, α ) Hn(0, 1, 0) = α 2 n log n− α 2 ( 1− 2t2 + log 2 ) n+ α2 4 log n+ α2 4 log ( 2 √ 1− t2 ) + log G ( 1 + α 2 )2 G(1 + α) +O ( log n n ) , (1.6) where G is Barnes’ G-function. Let us denote by x (v,α) min and x (v,α) max the smallest and largest eigenvalue in GUE(v, α), respec- tively. The probability of observing no eigenvalues in (−∞, v), denoted by P ( x (v,α) min ≥ v ) , can be expressed as a ratio of Hankel determinants given by (1.1) with s = 0 and s = 1. From a direct integration of (1.4), and from the symmetry w(x; v, 1, α) = w(−x;−v, 1, α), we have P ( x (v,α) min ≥ v ) = Hn(v, 0, α) Hn(v, 1, α) = P ( x(−v,α) max ≤ −v ) . (1.7) Asymptotics for Hankel Determinants Associated to a Hermite Weight 3 It is well-known that the empirical spectral distribution of the eigenvalues (after proper rescaling) in the GUE converges weakly almost surely to the Wigner semi-circle distribution, i.e., 1 n n∑ i=1 δ xi√ 2n → 2 π √ 1− x2dx, see for instance [1, Chapter 2]. It is also known that the smallest eigenvalue x (0,0) min is usually located near − √ 2n and the properly rescaled fluctuations of x (0,0) min around − √ 2n follow the Tracy–Widom distribution. The ratio (1.7) with v = − √ 2n ( 1 + w 2n2/3 ) and w in a compact subset of R, i.e., when v is near the edge of the spectrum, has been recently studied for α 6= 0 in [41] (including general s > 0). In this paper we investigate the probability of a large deviation of x (v,α) min , i.e., the proba- bility (1.7) when v is sufficiently far from − √ 2n. In the particular case of v = 0, (1.7) is the probability that a matrix M drawn from the GUE(0, α) is positive definite. Large n asymptotics for this probability have been obtained in [15]: log Hn(0, 0, α) Hn(0, 1, α) = − log 3 2 n2 − α log 3 2 n+ ( α2 4 − 1 12 ) log n+ c0 +O ( n−1 ) , (1.8) where c0 = α 2 log(2π) + ( α2 4 − 1 6 ) log 2 + ( 1 8 − α2 2 ) log 3 + ζ ′(−1)− log [ G ( 1 + α 2 )2 ] . Note that the denominator Hn(0, 1, α) can be obtained from (1.5) and (1.6) with t = 0. One of the goals of the present paper is to generalize this result for v 6= 0, i.e., to obtain strong large n asymptotics of Hn (√ 2nt, 0, α ) , when t is in a compact subset of (−1,∞). In particular, it is possible to deduce from our result the probability (1.7). Assume we thin the eigenvalues x1, . . . , xn from (1.4) by removing each of them independently with a certain probability s ∈ [0, 1]. The resulting point process is called the thinned GUE(v, α), whose spectrum is denoted by y1, . . . , ym, where m is itself a random variable following the Binomial distribution Bin(n, 1 − s). Thinning was introduced in random matrix theory by Bohigas and Pato [5], and we refer to [9, 11] for analogous situations and an overview of the theory of thinning. Let us denote by y (v,s,α) min for the smallest thinned eigenvalue (note that we always have y (v,s,α) min ≥ x (v,α) min = y (v,0,α) min ). The ratio Hn(v,s,α) Hn(v,1,α) is a one-parameter generalisation of (1.7) and represents the probability of observing no eigenvalue of the thinned spectrum in (−∞, v), i.e., we have P ( y (v,s,α) min ≥ v ) = Hn(v, s, α) Hn(v, 1, α) . (1.9) In the regime when s is in a compact subset of (0, 1] and v = √ 2nt with t in a compact subset of (−1, 1), asymptotics of this probability have been obtained rigorously in [27] for α = 0 and in [9] for α 6= 0. Note that asymptotics for integer α were obtained previously in [25], based on the work [7]. As n→∞, they are given by log Hn (√ 2nt, s, α ) Hn (√ 2nt, 1, α ) = n ∫ t −1 2 π √ 1− x2dx log s+ (log s)2 4π2 log n+ c̃0 +O ( log n n ) = [ 2 arcsin t+ 2t √ 1− t2 + π ] log s 2π n+ (log s)2 4π2 log n+ c̃0 +O ( log n n ) , (1.10) 4 C. Charlier and A. Deaño where the constant c̃0 = c̃0(t, s, α) is explicit: c̃0 = 3(log s)2 4π2 log ( 2 √ 1− t2 ) + α log s 2π arcsin t+ log G ( 1 + α 2 + log s 2πi ) G ( 1 + α 2 − log s 2πi ) G ( 1 + α 2 )2 . The second goal of the present paper is to obtain large n asymptotics of Hn (√ 2nt, s, α ) when s = s(n)→ 0 as n→∞, and to observe a transition in the large n asymptotics between (1.10), where s is bounded away from 0, and the case Hn (√ 2nt, 0, α ) . Another motivation for the study of the Hankel determinant (1.1) comes from solutions of the Painlevé IV differential equation (PIV). The PIV equation finds interesting applications in many different areas of physics, such as non-linear optics, dispersive long-wave equations, fluid dynamics and plasma physics (see, e.g., [13, Section 10.1], [40]). This equation depends on two parameters A,B ∈ C and is given by q′′(z) = 1 2q(z) q′(z)2 + 3 2 q(z)3 + 4zq(z)2 + 2 ( z2 −A ) q(z) + B q(z) . In this paper, we focus on special function solutions of PIV when the parameters A and B satisfy suitable constraints. It is known (see, e.g., [14, Theorem 3.4] and [26, Theorem 25.2]) that PIV has solutions expressible in terms of parabolic cylinder functions U(a, z) (these functions are defined in [36, Chapter 12]) if and only if either B = −2(2n+ 1 + εA)2 or B = −2n2, (1.11) with ε = ±1 and n ∈ Z. In this situation, we are considering the so-called special function solutions of PIV. In general, the standard method to derive special function solutions is by considering asso- ciated Riccati equations. We refer the reader to [13, Section 7] for the general theory or [36, Section 32.10] for a summary of special function solutions. For n = 0 in (1.11), the Riccati equation of PIV is q′(z) = εq(z)2 + 2εzq(z) + 2ν, with ν = −(1 + εA). If we set q(z) = −εϕ′ν(z)/ϕν(z) to linearise this equation, then ϕν satisfies ϕ′′ν(z)− 2εzϕ′ν(z) + 2ενϕν(z) = 0, (1.12) whose general solutions can be written in terms of the parabolic cylinder function U(a, z), see [36, Section 12.2]. If ν /∈ Z, then ϕν(z) = ϕν(z; ε) can be written in the form ϕν(z; ε) = {{ C1U ( −ν − 1 2 , √ 2z ) + C2U ( −ν − 1 2 ,− √ 2z )} exp ( 1 2z 2 ) , if ε = 1,{ C1U ( ν + 1 2 , √ 2z ) + C2U ( ν + 1 2 ,− √ 2z )} exp ( −1 2z 2 ) , if ε = −1, (1.13) where C1, C2 ∈ C. We comment on the case ν ∈ Z at the end of this section. For general n ∈ N, it is a remarkable fact that the special function solutions can be constructed explicitly in terms of the following Wronskian determinants (see [23, 35] and [14, Theorem 3.5]): τn,ν(z; ε) =W ( ϕν(z; ε), dϕν(z; ε) dz , . . . , dn−1ϕν(z; ε) dzn−1 ) , n ≥ 1, with τ0,ν(z; ε) = 1. This Wronskian determinant can in turn be written in terms of the Hankel determinant Hn(v, s, α), we obtain for n ≥ 0 Hn(v, s, α) = Γ(1 + α)n2−n 2−n(α−1) 2 τn,α(v,−1) Asymptotics for Hankel Determinants Associated to a Hermite Weight 5 = e−nv 2 Γ(1 + α)n2−n 2−n(α−1) 2 τn,−α−1(v, 1), (1.14) where C1 = 1 and C2 = s. This relation relies on the integral representation for the parabolic cylinder function [36, equation (12.5.1)]: U(a, z) = e− z2 4 Γ(a+ 1 2) ∫ ∞ 0 xa− 1 2 e− x2 2 −zxdx, < a > −1 2 . (1.15) By a change of variables, we can write the moment of order 0 of the weight function w(x) = w(x; v, s, α) given by (1.2) in terms of U as follows:∫ ∞ −∞ w(x)dx = 2− 1+α 2 e−v 2 [ s ∫ ∞ 0 xαe− x2 2 + √ 2vxdx+ ∫ ∞ 0 xαe− x2 2 − √ 2vxdx ] = 2− 1+α 2 e− v2 2 Γ(1 + α) [ sU ( α+ 1 2 ,− √ 2v ) + U ( α+ 1 2 , √ 2v )] . Therefore, from (1.13) with C1 = 1 and C2 = s, we have∫ ∞ −∞ w(x)dx = 2− 1+α 2 Γ(1 + α)ϕα(v;−1) = 2− 1+α 2 e−v 2 Γ(α+ 1)ϕ−α−1(v; 1). (1.16) Differentiating j times (1.16) with respect to v, we get ∂jvϕα(v;−1) = 2 1+α 2 Γ(1 + α) ∂jv (∫ ∞ −∞ w(x)dx ) = (−1)j2j2 1+α 2 Γ(1 + α) ∫ ∞ −∞ xjw(x)dx, ∂jvϕ−α−1(v; 1) = 2 1+α 2 Γ(1 + α) ∂jv ( ev 2 ∫ ∞ −∞ w(x)dx ) = (−1)j2j2 1+α 2 Γ(1 + α) ev 2 ∫ ∞ −∞ (x− v)jw(x)dx. After taking the determinant, this establishes the formula (1.14). Thus, the results presented in Section 2 imply large n asymptotics for special solutions of PIV expressed in terms of τn,ν , uniformly for C2 small. As explained in [14], when the parameter ν = m is an integer, we need to take a different combination of independent solutions of equation (1.12), since both parabolic cylinder functions in the seed function become Hermite polynomials. This leads to the so-called generalised Hermite polynomials, which are rational solutions of PIV, obtained as particular cases of the special function solutions, see the parameter plane of PIV in [13, Section 5.5] or [26]. We refer the reader to the work of Kawijara and Ohta [28, Definition 3.1] for more information on the rational solutions of PIV. We also note that in the previous discussion, since ν = α, the integral representation (1.15) imposes the restriction α > −1. This restriction can be lifted at the price of taking complex integration and a complex weight function. This defines an associated family of orthogonal polynomials only formally. We do not pursue this route in this paper, but we note that it has been used in the literature, for example in the analysis of the asymptotic behavior and pole structure of rational solutions of PII by Bertola and Bothner [3] and more recently of rational solutions of PIV by Buckingham [8]. 2 Main results Observe that if we naively take the limit s → 0 in (1.10), the asymptotics on the right-hand side blow up, due to the presence of log s terms. Therefore, a critical transition is expected when n→∞ and simultaneously s→ 0 in a suitable double scaling limit. As mentioned earlier, the contribution of this paper is to analyse 1) the case s = 0 (which is known only for v = 0, see (1.8)), and 2) the transition between the situation when s = 0 and the situation when s is in a compact subset of (0, 1], given by (1.10). We obtain the following results. 6 C. Charlier and A. Deaño Theorem 2.1. Let α ∈ (−1,∞) and t ∈ (−1,∞). As n→∞, we have log Hn (√ 2nt, 0, α ) Hn(0, 0, α) = C1(t)n2 + C2(t, α)n+ C3(t, α) +O ( n−1 ) , (2.1) where the coefficients are given by C1(t) = −2t3 27 (√ 3 + t2 − t ) − ( 4 3 t2 + 5 9 t √ 3 + t2 ) − log ( t+ √ 3 + t2√ 3 ) , (2.2) C2(t, α) = αt 3 ( t− √ 3 + t2 ) − α log ( t+ √ 3 + t2√ 3 ) , (2.3) C3(t, α) = 1− 3α2 6 log ( t+ √ 3 + t2√ 3 ) − 1 48 log ( 3 + t2 3 ) − 1 16 ( 1− 4α2 ) log ( 3 + 5t2 + 4t √ 3 + t2 3 ) . (2.4) Furthermore, the error term O ( n−1 ) is uniform for t in a compact subset of (−1,∞). Note that there is no critical transition in (2.1) as t→ 1. Remark 2.2. The probability (1.7) with v = √ 2nt can be rewritten as P ( x ( √ 2nt,α) min ≥ √ 2nt ) = Hn( √ 2nt, 0, α) Hn(0, 0, α) Hn(0, 0, α) Hn(0, 1, α) Hn(0, 1, α) Hn( √ 2nt, 1, α) . Large n asymptotics of these three ratios are given up to the constant term by Theorem 2.1 (for t > −1), (1.8) and (1.6) (for t ∈ (−1, 1)), respectively. Putting these asymptotics together, we obtain as n→∞ and for t ∈ (−1, 1), logP ( x ( √ 2nt,α) min ≥ √ 2nt ) = ( C1(t)− log 3 2 ) n2 + ( C2(t, α)− α log 3 2 − αt2 ) n + ( α2 4 − 1 12 ) log n+ C3(t, α) + c0 − α2 8 log ( 1− t2 ) +O ( log n n ) . (2.5) It can be checked from (2.2) that C1(−1) = log 3 2 and C ′1(t) = − 8 27 (√ 3 + t2 − t )( 3 + 5t2 + 4t √ 3 + t2 ) < 0, for t > −1, which shows that the leading term in (2.5) is negative. This implies that the above probability decays super exponentially fast as n→∞ for t ∈ (−1, 1). To observe a transition in the large n asymptotics of Hn (√ 2nt, s, α ) when s = 0 and when s is in a compact subset of (0, 1], we couple the parameter s with n in the form s = e−λn, λ ≥ 0. Large n asymptotics of Hn (√ 2nt, e−λn, α ) will depend on whether λ is greater or smaller than a critical value λc(t), which is explicit and given by λc(t) = 2t√ 3 √ 3 + t2 + 2t √ 3 + t2 + 2 log ( 2 + t2 + t √ 3 + t2 + √ 3 + t2 + t√ 3 √ 3 + t2 + 2t √ 3 + t2 ) . (2.6) Asymptotics for Hankel Determinants Associated to a Hermite Weight 7 Theorem 2.3. Let s = e−λn with λ ∈ [0,∞), we have the following asymptotic results. (1) If t ∈ (−1, 1) and λ ≥ λc(t), then log Hn (√ 2nt, e−λn, α ) Hn (√ 2nt, 0, α ) = O ( n−1/2e−n(λ−λc(t))), as n→∞, (2.7) and large n asymptotics for logHn (√ 2nt, 0, α ) are given by Theorem 2.1. Furthermore, the O term in (2.7) is uniform for t in a compact subset of (−1, 1) and for λ ≥ λc(t). (2) If t ∈ (−1, 1) and 0 ≤ λ ≤ λc(t) are fixed, then lim n→∞ 1 n2 log Hn (√ 2nt, e−λn, α ) Hn (√ 2nt, 1, α ) = − ∫ λ 0 Ω ( t, λ̃ ) dλ̃, (2.8) where Ω(t, λ) = ∫ b a ρ(x; t, λ)dx, ρ(x; t, λ) = 2 π √ c− x √ x− b x− t √ x− a, and a < b < t < c, with a, b and c depending on λ and t, are uniquely determined by the following equations: t = a+ b+ c, 2 = a2 + b2 + c2 − t2, λ = 4 ∫ t b √ c− x√ t− x √ x− b √ x− adx. Remark 2.4. In Theorem 2.3, we restrict ourselves to the case t ∈ (−1, 1). With increasing effort, this result can be extended for t ∈ (−1,∞). If t ≥ 1, a new region appears in the (t, λ) plane which deserves a separate analysis (which we expect to be straightforward but long). Therefore, we decided not to proceed in this direction. Remark 2.5. Note that the denominators on the left hand sides of (2.7) and (2.8) are different. We can use Theorem 2.3 to obtain information about the large deviation of the smallest thinned GUE (√ 2nt, α ) eigenvalue as follows. By (1.9) and (1.7) with v = √ 2nt and s = e−λn, we have P ( y ( √ 2nt,e−λn,α) min ≥ √ 2nt ) = Hn (√ 2nt, e−λn, α ) Hn (√ 2nt, 0, α ) P ( x ( √ 2nt,α) min ≥ √ 2nt ) . Therefore, for t ∈ (−1, 1) and λ ≥ λc(t), by (2.7), as n→∞ we have logP ( y ( √ 2nt,e−λn,α) min ≥ √ 2nt ) = logP ( x ( √ 2nt,α) min ≥ √ 2nt ) +O ( n−1/2e−n(λ−λc(t))), (2.9) and large n asymptotics of logP ( x ( √ 2nt,α) min ≥ √ 2nt ) are given by (2.5). In the regime t ∈ (−1, 1) and 0 ≤ λ ≤ λc(t), (2.8) implies logP ( y ( √ 2nt,e−λn,α) min ≥ √ 2nt ) = ( − ∫ λ 0 Ω(t, λ̃)dλ̃ ) n2 + o ( n2 ) . (2.10) Since (2.9) and (2.10) are both valid for λ = λc(t), by equalling the leading term, we have − ∫ λc(t) 0 Ω(t, λ)dλ = C1(t)− log 3 2 . (2.11) We will give an independent and more direct proof of this formula at the end of Section 7. 8 C. Charlier and A. Deaño Remark 2.6. Note that the limit (2.8) is independent of α. The subleading terms in the large n asymptotics of Hn( √ 2nt,e−λn,α) Hn( √ 2nt,1,α) are expected to depend on α and to be oscillatory and described in terms of elliptic θ-functions. These functions appear in our analysis (see, e.g., (6.12)). This heuristic is also supported by the analogy of our situation with [6], where the authors obtained θ-functions in the subleading terms. Outline The orthogonal polynomials (OPs) with respect to the weight (1.2) play a central role in our analysis. In Section 3, we obtain identities for ∂vHn(v, 0, α) and for ∂sHn(v, s, α) in terms of these OPs. The Riemann–Hilbert (RH) problem which characterizes these OPs is presented in Section 4. We obtain large n asymptotics for the OPs via a Deift–Zhou steepest descent method on this RH problem. The first steps of the steepest descent method are the same regardless of the value of the parameter s ∈ [0, 1), and are also presented in Section 4. For the last steps, the analysis will then depend on the speed of convergence of s to 0. By writing s = e−λn, λ ∈ (0,∞], we distinguish two different regimes in λ which are separated by the critical value λc(t) > 0. We study the situation λ ≥ λc(t) in Section 5 (the case s = 0 corresponds to the special case of λ = +∞), and the situation 0 < λ < λc(t) in Section 6. We integrate the differential identities and prove Theorem 2.1 and Theorem 2.3 in Section 7. 3 Orthogonal polynomials and differential identities 3.1 Orthogonal polynomials We consider the family of orthonormal polynomials pj of degree j with respect to w defined in (1.2), characterized by the orthogonality conditions∫ R pj(x)pk(x)w(x)dx = δjk, j, k = 0, 1, 2, . . . , (3.1) and κj > 0 is the leading coefficient of pj , that is πj(x) = κ−1 j pj(x) is the monic orthogonal polynomial of degree j which satisfies∫ R πj(x)πk(x)w(x)dx = hjδjk, j, k = 0, 1, 2, . . . , (3.2) where hj is the squared norm of πj . From (3.1), we have hj = κ−2 j . It is well-known (see, e.g., [38]) that these OPs satisfy the recurrence relation xπj(x) = πj+1(x) + βjπj(x) + γ2 j πj−1(x), j ≥ 0, (3.3) with π−1(x) := 0. Note that if we write πj(x) = xj + σjx j−1 + · · · , then from (3.3) we get the relation σj − σj+1 = βj , j ≥ 0, where σ0 := 0. (3.4) 3.2 Differential identity in v for s = 0 From the determinantal representation for OPs (see, e.g., [38]) and (1.3), the Hankel determinant Hn(v, 0, α) can be written in terms of the norms of the OPs, one has Hn(v, 0, α) = n−1∏ j=0 hj . (3.5) Asymptotics for Hankel Determinants Associated to a Hermite Weight 9 If we differentiate with respect to v the relation (3.2) with k = j, we obtain ∂vhj = ∂v (∫ ∞ v π2 j (x)w(x)dx ) = ∂v (∫ ∞ 0 π2 j (x+ v)w(x+ v)dx ) . Since w(x+ v) = xαe−(x+v)2 , we get ∂vhj = 2 ∫ ∞ 0 πj(x+ v)∂v(πj(x+ v))w(x+ v)dx− 2 ∫ ∞ 0 (x+ v)π2 j (x+ v)w(x+ v)dx = −2 ∫ ∞ v xπ2 j (x)w(x)dx, where we have used the orthogonality (3.2) and the fact that ∂v(πj(x + v)) is a polynomial of degree at most j − 1. From the recurrence relation (3.3), we obtain ∂vhj = −2βjhj . As a consequence of this, by taking the log in (3.5) and differentiating it with respect to v, we have ∂v logHn(v, 0, α) = n−1∑ j=0 ∂v log hj = −2 n−1∑ j=0 βj , This can be simplified by using (3.4), and gives ∂v logHn(v, 0, α) = 2σn(v), (3.6) where σn is the subleading coefficient of the polynomial πn(x), defined after (3.3), and we have explicitly written the dependence of σn on v. 3.3 Differential identity in s Suppose that the thinned eigenvalues y1, . . . , ym are observed and that ]{yi : yi < v} = 0. From Bayes’ formula, using (1.4) and (1.9), the distribution of the whole spectrum x1, . . . , xn conditionally on this event is given by 1 n!Hn(v, s, α) ∆(x)2 n∏ i=1 w(xi; v, s, α)dxi. Such point processes are called conditional, and were first considered in [11] on the unit circle and then on the real line in [9]. This point process is determinantal [16], and its correlation kernel is given by Kn(x, y) =  √ w(x)w(y) κn−1 κn pn−1(y)pn(x)− pn−1(x)pn(y) x− y , if x 6= y, w(x) κn−1 κn ( p′n(x)pn−1(x)− pn(x)p′n−1(x) ) , if x = y, (3.7) where the OPs pj are orthonormal with respect to w(x; v, s, α) and are defined in (3.1). The expected number of points on (−∞, v) in this point process is denoted by En(v, s, α). It is also known [16] that En(v, s, α) can be expressed in terms of the one-point correlation function Kn(x, x), we have En(v, s, α) = ∫ v −∞ Kn(x, x)dx. (3.8) 10 C. Charlier and A. Deaño The quantity En(v, s, α) can also be expressed in terms of the logarithmic derivative of Hn(v, s, α) with respect to s. Consider the following partition of Rn: Ak = { (x1, . . . , xn) ∈ Rn : ]{xi : xi < v} = k } , n⊔ k=0 Ak = Rn. By definition of En(v, s, α) we have En(v, s, α) = n∑ k=0 k n!Hn(v, s, α) ∫ Ak ∆(x)2 n∏ i=1 w(xi; v, s, α)dxi = n∑ k=0 ksk n!Hn(v, s, α) ∫ Ak ∆(x)2 n∏ i=1 \|xi − v\|αe−x 2 i dxi. Note that the n-fold integral (1.3) can be rewritten as Hn(v, s, α) = n∑ k=0 sk n! ∫ Ak ∆(x)2 n∏ i=1 \|xi − v\|αe−x 2 i dxi, and thus we have En(v, s, α) = s∂s logHn(v, s, α). Putting this together with (3.8), we obtain the differential identity s∂s logHn(v, s, α) = ∫ v −∞ Kn(x, x)dx. (3.9) We will also use later the well-known (see, e.g., [16]) formula for reproducing kernels∫ ∞ −∞ Kn(x, x)dx = n. (3.10) 4 A Riemann–Hilbert problem and renormalization of the problem We will perform the Deift–Zhou [20, 21] steepest descent method on a Riemann–Hilbert problem to get the large n asymptotics for pn. Consider the matrix valued function Y , defined by Y (z) =  κ−1 n pn(z) κ−1 n 2πi ∫ R pn(x)w(x) x− z dx −2πiκn−1pn−1(z) −κn−1 ∫ R pn−1(x)w(x) x− z dx  . (4.1) It is well-known [22] that Y is the unique solution of the following RH problem. RH problem for Y (a) Y : C \ R→ C2×2 is analytic. (b) The limits of Y (x ± iε) as ε > 0 approaches 0 exist, are continuous on R \ {v} and are denoted by Y+ and Y− respectively. Furthermore they are related by Y+(x) = Y−(x) ( 1 w(x) 0 1 ) , for x ∈ R \ {v}. (4.2) (c) As z →∞, we have Y (z) = ( I + Y1z −1 +O ( z−2 )) znσ3 , where σ3 = ( 1 0 0 −1 ) . Asymptotics for Hankel Determinants Associated to a Hermite Weight 11 (d) As z tends to v, the behaviour of Y is Y (z) = ( O(1) O(log(z − v)) O(1) O(log(z − v)) ) , if α = 0, Y (z) = ( O(1) O(1) +O((z − v)α) O(1) O(1) +O((z − v)α) ) , if α 6= 0. If s = 0, from condition (b) Y has no jump along (−∞, v) and thus Y is analytic in C \ [v,∞). Note also that Y11(z) = κ−1 n pn(z) = πn(z), and thus Y1,11 = σn(v), (4.3) where Y1,11 denotes the (1, 1) entry of the matrix Y1. 4.1 Normalization of the RH problem We define t = v√ 2n , and we normalize the RH problem for Y with the following transformation U(z) = (2n)−(α 4 +n 2 )σ3Y (√ 2nz ) (2n) α 4 σ3 . (4.4) The matrix U satisfies the following RH problem. RH problem for U (a) U : C \ R→ C2×2 is analytic. (b) U has the following jumps: U+(x) = U−(x) ( 1 w̃(x) 0 1 ) , for x ∈ R \ {t}, where w̃(x) = (2n)− α 2w( √ 2nx) = \|x− t\|αe−2nx2 { s, if x < t, 1, if x > t. (c) As z →∞, we have U(z) = ( I + U1z −1 +O ( z−2 )) znσ3 . (d) As z tends to t, the behaviour of U is U(z) = ( O(1) O(log(z − t)) O(1) O(log(z − t)) ) , if α = 0, U(z) = ( O(1) O(1) +O((z − t)α) O(1) O(1) +O((z − t)α) ) , if α 6= 0. The following lemma translates the differential identities (3.6) and (3.9) in terms of U . Lemma 4.1. We have the following differential identities ∂t logHn (√ 2nt, 0, α ) = 4nU1,11, (4.5) s∂s logHn (√ 2nt, s, α ) = ∫ t −∞ w̃(x) 2πi [ U−1(x)U ′(x) ] 21 dx. (4.6) 12 C. Charlier and A. Deaño Proof. The differential identity (4.5) is obtained by substituting (4.4) and (4.3) into (3.6). Similarly, using (3.7) and (4.1), the differential identity (3.9) can be rewritten as s∂s logHn (√ 2nt, s, α ) = ∫ √2nt −∞ w(x) 2πi [ Y −1(x)Y ′(x) ] 21 dx, which gives (4.6) after using (4.4) and a change of variables. � Remark 4.2. Note that [ Y −1(z)Y ′(z) ] 21 only involves the first column of Y , which is entire (see (4.1) or equivalently (4.2)). Thus [ Y −1 + (x)Y ′+(x) ] 21 = [ Y −1 − (x)Y ′−(x) ] 21 for x ∈ R, and we simply denote it by [ Y −1(x)Y ′(x) ] 21 without ambiguity. The same remark holds for U . 4.2 Equilibrium measure We introduce a new parameter λ ∈ [0,+∞], defined through s = e−λn, which characterizes the speed of convergence of s to 0 as n → ∞. An essential tool in the RH analysis is the so-called equilibrium measure. In our case, the equilibrium measure µV is the unique minimizer of the functional∫∫ R2 log \|x− y\|−1dµ(x)dµ(y) + ∫ R V (x)dµ(x), among all Borel probability measures µ on R, where the potential V is defined by V (x) = { 2x2 + λ, if x < t, 2x2, if x ≥ t, and where the parameter t has been defined above (4.4). The equilibrium measure is absolutely continuous with respect to the Lebesgue measure and its density will be denoted by ρ(x). The equilibrium measure and its support, denoted by S, are completely determined by the following Euler–Lagrange variational conditions [37]: 2 ∫ S log \|x− y\|ρ(y)dy = V (x)− `, for x ∈ S, (4.7) 2 ∫ S log \|x− y\|ρ(y)dy ≤ V (x)− `, for x ∈ R \ S, (4.8) where ` is a constant. Proposition 4.3 below shows that the equilibrium measure depends crucially on whether λ ≥ λc(t) or 0 < λ < λc(t). If λ = 0, the potential is simply V (x) = 2x2 and the equilibrium measure is the semicircle law supported on (−1, 1) (see, e.g., [37]). For convenience we also include it in Proposition 4.3 (see case (3)), but without giving a proof of it. Proposition 4.3. (1) If t ∈ (−1,∞) and λ ≥ λc(t), the density of the equilibrium measure ρ(x) = ρ(x; t) is independent of λ and is given by ρ(x) = 2 π (x− b) √ c− x√ x− t , (4.9) supported on S = [t, c], with b = b(t) = t− √ 3 + t2 3 , c = c(t) = t+ 2 √ 3 + t2 3 . Asymptotics for Hankel Determinants Associated to a Hermite Weight 13 The constant ` = `(t) in the variational conditions (4.7) and (4.8) is given by ` = 1 + 2 3 t (√ 3 + t2 + 2t ) + 2 log ( 2 ( t+ √ 3 + t2 )) . (4.10) Furthermore, the variational inequality (4.8) is strict for all x ∈ R \ S if λ > λc, and if λ = λc then (4.8) is strict for all x ∈ R \ (S ∪ {b}) and (4.8) is an equality at x = b. (2) If t ∈ (−1, 1) and 0 < λ < λc(t), the density of the equilibrium measure ρ(x) = ρ(x; t, λ) is given by ρ(x) = 2 π √ c− x √ x− b x− t √ x− a, (4.11) supported on two disjoint intervals S = [a, b] ∪ [t, c], a < b < t < c, and a, b, c, depending on λ and t, are uniquely determined by the following equations: t = a+ b+ c, (4.12) 2 = a2 + b2 + c2 − t2, (4.13) λ = 4 ∫ t b √ c− x√ t− x √ x− b √ x− adx. (4.14) Furthermore, for a fixed t, the function λ 7→ b = b(a(λ), c(λ), λ) given by the system (4.12)–(4.14) is strictly decreasing from λ ∈ (0, λc(t)) to b ∈ (b, t). The constant ` = `(t, λ) is given by ` = −2 ∫ S log \|x− t\|ρ(x)dx+ 2t2 = −2 log \|c\|+ 2c2 + ∫ ∞ c ( 4x− 2 x − 4 √ x− c√ x− t √ x− b √ x− a ) dx, (4.15) and (4.8) is strict. (3) If t ∈ (−1, 1) and λ = 0, then we have the semi-circle law ρ(x) = 2 π √ 1− x2, S = [−1, 1], ` = 1 + log 4, (4.16) and (4.8) is strict for x ∈ R \ S. Proof. We will start by proving case (2). From (4.12) and (4.13), we can express a and c in terms of b and t as follows a = − b 2 + t 2 − √ 4− 3b2 + 2tb+ t2 2 , c = − b 2 + t 2 + √ 4− 3b2 + 2tb+ t2 2 . (4.17) For t ∈ (−1, 1) fixed and b ∈ (b, t), a direct check from (4.17) shows that a < b < t < c, ∂bc < 0 and −1 < ∂ba. This implies from (4.14) that ∂bλ = 4 ∫ t b √ c− x√ t− x √ x− b √ x− a ( ∂bc 2(c− x) − 1 2(x− b) − ∂ba 2(x− a) ) dx < 0. 14 C. Charlier and A. Deaño If b↗ t, equation (4.14) implies λ→ 0. On the other hand, if b↘ b, equations (4.12) and (4.13) imply a→ b and c→ c. Again from (4.14), we thus have λ→ 4 ∫ t b √ c− x√ t− x (x− b)dx = λc(t), as b→ b, (4.18) where λc(t) is given by (2.6). This proves that the function λ 7→ b(a(λ), c(λ), λ) is a decreasing bijection from λ ∈ (0, λc(t)) to b ∈ (b, t). In particular, given t ∈ (−1, 1) and 0 < λ < λc(t), a, b and c are uniquely determined by the equations (4.12)–(4.14). Equations (4.12) and (4.13) imply also that ρ is a density. Indeed, with a contour deformation and a residue calculation at ∞, we obtain∫ S ρ(x)dx = a2 + b2 + c2 − 2ab− 2ac− 2bc+ 2(a+ b+ c)t− 3t2 4 = 1. Now, we define f(x) = 2 ∫ S log \|x− y\|ρ(y)dy − 2x2, where ρ is given by (4.11). Its derivative f ′(x) can be explicitly evaluated. Consider the function ρ̃(z) = 2 π √ z − c √ z−b√ z−t √ z − a, such that ρ̃±(x) = ±iρ(x) for x ∈ S. From (4.2), we have that f ′(x) = 1 2πi ∫ Σ −2πρ̃(w) x− w dw − 4x, where Σ consists of two circles surrounding S in the counter-clockwise direction, and if x /∈ S, then x does not lie in the interior region of Σ. Also, note that Res ( −2πρ̃(w) x− w ,w =∞ ) = −2(a+ b+ c− t− 2x) = 4x, where we have used (4.12). Therefore, f ′(x) = 0 for x ∈ S and from a contour deformation and a residue calculation, we obtain f ′(x) =  4 √ c− x√ t− x √ b− x √ a− x, x < a, 0, a ≤ x ≤ b, −4 √ c− x√ t− x √ x− b √ x− a, b < x < t, 0, t ≤ x ≤ c, −4 √ x− c√ x− t √ x− b √ x− a, c < x. (4.19) Note furthermore that from (4.7), we have λ = − ∫ t b f ′(x)dx = −(f(t)− f(b)). This implies that (4.7) and (4.8) are satisfied, with a strict inequality in (4.8). From (4.7), we have −` = f(x), for any x ∈ [t, c]. In particular we have ` = −f(t) = −2 ∫ S log \|x− t\|ρ(x)dx+ 2t2, (4.20) Asymptotics for Hankel Determinants Associated to a Hermite Weight 15 which is (4.15). To prove the other expression for ` in (4.15), we note that f(x) = 2 log \|x\| − 2x2 + f̃(x), (4.21) where f̃(x) = 2 ∫ S log ∣∣1− y x ∣∣ ρ(y)dy. Using the fact that lim x→∞ f̃(x) = 0 and the equations (4.19) and (4.21), we obtain 0 = f̃(c) + ∫ ∞ c f̃ ′(x)dx = −`− 2 log \|c\|+ 2c2 + ∫ ∞ c ( 4x− 2 x + f ′(x) ) dx, from which we find the second expression in (4.15). This finishes the proof of case (2). The case (1) is similar and simpler. In this case, we define a function f as in (4.2), where ρ(x) = 2 π (x− b) √ c−x√ x−t . The derivative of f is equal to f ′(x) =  −4 √ c− x√ t− x (x− b), x < t, 0, t ≤ x ≤ c, −4 √ x− c√ x− t (x− b), c < x. (4.22) The Euler–Lagrange constant can also be written as in (4.20), but in case (1) the integral can be explicitly evaluated with a primitive and gives (4.10). Since λ ≥ λc(t), from (4.22) and by the formula for λc(t) given by (4.18), (4.8) is strictly satisfied in case (1) except at x = b if λ = λc. This finishes the proof of Proposition 4.3. � Remark 4.4. Here we just comment briefly on what happens for t ≥ 1 in parts (2) and (3) of Proposition 4.3, even though we will only focus on t ∈ (−1, 1) in the present paper, as mentioned in Remark 2.4. If t > 1, a critical situation occurs if the endpoints t and c merge together. From equations (4.12) and (4.13), in this case we have a = −1 and b = 1. From (4.14), this corresponds to λ = 4 ∫ t 1 √ x2 − 1dx = 2t √ t2 − 1− 2 log ( t+ √ t2 − 1 ) =: ε(t). With increasing effort (by adapting the above proof), it is possible to show that part (2) of Proposition 4.3 remains valid for t ≥ 1, provided that λ ∈ (ε(t), λc(t)), and that part (3) remains valid for t ≥ 1, provided that λ ∈ [0, ε(t)]. Remark 4.5. We will prove part 2 of Theorem 2.3 (that is, (2.8)) by using the differential identity (4.6) (after the change of variables s = e−λn). We a priori need large n asymptotics of ∂λ logHn (√ 2nt, e−λn, α ) uniformly in λ ∈ [0, λc(t)]. As it can be seen from Proposition 4.3, in this case the support of the equilibrium measure consists of two intervals. The region where λc(t)− λ > 0 is small corresponds to the “birth of a cut”, which was studied in [4, 12, 34], and when λc(t)− λ > 0 becomes larger, θ-functions appear in the analysis. It is a technical task to obtain uniform asymptotics in these regions as λ approaches λc(t). Nevertheless, following [11], we will only need pointwise convergence in λ ∈ (0, λc(t)) and apply Lebesgue’s dominated convergence theorem. 4.3 First transformation: U 7→ T The first step of the steepest descent analysis consists of normalizing the RH problem at ∞, which can be done by using a so-called g-function. We define it by g(z) = ∫ S log(z − y)ρ(y)dy, (4.23) 16 C. Charlier and A. Deaño where the principal branch is chosen for the logarithm. The density ρ and its support S are defined in Proposition 4.3. The g-function is analytic in C \ (−∞, supS] and possesses the following properties g+(x) + g−(x) = 2 ∫ S log \|x− y\|ρ(y)dy, x ∈ R, (4.24) g+(x)− g−(x) = 2πi, x < inf S, (4.25) g+(x)− g−(x) = 2πi ∫ supS x ρ(y)dy, x ∈ [inf S, supS], (4.26) g+(x)− g−(x) = 0, supS < x, (4.27) Furthermore, by expanding the g-function as z →∞ in (4.23), we have eng(z) = zn ( 1− n z ∫ S xρ(x)dx+O ( z−2 )) , z →∞. (4.28) We apply a first transformation on U by T (z) = e n` 2 σ3U(z)e−ng(z)σ3e− n` 2 σ3 . T satisfies the following RH problem. RH problem for T (a) T : C \ R→ C2×2 is analytic. (b) T has the following jumps: T+(x) = T−(x)JT (x), for x ∈ R \ {t}, where JT (x) = ( e−n(g+(x)−g−(x)) \|x− t\|αen(g+(x)+g−(x)+`−V (x)) 0 en(g+(x)−g−(x)) ) , for x ∈ R \ {t}. (c) As z →∞, we have T (z) = I +O ( z−1 ) . (d) As z tends to t, the behaviour of T is T (z) = ( O(1) O(log(z − t)) O(1) O(log(z − t)) ) , if α = 0, T (z) = ( O(1) O(1) +O((z − t)α) O(1) O(1) +O((z − t)α) ) , if α 6= 0. From now on, we will separate the analysis into two parts, depending on whether λ ≥ λc(t) or 0 < λ < λc(t). 5 RH analysis for λ ≥ λc(t) In this section, t lies in a compact subset of (−1,∞) and λ lies in [λc(t),∞] as n → ∞. The parameter λ is not necessarily bounded, and the case λ = +∞ (i.e., s = 0) is also included in this analysis. First, we express the jumps for T in terms of ξ(z), which is defined by ξ(z) = −π ∫ z c ρ̃(w)dw, (5.1) Asymptotics for Hankel Determinants Associated to a Hermite Weight 17 where the path of integration lies in C \ (−∞, c) and ρ̃ is given by ρ̃(z) = 2 π (z − b) √ z − c√ z − t , (5.2) where in the above expression the principal branch is chosen for each square root. The function ξ is analytic in C \ (−∞, c], and since ρ̃±(x) = ±iρ(x) for x ∈ S = [t, c], we have 2ξ±(x) = ±(g+(x)− g−(x)) = 2g±(x) + `− 2x2, x ∈ S, (5.3) where we have used (4.7), (4.24) and (4.26). Thus the function ξ(z)− g(z) has no jump along S and can be analytically continued on C. This implies the following relation ξ(z) = g(z) + ` 2 − z2, for all z ∈ C \ (−∞, c). (5.4) The jump matrix JT can be rewritten in terms of ξ as follows: JT (x) =  ( 1 \|x− t\|αen(ξ+(x)+ξ−(x)−λ) 0 1 ) , if x < t,( e−2nξ+(x) \|x− t\|α 0 e2nξ+(x) ) , if t < x < c,( 1 \|x− t\|αe2nξ(x) 0 1 ) , if c < x. (5.5) As ξ(z) appears in the jumps for T (and in the subsequent transformations), it will be useful for us to make the following observations. From (4.8) together with Proposition 4.3, and from (4.24) and (5.4), we have that ξ(x) < 0, for x > c, (5.6) ξ+(x) + ξ−(x)− λ < 0, for x < t, (5.7) except if λ = λc(t) and x = b, in which case (5.7) becomes an equality. Also, if x ∈ (t, c), from the definition (5.1) we have ξ±(x) ∈ ±iR+ and by Cauchy-Riemann equations we have ∂x=ξ+(x) = −πρ(x) = −∂y<ξ(x+ iy) ∣∣ y=0 , ∂x=ξ−(x) = πρ(x) = ∂y<ξ(x− iy) ∣∣ y=0 . In particular, this implies that there exists an open neighbourhood W of (t, c) such that we have <ξ(z) > 0, for z ∈W \ (t, c). (5.8) 5.1 Second transformation: T 7→ S We will use the following factorization of JT (x) for x ∈ S( e−2nξ+(x) \|x− t\|α 0 e−2nξ−(x) ) = ( 1 0 \|x− t\|−αe−2nξ−(x) 1 ) × ( 0 \|x− t\|α −\|x− t\|−α 0 )( 1 0 \|x− t\|−αe−2nξ+(x) 1 ) . We open the lenses with γ+ and γ− around S as illustrated in Fig. 1, such that γ+ ∪ γ− ⊂ W 18 C. Charlier and A. Deaño S t c γ+ γ− Figure 1. Jump contour for S. and we define S(z) = T (z)  ( 1 0 −(z − t)−αe−2nξ(z) 1 ) , if z is inside the lenses, =z > 0,( 1 0 (z − t)−αe−2nξ(z) 1 ) , if z is inside the lenses, =z < 0, I, if z is outside the lenses, where the principal branch is taken for (z − t)−α. S satisfies the following RH problem. RH problem for S (a) S : C \ (R ∪ γ+ ∪ γ−)→ C2×2 is analytic, where γ+ and γ− are shown in Fig. 1. (b) S has the following jumps: S+(z) = S−(z) ( 1 \|z − t\|αen(ξ+(z)+ξ−(z)−λ) 0 1 ) , if z < t, S+(z) = S−(z) ( 1 \|z − t\|αe2nξ(z) 0 1 ) , if c < z, S+(z) = S−(z) ( 0 \|z − t\|α −\|z − t\|−α 0 ) , if t < z < c, S+(z) = S−(z) ( 1 0 (z − t)−αe−2nξ(z) 1 ) , if z ∈ γ+ ∪ γ−. (c) As z →∞, we have S(z) = I +O ( z−1 ) . (d) As z tends to t, we have S(z) =  ( O(1) O ( log(z − t) ) O(1) O ( log(z − t) )) , z outside the lenses,( O ( log(z − t) ) O ( log(z − t) ) O ( log(z − t) ) O ( log(z − t) )) , z inside the lenses, if α = 0, S(z) =  ( O(1) O(1) O(1) O(1) ) , z outside the lenses,( O ( (z − t)−α ) O(1) O ( (z − t)−α ) O(1) ) , z inside the lenses, if α > 0, S(z) = ( O(1) O ( (z − t)α ) O(1) O ( (z − t)α )) , if α < 0. (5.9) As z tends to c, we have S(z) = O(1). Asymptotics for Hankel Determinants Associated to a Hermite Weight 19 From (5.6), (5.7) and (5.8), we have that the jumps for S(z) on the boundary of the lenses γ+∪γ− tend to the identity matrix as n→∞, and that the (1, 2) entry of the jumps on R \ ([t, c]∪{b}) tends to 0 as n → ∞. This convergence is slower when z approaches t and c, and also when z approaches b if λ = λc(t). The jump for S on (t, c) is independent of n and different from the identity matrix. 5.2 Global parametrix Ignoring the exponentially small terms as n→∞ in the jumps of S and a small neighbourhood of b, t and c, we are left to consider the following RH problem, whose solution P (∞) is a good approximation of S away from a neighbourhood of b, t and c. RH problem for P (∞) (a) P (∞) : C \ [t, c]→ C2×2 is analytic. (b) P (∞) has the following jumps: P (∞) + (z) = P (∞) − (z) ( 0 \|z − t\|α −\|z − t\|−α 0 ) , if t < z < c. (5.10) (c) As z →∞, we have P (∞)(z) = I +O ( z−1 ) . (d) As z tends to t, we have P (∞)(z) = O ( (z − t)−1/4 ) (z − t)− α 2 σ3 . As z tends to c, we have P (∞)(z) = O ( (z − c)−1/4 ) . The construction of the solution of the above RH problem is now standard, and can be done similarly as in [2, 29, 30]. We define β(z) = 4 √ z−t z−c , analytic on C \ [t, c] and such that β(z) ∼ 1 as z →∞. It can be checked that the unique solution of the above RH problem is given by P (∞)(z) = 1 2 ( 4 c− t )−α 2 σ3 ( β(z) + β−1(z) i(β(z)− β−1(z)) −i(β(z)− β−1(z)) β(z) + β−1(z) ) × ϕ ( 2 c− t ( z − c+ t 2 ))α 2 σ3 (z − t)− α 2 σ3 , (5.11) where ϕ(z) = z + √ z2 − 1 is analytic in C \ [−1, 1] and such that ϕ(z) ∼ 2z as z →∞. We will later need the following expansion as z →∞: P (∞) 11 (z) = 1− α(c− t) 4z +O ( z−2 ) . (5.12) 5.3 Local parametrix near t Note that the assumption at the beginning of the section, i.e., that t lies in a compact subset of (−1,∞) and λ ≥ λc(t) as n → ∞, implies from Proposition 4.3 that there exists a constant δ > 0 independent of n such that δ < min(t− b, c− t). Inside a disk Dt around t, of radius fixed but smaller than δ/3, we want the local parametrix P to satisfy exactly the same jumps as S and to have the same behaviour as S near t. Furthermore, the local parametrix P should be close to the global parametrix on the boundary of the disk. 20 C. Charlier and A. Deaño RH problem for P (a) P : Dt \ (R ∪ γ+ ∪ γ−)→ C2×2 is analytic. (b) P has the following jumps: P+(z) = P−(z) ( 1 \|z − t\|αen(ξ+(z)+ξ−(z)−λ) 0 1 ) , if z ∈ (−∞, t) ∩Dt, (5.13) P+(z) = P−(z) ( 0 \|z − t\|α −\|z − t\|−α 0 ) , if z ∈ (t,∞) ∩Dt, P+(z) = P−(z) ( 1 0 (z − t)−αe−2nξ(z) 1 ) , if z ∈ (γ+ ∪ γ−) ∩Dt. (c) As n→∞, we have P (z) = ( I +O ( n−1 )) P (∞)(z) uniformly for z ∈ ∂Dt. (d) As z tends to t, we have S(z)P (z)−1 = O(1). The construction of a local parametrix associated with a FH singularity has been studied in [27] when the singularity is a pure jump, and then in [24] and [17] for the general case, and involves hypergeometric functions. On the other hand, the construction of a local parametrix associated to a pure root-type FH singularity involves Bessel functions [30]. In the present case, we are in a presence of a FH singularity of both root-type and jump-type, but the significant difference is that the parameter s (which parametrizes the jump) is exponentially small as n → ∞. The solution of the present local parametrix will be expressed in terms of Bessel functions, exactly as for a pure root-type singularity. Nevertheless, as the (1, 2) element of the jump matrix for P in (5.13) is not zero (if λ 6= +∞, i.e., s 6= 0), the construction of the solution of the above RH problem is not standard. It was done in [10] for the case α = 0. We generalize here the construction for a general α > −1. We will need a modified version of the Bessel model RH problem PBe, which is presented in Appendix B. We search for a matrix function P̂Be which satisfies the same jumps as PBe, see (B.1), and an extra jump on R+ given by P̂Be(z)+ = P̂Be(z)− ( 1 e−λn 0 1 ) , z ∈ (0,∞), where the orientation of (0,∞) is taken from 0 to ∞. Modified Bessel model RH problem We define F (z) = PBe(z)K(z)−1 ( 1 −h(z) 0 1 ) z− α 2 σ3 , (5.14) where K(z) =  I, \| arg z\| < 2π 3 ,( 1 0 −eπiα 1 ) , 2π 3 < arg z < π,( 1 0 e−πiα 1 ) , −π < arg z < −2π 3 , h(z) =  1 2i sin(πα) , if α /∈ N, (−1)α 2πi log z, if α ∈ N. Asymptotics for Hankel Determinants Associated to a Hermite Weight 21 From the jumps for PBe, given by (B.1), it can be checked that F has no jumps at all on C. Also, the behaviour of PBe(z) as z → 0, given by (B.3), implies that 0 is a removable singularity of F , and thus F is an entire function. We define P̂Be by P̂Be(z) = (I +A(z))PBe(z), where A(z) = −e−λnh(−z)(−z)αF (z) ( 0 1 0 0 ) F−1(z), (5.15) and if α /∈ Z, (−z)α is chosen with a branch cut on [0,∞) such that (−z)α > 0 for z < 0. Since F is entire, A is analytic on C \R+. Therefore, it can be checked that P̂Be is the solution of the following RH problem. RH problem for P̂Be (a) P̂Be : C \ (ΣB ∪ (0,∞)) → C2×2 is analytic, where the orientation of (0,∞) is from 0 towards ∞ and ΣB is the jump contour for PBe, shown in Fig. 7. (b) P̂Be satisfies the jump conditions P̂Be,+(z) = P̂Be,−(z) ( 0 1 −1 0 ) , z ∈ R−, P̂Be,+(z) = P̂Be,−(z) ( 1 0 eπiα 1 ) , z ∈ e 2πi 3 R+, P̂Be,+(z) = P̂Be,−(z) ( 1 0 e−πiα 1 ) , z ∈ e− 2πi 3 R+, P̂Be,+(z) = P̂Be,−(z) ( 1 e−λn 0 1 ) , z ∈ R+. (c) As z →∞, z /∈ ΣB ∪ (0,∞), we have P̂Be(z) = (I +A(z)) ( 2πz 1 2 )−σ3 2 N ( I +O ( z− 1 2 )) e2z 1 2 σ3 , where N = 1√ 2 ( 1 i i 1 ) . (d) As z tends to 0, the behaviour of P̂Be(z) is PBe(z) −1P̂Be(z) = O(log z), if α ∈ N, PBe(z) −1P̂Be(z) = O(1), if α /∈ N. Construction of the local parametrix We consider the function f(z) = −1 4 ξ̃(z)2, where ξ̃(z) = { ξ(z)− ξ+(t), if =z > 0, ξ(z)− ξ−(t), if =z < 0. (5.16) This a conformal map from Dt to a neighbourhood of 0, and as z → t, we have f(z) = k2 1(z − t)(1 +O(z − t)), where k1 = 2(t− b) √ c− t. (5.17) 22 C. Charlier and A. Deaño To construct the solution P of the above RH problem, it is important to note that A(−n2f(z)) remains small as n → ∞ uniformly for z ∈ ∂Dt. More precisely, from (5.14) and (B.2), as n→∞ we have F ( −n2f(z) ) = O(log n)O ( n\|α\| ) O ( PBe ( −n2f(z) )) = O ( e(d+ε)n ) , (5.18) where ε > 0 can be chosen arbitrary small but fixed, and d = max z∈∂Dt ∣∣2√−f(z) ∣∣. Similarly, we have the estimate F−1 ( −n2f(z) ) = O ( e(d+ε)n ) . We choose the radius of Dt fixed but sufficiently small such that d < λc(t) 3 ≤ λ 3 . This implies from (5.15) that A ( −n2f(z) ) = O ( e− λ 3 n ) as n→∞, uniformly for z ∈ ∂Dt. The local parametrix is given by P (z) = E(z)σ3P̂Be ( −n2f(z) ) σ3e −nξ(z)σ3e πiα 2 θ̃(z)σ3(z − t)− α 2 σ3 , (5.19) where θ̃(z) = { +1, if =z > 0, −1, if =z < 0, (5.20) and the function E(z) is defined for z ∈ Dt by E(z) = (−1)nP (∞)(z)(z − t) α 2 σ3e− πiα 2 θ̃(z)σ3N ( 2πn(−f(z))1/2 )σ3/2. (5.21) It can be checked directly from the jumps for P (∞) (5.10) that E has no jump at all in Dt. Furthermore, from the behaviour of P (∞)(z) near t and from (5.17), one has E(z) = O ( (z−t)− 1 2 ) as z → t. Thus, t is a removable singularity of E and E is analytic in the whole disk Dt. Since A(−n2f(z)) is exponentially small in n uniformly for z ∈ ∂Dt, it doesn’t contribute to the n−1 term in the condition (c) of the RH problem for P . Using the large ζ asymptotics for the Bessel model RH problem given by (B.2), we obtain as n→∞ that P (z)P∞(z)−1 = I + 1 n(−f(z))1/2 P (∞)(z)(z − t) α 2 σ3e− πiα 2 θ̃(z)σ3σ3B1σ3 × e πiα 2 θ̃(z)σ3(z − t)− α 2 σ3P (∞)(z)−1 +O ( n−2 ) , (5.22) uniformly for z ∈ ∂Dt, where B1 = 1 16 ( −(1+4α2) −2i −2i 1+4α2 ) . 5.4 Local parametrix near c Inside a disk Dc around c, of radius fixed but smaller than δ/3, we want the local parametrix P to satisfy the following RH problem. RH problem for P (a) P : Dc \ (R ∪ γ+ ∪ γ−)→ C2×2 is analytic. (b) P has the following jumps: P+(z) = P−(z) ( 0 \|z − t\|α −\|z − t\|−α 0 ) , if z ∈ (−∞, c) ∩Dc, P+(z) = P−(z) ( 1 \|z − t\|αe2nξ(z) 0 1 ) , if z ∈ (c,∞) ∩Dc, P+(z) = P−(z) ( 1 0 (z − t)−αe−2nξ(z) 1 ) , if z ∈ (γ+ ∪ γ−) ∩Dc. Asymptotics for Hankel Determinants Associated to a Hermite Weight 23 (c) As n→∞, we have P (z) = ( I +O ( n−1 )) P (∞)(z) uniformly for z ∈ ∂Dc. (d) As z tends to c, we have P (z) = O(1). The solution P of the above RH problem is standard [18] and can be constructed in terms of Airy functions and the associated Airy model RH problem, whose solution is denoted PAi and is presented in Appendix A. Let us first define the function f(z) = ( −3 2 ξ(z) )2/3 . This is a conformal map from Dc to a neighbourhood of 0, and as z → c we have f(z) = k 2/3 2 (z − c) [ 1 + 2 5 k3(z − c) +O ( (z − c)2 )] , with k2 = 2 c− b√ c− t , k3 = 1 c− b − 1 2(c− t) . It can be verified that P (z) = E(z)PAi ( n2/3f(z) ) e−nξ(z)σ3(z − t)− α 2 σ3 , satisfies the above RH problem, where E(z) = P (∞)(z)(z − t) α 2 σ3N−1f(z) σ3 4 n σ3 6 . Again, one can show that E has no jump at all inside Dc and has a removable singularity at c, and therefore E is analytic in the whole disk Dc. We will also need explicitly the first term in the large n expansion of P (z)P (∞)(z)−1 on ∂Dc. As n→∞, by (A.1) we have P (z)P (∞)(z)−1 = I + 1 nf(z)3/2 P (∞)(z)(z − t) α 2 σ3A1(z − t)− α 2 σ3P (∞)(z)−1 +O ( n−2 ) . (5.23) uniformly for z ∈ ∂Dc, and where A1 = 1 8 ( 1 6 i i − 1 6 ) , 5.5 Local parametrix near b The local parametrix P in a fixed disk Db around b can be constructed explicitly. The construc- tion is similar to the one done in [10] and it is valid for every λ ≥ λc but is only needed for λ close to λc. RH problem for P (a) P : Db \ R→ C2×2 is analytic. (b) P has the following jumps: P+(z) = P−(z) ( 1 \|z − t\|αen(ξ+(z)+ξ−(z)−λ) 0 1 ) , if z ∈ R ∩Db. (c) As n→∞, we have P (z) = (I + o(1))P (∞)(z) uniformly for z ∈ ∂Db. 24 C. Charlier and A. Deaño ΣR b t c Figure 2. Jump contours for the RH problem for R. The circles are oriented in clockwise direction. Note that for z ∈ Db, we have φ(z) = ξ+(z) + ξ−(z) = 2π ∫ t z ρ̃(w)dw, (5.24) and φ is analytic in Db. The unique solution of the above RH problem is given by P (z) = P (∞)(z) ( 1 l(z) 0 1, ) where l(z) = (z − t)αe−πiαθ̃(z)e−n(λ−λc) 2πi ∫ R∩Db en(φ(x)−λc) x− z dx. From (5.7) and (5.24), we have φ(b) = λc, φ ′(b) = 0 and φ′′(b) < 0. Thus, as n → ∞ we have l(z) = O(n−1/2e−n(λ−λc)) and P (z) = ( I +O ( n−1/2e−n(λ−λc)))P (∞)(z), uniformly for z ∈ ∂Db. 5.6 Small norm RH problem The final transformation of the RH analysis is given by R(z) = { S(z)P (∞)(z)−1, z ∈ C \ (Dt ∪Dc ∪Db), S(z)P (z)−1, z ∈ Dt ∪Dc ∪Db. Since P has exactly the same jumps as S inside Dt ∪Dc ∪Db, and the same behaviour near t and c, R has no jumps and is analytic inside these disks, except possibly at b, c and t. From the RH problem for S, and from the local parametrices around b and c, we have that S(z) and P (z) are bounded as z → b and as z → c. From (5.9), (5.19) and (B.3), as z → t from outside the lenses, we have R(z) =  ( O ( log(z − t) ) O ( log(z − t) ) O ( log(z − t) ) O ( log(z − t) )) , α = 0,( O(1) O(1) O(1) O(1) ) , α > 0,( O ( (z − t)α ) O ( (z − t)α ) O ( (z − t)α ) O ( (z − t)α )) , α < 0. (5.25) By (5.25), and since R(z) is bounded as z → b and as z → c, the three isolated singularities at b, c and t are removable. On the circles ∂Dt, ∂Dc and ∂Dc, we choose the clockwise orientation, as shown in Fig. 2. R satisfies the following RH problem: Asymptotics for Hankel Determinants Associated to a Hermite Weight 25 RH problem for R (a) R : C \ ΣR → C2×2 is analytic, where the contour ΣR is shown in Fig. 2. (b) The jumps JR(z) := R−1 − (z)R+(z) satisfy the following large n asymptotics for z ∈ ΣR: JR(z) = I +O ( e−cn ) , uniformly for z ∈ (γ+ ∪ γ− ∪ R) \ (S ∪Dt ∪Dc ∪Db), JR(z) = I +O ( n−1 ) , uniformly for z ∈ ∂Dt ∪ ∂Dc, JR(z) = I +O ( n−1/2e−n(λ−λc)), uniformly for z ∈ ∂Db, where c > 0 is a constant. (c) As z →∞, we have R(z) = I +O ( z−1 ) . From the standard theory for small-norm RH problems [18], R exists for all n sufficiently large and as n→∞ we have R(z) = I +O ( n−1 ) +O ( n−1/2e−n(λ−λc)), R′(z) = O ( n−1 ) +O ( n−1/2e−n(λ−λc)), (5.26) uniformly for z ∈ C \ ΣR, and uniformly for t in compact subsets of (−1,∞) and for λ ≥ λc(t). In the rest of this section, we consider the case when λ lies in a compact subset of (λc(t),∞] as n → ∞, i.e., when λ is bounded away from λc(t). In this case, the jumps for R on ∂Db are exponentially small. On the other hand, the jumps for R on ∂Dt ∪ ∂Dc have a series expansion of the form JR(z) = I + r∑ j=1 J (j) R (z)n−j +O ( n−r−1 ) , for any r ∈ N, where J (j) R (z) = O(1) as n → ∞ uniformly for z ∈ ∂Dt ∪ ∂Dc. Thus, R admits a series expansion of the form R(z) = I + r∑ j=1 R(j)(z)n−j +O ( n−r−1 ) , as n→∞, for any r ∈ N, where R(j)(z) = O(1) as n → ∞ uniformly for z ∈ C \ ΣR. By a perturbative analysis of the RH problem for R, the first term R(1)(z) is given by R(1)(z) = 1 2πi ∫ ∂Dt∪∂Dc J (1) R (w) w − z dw. We can evaluate this integral explicitly. When z is outside the disks, we have from (5.22) and (5.23) that( 4 c− t )α 2 σ3 R(1)(z) ( 4 c− t )−α 2 σ3 = √ c− t z − t 4α2 − 1 32k1 ( 1 i i −1 ) + 1 (z − c)2 5 √ c− t 96k2 ( −1 i i 1 ) (5.27) + 1 z − c 1 64 √ c− tk2 ( −8α2 + 2(c− t)k3 + 3 i 3 ( 24(α+ 2)α− 6(c− t)k3 + 19 ) i 3 ( 24(α− 2)α− 6(c− t)k3 + 19 ) 8α2 − 2(c− t)k3 − 3 ) . 26 C. Charlier and A. Deaño 6 RH analysis for 0 < λ < λc(t) In this section, we analyse the case when (t, λ) is in a compact subset of R = {(t, λ) : t ∈ (−1, 1) and 0 < λ < λc(t)} as n→∞. We define the function ρ̃(z) = 2 π √ z − c √ z − b √ z − a√ z − t , analytic in C \ ([a, b]∪ [t, c]) such that ρ̃(z) ∼ 2 πz as z → +∞. We will adopt the same approach as in Section 5, and we will rewrite the jumps for T in terms of the following two functions: ξ1(z) = −π ∫ z c ρ̃(w)dw, ξ2(z) = −π ∫ z b ρ̃(w)dw. (6.1) For ξ1 the path of integration is chosen to be in C \ (−∞, c), and for ξ2 the path lies in C \ ((−∞, b) ∪ [t,∞)). Therefore, ξ1 is analytic in C \ (−∞, c), satisfies ξ1(z) < 0 for z > c and ξ2 is analytic in C \ ((−∞, b) ∪ [t,∞)). Similarly to (5.3), note that ξ1(z)− g(z) satisfies ξ1,+(z)− g+(z) = ξ1,−(z)− g−(z) = ` 2 − z2, z ∈ (t, c). (6.2) Analytically continuing ξ1(z)− g(z) in (6.2) for z outside (−∞, t), we have ξ1(z) = g(z) + ` 2 − z2, z ∈ C \ (−∞, c]. (6.3) It will be useful later to notice the connection formula between g(z), ξ1(z) and ξ2(z) for z ∈ (b, t): g+(z) + g−(z) + `− 2z2 = ξ1,+(z) + ξ1,−(z) = 2ξ2(z) + 2π ∫ t b ρ̃(x)dx = 2ξ2(z) + λ, (6.4) where we have used (4.14), (4.2), (4.19) and (4.24). Furthermore, by (4.7) and (4.24), we have ξ2,+(z) + ξ2,−(z) = 0 = g+(z) + g−(z)− 2z2 + `− λ, z ∈ (a, b), and by (4.19), we also have (ξ2,+(z) + ξ2,−(z))′ = −2πρ̃(z) = (g+(z) + g−(z)− 2z2 + `− λ)′, z < a. Thus, we have the following identity between ξ2 and g on (−∞, a): ξ2,+(z) + ξ2,−(z) = g+(z) + g−(z)− 2z2 + `− λ < 0, z < a. (6.5) The mass of ρ on the interval (a, b) will play an important role later and will appear in the jumps of the subsequent RH problems, and we denote it as follows: Ω(t, λ) = ∫ b a ρ(x; t, λ)dx. (6.6) We will sometimes omit the dependence of Ω(t, λ) in t and λ and simply write Ω when there is no confusion. For z ∈ (a, b), by the relation (4.26), we have this identity 2ξ2,+(z) = 2πi ∫ b z ρ(x)dx = g+(z)− g−(z) + 2πiΩ− 2πi. Asymptotics for Hankel Determinants Associated to a Hermite Weight 27 Also, by Cauchy–Riemann equations, we can show similarly to equation (5.8) that there exists an open neighbourhood W1 of (t, c) and an open neighbourhood W2 of (a, b) such that <ξ1(z) > 0, for z ∈W1 \ (t, c), (6.7) <ξ2(z) > 0, for z ∈W2 \ (a, b). (6.8) The jumps for T can now be rewritten as JT (x) =  ( 1 \|x− t\|αen(ξ2,+(x)+ξ2,−(x)) 0 1 ) , if x < a,( e2πiΩne−2nξ2,+(x) \|x− t\|α 0 e−2πiΩne2nξ2,+(x) ) , if a < x < b,( e2πiΩn \|x− t\|αen(ξ1,+(x)+ξ1,−(x)−λ) 0 e−2πiΩn ) , if b < x < t,( e−2nξ1,+(x) \|x− t\|α 0 e2nξ1,+(x) ) , if t < x < c,( 1 \|x− t\|αe2nξ1(x) 0 1 ) , if c < x. For x ∈ (b, t), by (6.4), one can also rewrite JT (x) as JT (x) = ( e2πiΩn \|x− t\|αe2nξ2(x) 0 e−2πiΩn ) . 6.1 Second transformation: T 7→ S We proceed to the opening of the lenses with γ+, γ−, γ̃+, and γ̃− as shown in Fig. 3, such that γ+ ∪ γ− ⊂W1 and γ̃+ ∪ γ̃− ⊂W2. The next transformation S is defined by S(z) = T (z)  ( 1 0 −(z − t)−αe−2nξ1(z) 1 ) , if z ∈ I1,( 1 0 (z − t)−αe−2nξ1(z) 1 ) , if z ∈ I2,( 1 0 −(z − t)−αe−2nξ2(z)e2πiΩn 1 ) , if z ∈ Ĩ1,( 1 0 (z − t)−αe−2nξ2(z)e−2πiΩn 1 ) , if z ∈ Ĩ2, I, if z is outside the lenses, (6.9) where the sectors I1, I2, Ĩ1, and Ĩ2 are inside the lenses and are shown in Fig. 3. S satisfies the following RH problem. RH problem for S (a) S : C \ (R ∪ γ+ ∪ γ− ∪ γ̃+ ∪ γ̃−)→ C2×2 is analytic, see Fig. 3. 28 C. Charlier and A. Deaño t c γ+ γ− ba γ̃+ γ̃− I1 I2 Ĩ1 Ĩ2 Figure 3. Jump contour for S. (b) S has the following jumps: S+(z) = S−(z) ( 1 \|z − t\|αen(ξ2,+(z)+ξ2,−(z)) 0 1 ) , if z < a, S+(z) = S−(z) ( e2πiΩn \|z − t\|αen(ξ1,+(z)+ξ1,−(z)−λ) 0 e−2πiΩn ) , if b < z < t, S+(z) = S−(z) ( 1 \|z − t\|αe2nξ1(z) 0 1 ) , if c < z, S+(z) = S−(z) ( 0 \|z − t\|α −\|z − t\|−α 0 ) , if z ∈ S, S+(z) = S−(z) ( 1 0 (z − t)−αe−2nξ2(z)e2πiΩn 1 ) , if z ∈ γ̃+, S+(z) = S−(z) ( 1 0 (z − t)−αe−2nξ2(z)e−2πiΩn 1 ) , if z ∈ γ̃−, S+(z) = S−(z) ( 1 0 (z − t)−αe−2nξ1(z) 1 ) , if z ∈ γ+ ∪ γ−. (c) As z →∞, we have S(z) = I +O ( z−1 ) . (d) As z tends to t, we have S(z) =  ( O(1) O ( log(z − t) ) O(1) O ( log(z − t) )) , zoutside the lenses,( O ( log(z − t) ) O ( log(z − t) ) O ( log(z − t) ) O ( log(z − t) )) , z inside the lenses, if α = 0, S(z) =  ( O(1) O(1) O(1) O(1) ) , z outside the lenses,( O ( (z − t)−α ) O(1) O ( (z − t)−α ) O(1) ) , z inside the lenses, if α > 0, S(z) = ( O(1) O ( (z − t)α ) O(1) O ( (z − t)α )) , if α < 0. As z tends to a, b or c, we have S(z) = O(1). From (6.5), (6.7), (6.8) and the fact that ξ1(z) < 0 for z > c, the jumps for S on the boundary of the lenses are exponentially close to the identity matrix as n → ∞ and the (1, 2) entries of the jumps on R \ S are exponentially small as n→∞, but these convergences are not uniform Asymptotics for Hankel Determinants Associated to a Hermite Weight 29 t cba B A Figure 4. The cycles A and B. The solid line of A is in the first sheet and the dashed line is in the second sheet. The cycle B lies on the first sheet. for z in a neighbourhood of a, b, t and c. By ignoring the exponentially small terms in the jumps for S, we are left with a simpler RH problem, whose solution P (∞) will be a good approximation of S away from neighbourhoods of a, b, t and c. We construct P (∞) explicitly in Section 6.2. 6.2 Global parametrix If we ignore the exponentially small terms as n → ∞ in the jumps for S and a small neigh- bourhood of a, b, t and c, we are left with a simpler RH problem, whose solution P (∞) is called the global parametrix, and is a good approximation of S away from a neighbourhood of a, b, t and c. RH problem for P (∞) (a) P (∞) : C \ [a, c]→ C2×2 is analytic. (b) P (∞) has the following jumps: P (∞) + (z) = P (∞) − (z) ( 0 \|z − t\|α −\|z − t\|−α 0 ) , if z ∈ S, P (∞) + (z) = P (∞) − (z)e2πiΩnσ3 , if z ∈ (b, t). (c) As z →∞, we have P (∞)(z) = I +O ( z−1 ) . (d) As z tends to z̃ ∈ {a, b, c}, we have P (∞)(z) = O ( (z − z̃)−1/4 ) . As z tends to t, we have P (∞)(z) = O ( (z − t)−1/4 ) (z − t)− α 2 σ3 . The construction of P (∞) has been done in similar situations in [19] for α = 0 and in [31] for α 6= 0. It involves θ-functions and quantities related to a Riemann surface. Let X be the two sheeted Riemann surface of genus one associated to √ R(z), with R(z) = (z − c)(z − t)(z − b)(z − a), and we let √ R(z) ∼ z2 as z → ∞ on the first sheet. We also define cycles A and B such that they form a canonical homology basis of X. The upper part of the cycle A (the dashed line in Fig. 4) lies on the second sheet, and the lower part lies on the first sheet. The cycle B surrounds (a, b) in the clockwise direction, and lie in the first sheet. The unique A-normalized holomorphic one-form ω on X is given by ω = c0dz√ R(z) , c0 = (∫ A 1√ R(z) dz )−1 . 30 C. Charlier and A. Deaño By construction ∫ A ω = 1 and the lattice parameter is given by τ = ∫ B ω. A direct calculation shows that c0 = (∫ t b 2√ \|R(x)\| dx )−1 ∈ R+, τ = ∫ b a 2ic0√ \|R(x)\| dx ∈ iR+. The associated θ-function of the third kind θ(z) = θ(z; τ) is given by θ(z) = ∞∑ m=−∞ e2πimzeπim 2τ . It is an entire function which satisfies θ(z + 1) = θ(z), θ(z + τ) = e−2πize−πiτθ(z), for all z ∈ C. (6.10) We also need the function u(z) = ∫ z c ω, where the path of integration lies in C\ [a, c). Since ∫ C ω = 0 for any circle C winding around S, u is a single valued function for z ∈ C \ [a, c). For z on the first sheet, it satisfies u+(z) + u−(z) = 0, z ∈ (t, c), u+(z) + u−(z) = 1, z ∈ (a, b), u+(z)− u−(z) = τ, z ∈ (b, t), lim z→∞ u(z) = u∞ ∈ C. We define β(z) = 4 √ z−a z−b z−t z−c , such that β(z) ∼ 1 as z → ∞ on the first sheet. On this first sheet, it can be verified that the function β(z) + β−1(z) never vanishes, while β(z) − β−1(z) vanishes at a single point z?, given by z? = c− t c− t+ b− a b+ b− a c− t+ b− a t ∈ (b, t). This observation will be useful later, because the solution P (∞) will involve functions of the form 1/θ(u(z)± d), where d = −1 2 − τ 2 + ∫ z? c ω. The function 1/θ(u(z) + d) has no pole on the first sheet, while the function 1/θ(u(z)− d) has a pole at z?, see [19]. Therefore, the functions β(z)±β−1(z) θ(u(z)±d) are analytic on the first sheet. Before stating the solution P (∞), we follow [31] and introduce a scalar Szegő function D which satisfies (a) D : C \ [a, c]→ C is analytic. (b) D has the following jumps: D+(z)D−(z) = \|z − t\|α, if z ∈ S, D+(z) = D−(z)e2πiα̃, if z ∈ (b, t). (c) As z →∞, we have D(z) = D∞ +O ( z−1 ) , D∞ 6= 0. (d) As z → t, D(z) = (z − t) α 2 dt + o ( (z − t) α 2 ) , dt 6= 0. As z tends to z̃ ∈ {a, b, c}, D(z) is bounded. Asymptotics for Hankel Determinants Associated to a Hermite Weight 31 The new parameter α̃ is not arbitrary and is completely determined to ensure the existence of D. The solution is given by D(z) = exp (√ R(z) [ 1 2πi ∫ S α log \|x− t\|√ R(x)+ dx x− z + α̃ ∫ t b 1√ R(x) dx x− z ]) , (6.11) see [31] for a detailed proof of it, and α̃ is chosen such that D(z) remains bounded as z →∞: α̃ = − (∫ t b 1√ R(x) dx )−1 1 2πi ∫ S α log \|x− t\|√ R(x)+ dx. From (6.11), by expanding 1 x−z = −1 z ( 1 + x z + O ( z−2 )) as z → ∞, we can evaluate D∞. We obtain D∞ = exp ( − 1 2πi ∫ S αx log \|x− t\|√ R(x)+ dx− α̃ ∫ t b x√ R(x) dx ) . The solution P (∞) is given by (see [31, Section 4], and [19, Lemma 4.3]) P (∞)(z) = Dσ3 ∞ 2 ( β(z) + β−1(z)Θ11(z) i(β(z)− β−1(z))Θ12(z) −i(β(z)− β−1(z))Θ21(z) β(z) + β−1(z)Θ22(z) ) D(z)−σ3 , (6.12) where Θ11(z) = θ(u∞ + d)θ(u(z) + d− Ωn− α̃) θ(u∞ + d− Ωn− α̃)θ(u(z) + d) , Θ12(z) = θ(u∞ + d)θ(u(z)− d+ Ωn+ α̃) θ(u∞ + d− Ωn− α̃)θ(u(z)− d) , Θ21(z) = θ(u∞ + d)θ(u(z)− d− Ωn− α̃) θ(u∞ + d+ Ωn+ α̃)θ(u(z)− d) , Θ22(z) = θ(u∞ + d)θ(u(z) + d+ Ωn+ α̃) θ(u∞ + d+ Ωn+ α̃)θ(u(z) + d) . 6.3 Local parametrices By the assumption made at the beginning of Section 6, i.e., that (t, λ) is in a compact subset of R as n→∞, by Proposition 4.3, there exists δ > 0 independent of n such that δ < min{b− a, t− b, c− t}. We consider some disks Da, Db, Dt, Dc around a, b, t and c respectively, such that the radii are smaller than δ/3 but fixed. Inside these disks, we require the local parametrices P to have the same jumps as S, and the same behaviour near a, b, t and c. Furthermore, uniformly for z on the boundary of these disks, P satisfies the matching condition P (z) = ( I +O ( n−1 )) P (∞)(z), as n→∞. The solution P of the local parametrices around a, b and c is constructed in terms of the model Airy RH problem, and the local parametrix P around t in terms of the modified Bessel model RH problem. As these constructions do not present any additional technicalities other than the ones in Section 5, and as we will not use them explicitly later, we have decided not to include them in the present article. 6.4 Small norm RH problem The final transformation is given by R(z) = { S(z)P (∞)(z)−1, z ∈ C \D, S(z)P (z)−1, z ∈ D, where D = Da∪Db∪Dt∪Dc. Since P has exactly the same jumps as S inside D, and the same behaviour near a, b, t and c, we show in a similar way as done at the beginning of Section 5.6 that R is analytic inside these disks and it satisfies the following RH problem. 32 C. Charlier and A. Deaño ΣR t cba Figure 5. Jump contour for R. The orientation on the circles is clockwise. RH problem for R (a) R : C \ ΣR → C2×2 is analytic, with the contour ΣR shown in Fig. 5. (b) The jumps JR(z) := R−1 − (z)R+(z) satisfy the following large n asymptotics for z ∈ ΣR: JR(z) = I +O ( e−cn ) , uniformly for z ∈ (γ+ ∪ γ− ∪ γ̃+ ∪ γ̃− ∪ R) \ (S ∪D), JR(z) = I +O ( n−1 ) , uniformly for z ∈ ∂D, where c > 0 is a constant. (c) As z →∞, we have R(z) = I +O ( z−1 ) . Again, from standard theory for small-norm RH problems [18], R exists for sufficiently large n and we have R(z) = I +O ( n−1 ) , R′(z) = O ( n−1 ) , (6.13) uniformly for z ∈ C \ ΣR, and uniformly for (t, λ) in a compact subset of R. 7 Asymptotics for the Hankel determinant Hn(v, s, α) 7.1 Proof of Theorem 2.1 In this section, we use the RH analysis done in Section 5 with λ = +∞ (i.e., s = 0) and the differential identity ∂t logHn (√ 2nt, 0, α ) = 4nU1,11, (7.1) which was obtained in (4.5). Inverting the transformations R 7→ S 7→ T 7→ U , we obtain for z outside the disks and outside the lenses that U(z) = e− n` 2 σ3R(z)P (∞)(z)eng(z)σ3e n` 2 σ3 . In particular, the (1, 1) entry in the above expression is given by U11(z) = eng(z)P (∞) 11 (z) ( R11(z) +R12(z) P (∞) 21 (z) P (∞) 11 (z) ) . Thus, by equations (4.28) and (5.12), we have U1,11 = −n ∫ S xρ(x)dx− α(c− t) 4 + R (1) 1,11 n +O ( n−2 ) , as n→∞, Asymptotics for Hankel Determinants Associated to a Hermite Weight 33 where the O ( n−2 ) is uniform for t in compact subsets of (−1,∞), and R (1) 1,11 is the coefficient of the z−1 term in the large z expansion of R (1) 11 (z). From (5.27), it is given by R (1) 1,11 = √ c− t4α 2 − 1 32k1 + −8α2 + 2(c− t)k3 + 3 64 √ c− tk2 . On the other hand, by using (4.9) we can calculate explicitly the first moment of ρ, we have∫ c t xρ(x)dx = 2 27 (√ 3 + t2 − t )( 3 + 5t2 + 4t √ 3 + t2 ) . Thus, we can rewrite the differential identity (7.1) more explicitly: ∂t logHn (√ 2nt, 0, α ) = u1(t)n2 + u2(t, α)n+ u3(t, α) +O ( n−1 ) , (7.2) where u1(t) = − 8 27 (√ 3 + t2 − t )( 3 + 5t2 + 4t √ 3 + t2 ) , u2(t, α) = −2α 3 (√ 3 + t2 − t ) , u3(t, α) = (√ 3 + t2 − t )( t+ √ 3 + t2 ( 6α2 − 1 )) 12 ( 3 + t2 )( 2t+ √ 3 + t2 ) . Note that for t > −1 we have ∫ t 0 u1(x)dx = C1(t), ∫ t 0 u2(x, α)dx = C2(t, α) and ∫ t 0 u3(x, α)dx = C3(t, α), where C1(t), C2(t, α) and C3(t, α) have been defined in (2.2), (2.3) and (2.4) respec- tively. Since the O ( n−1 ) term in (7.2) is uniform for t in compact subsets of (−1,∞) and for λ ≥ λc(t), this gives the result. 7.2 Proof of Theorem 2.3(1) In this section, we again use the RH analysis done in Section 5. In order to use the differential identity (4.6), we need to obtain large n asymptotics for U uniformly on (−∞, t). This can be achieved by inverting the transformations R 7→ S 7→ T 7→ U in different regions. For z ∈ Db \R, we have U(z) = e− n` 2 σ3R(z)P (∞)(z) ( 1 l(z) 0 1 ) eng(z)σ3e n` 2 σ3 . (7.3) For z outside the lenses and outside the disks, z /∈ R, the expression for U in terms of R is U(z) = e− n` 2 σ3R(z)P (∞)(z)eng(z)σ3e n` 2 σ3 . (7.4) Thus, from (7.3) and (7.4), for all z such that z /∈ R and <z < t, z /∈ Dt, we obtain[ U−1(z)U ′(z) ] 21 =en`e2ng(z) [ P (∞)(z)−1R(z)−1R′(z)P (∞)(z)+P (∞)(z)−1P (∞)(z)′ ] 21 . (7.5) From (5.11), as n→∞ we have the following bounds for the global parametrix: P (∞)(z) = O(1), P (∞)(z)′ = O(1), uniformly for z outside the disks. Furthermore, by using the estimate for R given by (5.26), and by taking the limit z → x ∈ R in equation (7.5) (the limits from the upper and lower half plane are the same, see Remark 4.2), we obtain w̃(x) 2πi [ U−1(x)U ′(x) ] 21 = \|x− t\|αen(g+(x)+g−(x)+`−V (x))O(1), as n→∞, (7.6) where the O(1) is uniform for x ∈ (−∞, t), x /∈ Dt. It is more complicated to obtain similar asymptotics for w̃(x) 2πi [ U−1(x)U ′(x) ] 21 , uniformly for x ∈ (−∞, t)∩Dt. We will need the following lemma. 34 C. Charlier and A. Deaño Lemma 7.1. As n→∞, we have U+(x) ( 1 0 ) = e n` 2 eng+(x)e− n` 2 σ3 ( O ( n 1 2 +max(α,0) ) O ( n 1 2 +max(α,0) )) , (7.7) U ′+(x) ( 1 0 ) = e n` 2 eng+(x)e− n` 2 σ3 ( O ( n 5 2 +max(α,0) ) O ( n 5 2 +max(α,0) )) , (7.8) uniformly for x ∈ (−∞, t) ∩Dt. Proof. For z ∈ Dt, z outside the lenses, we have U(z) ( 1 0 ) = e n` 2 eng(z)e− n` 2 σ3R(z)P (z) ( 1 0 ) . (7.9) If furthermore, =z > 0, by (5.19) and (B.4) we have P (z) ( 1 0 ) = e πiα 2 (z − t)− α 2 e−nξ(z)E(z) ( Iα(2n √ −f(z)) −2πin √ −f(z)I ′α(2n √ −f(z)) ) . (7.10) Let x ∈ (−∞, t) ∩Dt. Note that ξ̃(x) > 0 (ξ̃ is defined in (5.16)) and thus √ −f(x)+ = 1 2 ξ̃(x). Inserting (7.10) into (7.9), we can take the limit z → x, this gives U+(x) ( 1 0 ) = (−1)ne n` 2 eng+(x)e−nξ̃(x)(t− x)− α 2 e− n` 2 σ3 ×R(x)E(x) ( Iα(nξ̃(x)) −πinξ̃(x)I ′α(nξ̃(x)) ) . (7.11) Since E is analytic in Dt, one has from (5.21) that as n→∞ E(z) = O(1)n σ3 2 , E′(z) = O(1)n σ3 2 , uniformly for z ∈ Dt. (7.12) To obtain a uniform bound from (7.11), we distinguish three cases. Let M > 0 be an arbitrary large but fixed constant and let m > 0 be an arbitrary small but fixed constant. Case (a): nξ̃(x) ≥ M as n → ∞. In this case we need large ζ asymptotics for Iα(ζ) and I ′α(ζ). From (B.2), we have Iα(ζ) = eζ√ 2πζ ( 1 +O ( ζ−1 )) , I ′α(ζ) = eζ√ 2πζ ( 1 +O ( ζ−1 )) , as ζ →∞. (7.13) If we insert (7.13) into (7.11), the result follows for Case (a) from (5.26), (7.12) and from the fact that (t− x)− α 2 = O(nmax(α,0)). Case (b): m ≤ nξ̃(x) ≤M as n→∞. In this case we have Iα(nξ̃(x))=O(1), nξ̃(x)I ′α(nξ̃(x)) = O(1), e−nξ̃(x) = O(1), (t− x)− α 2 = O(nα). Again from (5.26) and (7.12), we obtain U+(x) ( 1 0 ) = e n` 2 eng+(x)e− n` 2 σ3 ( O ( n 1 2 +α ) O ( n− 1 2 +α )) , (7.14) which is even slightly better than (7.7). Case (c): nξ̃(x) ≤ m as n→∞. From [36, formula (10.25.2)], we have Iα(ζ) = ( ζ 2 )α( 1 Γ(1 + α) +O ( ζ2 )) , Asymptotics for Hankel Determinants Associated to a Hermite Weight 35 I ′α(ζ) = ( ζ 2 )α−1( α Γ(1 + α) +O ( ζ2 )) , as ζ → 0. From the above expansion, we have for Case (c) that Iα(nξ̃(x)) (t− x) α 2 = O ( nα ) , nξ̃(x)I ′α(nξ̃(x)) (t− x) α 2 = O ( nα ) and e−nξ̃(x) = O(1). Thus, from (5.26) and (7.12) we obtain again (7.14), which finishes the proof of (7.7). We now turn to the proof of (7.8). From (7.11), we have U ′+(x) ( 1 0 ) = Ũ1(x) + Ũ2(x), where Ũ1(x) = n(g′+(x)− ξ̃′(x))U+(x) ( 1 0 ) + (−1)ne n` 2 eng+(x)e−nξ̃(x)(t− x)− α 2 e− n` 2 σ3 × (R′(x)E(x) +R(x)E′(x)) ( Iα(nξ̃(x)) −πinξ̃(x)I ′α(nξ̃(x)) ) , Ũ2(x) = (−1)ne n` 2 eng+(x)e−nξ̃(x)e− n` 2 σ3R(x)E(x)  ( Iα(nξ̃(x)) (t− x) α 2 )′ ( −πinξ̃(x)I ′α(nξ̃(x)) (t− x) α 2 )′  . The analysis of Ũ1(x) and Ũ2(x) can be done very similarly to the first part of the proof and we do not provide here all the details. From (5.4), one has g′+(x)− ξ̃′(x) = 2x and thus by (5.26), (7.7) and (7.12), Ũ1(x) = e n` 2 eng+(x)e− n` 2 σ3 ( O ( n 3 2 +max(α,0) ) O ( n 3 2 +max(α,0) )) . Again, by splitting the analysis into the same three cases as in the first part of the proof, we obtain the estimates( Iα(nξ̃(x)) (t− x) α 2 )′ = O ( n2 ( Iα(nξ̃(x)) (t− x) α 2 )) ,( −iπnξ̃(x)I ′α(nξ(x)) (t− x) α 2 )′ = O ( n2 ( −iπnξ̃(x)I ′α(nξ(x)) (t− x) α 2 )) , which yields Ũ2(x) = e n` 2 eng+(x)e− n` 2 σ3 ( O ( n 5 2 +max(α,0) ) O ( n 5 2 +max(α,0) )) and finishes the proof. � Note that g+(x) + g−(x)− 2x2 + ` is continuous on R and equal to 0 at x = t by (4.7). Thus, from (4.24) and (4.8) and the fact that V (x) has a jump discontinuity at x = t, we have lim x→t x<t g+(x) + g−(x)− V (x) + ` = −λ < −λc < 0. (7.15) Therefore, by using first Lemma 7.1 and then (7.15), there exists c ∈ (0, λc) such that w̃(x) 2πi [ U−1(x)U ′(x) ] 21 = \|x− t\|αen(g+(x)+g−(x)−V (x)+`)O ( n3+2 max(α,0) ) = \|x− t\|αO ( e−(λ−c)n), (7.16) 36 C. Charlier and A. Deaño as n → ∞ uniformly for x ∈ Dt ∩ (−∞, t). Now, we will split the integral of the differential identity (4.6) into two parts: s∂s logHn (√ 2nt, s ) = I1(s) + I2(s), I1(s) = ∫ (−∞,t)\Dt w̃(x) 2πi [ U−1(x)U ′(x) ] 21 dx, I2(s) = ∫ (−∞,t)∩Dt w̃(x) 2πi [ U−1(x)U ′(x) ] 21 dx. The first integral can be evaluated using (7.6). By (5.1), (5.2), (5.4) and (5.7) (see also the comment just after), we have g+(b) + g−(b) + `− V (b) = −(λ− λc) and (g+(x) + g−(x) + `− V (x))′ ∣∣ x=b = 0, (g+(x) + g−(x) + `− V (x)′′ ∣∣ x=b < 0. Therefore, we obtain∣∣I1 ( s = e−λn )∣∣ = O ( n−1/2e−n(λ−λc)), as n→∞. On the other hand, from (7.16), it immediately follows that∣∣I2 ( s = e−λn )∣∣ = O ( e−(λ−c)n), as n→∞, where c ∈ (0, λc). Therefore, the differential identity becomes ∂s logHn(v, s, α) ∣∣ s=e−λn = O ( n−1/2enλc ) , as n→∞, where in the above expression the O term is uniform for t in a compact subset of (−1,∞) and for λ ≥ λc(t). Thus, we can integrate it from s = 0 to s = e−λn, and it gives logHn (√ 2nt, e−λn, α ) = logHn (√ 2nt, 0, α ) +O ( n−1/2e−n(λ−λc(t))), as n→∞, which is the claim (2.7). 7.3 Proof of Theorem 2.3(2) In this section we use the RH analysis done in Section 6. Proposition 7.2. Let W ⊂ R be an arbitrary small but fixed neighbourhood of the four points {a, b, t, c}. We have as n→∞ w̃(x) 2πi [ U−1(x)U ′(x) ] 21 − nρ(x)χS(x) = en(g+(x)+g−(x)+`−V (x))O(1), uniformly for x ∈ R \W. Proof. We can assume without loss of generality that the disks of the local parametrices are sufficiently small such that D ⊂ W. Let z be outside the lenses and outside the disks. In this region, by inverting the transformations R 7→ S 7→ T 7→ U , we have U(z) = e− n` 2 σ3R(z)P (∞)(z)eng(z)σ3e n` 2 σ3 . Since the dependence in n of the global parametrix (6.12) appears only in the form nΩ ∈ R, and as an argument of the θ-function, by the periodicity property (6.10), as n→∞ we have P (∞)(z) = O(1), P (∞)(z)′ = O(1), uniformly for z outside the disks. Asymptotics for Hankel Determinants Associated to a Hermite Weight 37 Therefore, using also the large n asymptotics for R (6.13), we have[ U(z)−1U ′(z) ] 21 = en`e2ng(z)O(1), as n→∞, (7.17) uniformly for z outside the lenses and outside the disks. For x ∈ R \ (S ∪W), we can take the limit z → x in (7.17). As n→∞, we have w̃(x) 2πi [ U−1(x)U ′(x) ] 21 = \|x− t\|αen(g+(x)+g−(x)+`−V (x))O(1), uniformly for x ∈ R \ (S ∪W). Now, we consider the case when z is still outside the disks but inside I1, see (6.9) and Fig. 3. Inverting the transformations in this region, we get U(z) = e− n` 2 σ3R(z)P (∞)(z) ( 1 0 (z − t)−αe−2nξ1(z) 1 ) eng(z)σ3e n` 2 σ3 . Since P (∞)(z) = O(1) as n→∞ uniformly for z in this region, we have[ U−1(z)U ′(z) ] 21 = (z − t)−αen(2g(z)+`) ( −2nξ′1(z)e−2nξ1(z) +O(1) ) , as n→∞,(7.18) where we have also used (6.7) and <ξ1,+(x) = 0 for x ∈ (t, c). Note that from (6.1), we have ξ′1,+(x) = −πiρ(x) for x ∈ (t, c). Thus, if we let z → x ∈ (t, c)\W in (7.18), from (4.7) and (6.3), we have w̃(x) 2πi [ U−1(x)U ′(x) ] 21 = nρ(x) ( 1 +O ( n−1 )) , as n→∞, (7.19) where the O term in the above expression is uniform for x ∈ (t, c) \ W. For x ∈ (a, b) \ W, we can invert the transformations for z ∈ Ĩ1 and then take the limit z → x. The computations are similar and we obtain the same asymptotics as (7.19). � By (4.1) and (4.4), note that (3.10) can be rewritten as ∫ R w̃(x) 2πi [ U−1(x)U ′(x) ] 21 dx = n. Thus, a consequence of Proposition 7.2 (by taking W arbitrarily small) and (4.6) is that for fixed t ∈ (−1, 1) and fixed λ ∈ (0, λc(t)), we have lim n→∞ s n ∂s logHn (√ 2nt, s, α )∣∣ s=e−λn = lim n→∞ ∫ t −∞ w̃(x) 2πin [ U−1(x)U ′(x) ] 21 dx = Ω(t, λ). (7.20) A simple change of variables shows that s n ∂s logHn (√ 2nt, s, α )∣∣ s=e−λn = − 1 n2 ∂λ logHn (√ 2nt, e−λn, α ) . By (7.20), for every (t, λ) such that t ∈ (−1, 1) and λ ∈ (0, λc(t)), the right-hand side of the above expression converges to Ω(t, λ) as n→∞. Also, by (3.8) and (3.9), we have s n ∂s logHn(v, s, α) = En(v, s, α) n ≤ 1. Since the constant function 1 is integrable on any bounded interval, we can apply Lebesgue’s dominated convergence theorem, and we have lim n→∞ −1 n2 ∫ λ 0 ∂λ̃ logHn (√ 2nt, e−λ̃n, α ) dλ̃ = ∫ λ 0 Ω ( t, λ̃ ) dλ̃, which finishes the proof. Remark 7.3. As mentioned in Remark 4.5, we have indeed only used pointwise convergence for λ ∈ (0, λc(t)) of the quantity s n∂s logHn (√ 2nt, s, α )∣∣ s=e−λn to Ω(t, λ) as n→∞ and Lebesgue’s theorem. The technical RH analysis as λ→ 0 or λ→ λc(t) was thus not needed. 38 C. Charlier and A. Deaño 7.4 Direct proof of formula (2.11) In this section we suppose that t ∈ (−1, 1) and λ ∈ (0, λc(t)), but as we will have to integrate in λ over the interval [0, λc(t)], some quantities need also to be defined for λ = 0 and for λ = λc(t). The quantities ρ(x; t, λ) and `(t, λ) refer to (4.11) and (4.15) if λ ∈ (0, λc(t)), to (4.9) and (4.10) if λ = λc(t), and to (4.16) if λ = 0. Also, Ω(t, λ) is given by (6.6) for λ ∈ (0, λc(t)), and we define by continuity Ω(t, λc(t)) = 0. Lemma 7.4. For t ∈ (−1, 1) and λ ∈ (0, λc(t)), there holds a relation between Ω(t, λ), the density ρ(x; t, λ) given by (4.11), and the Euler–Lagrange constant `(t, λ) given by (4.15): Ω(t, λ) = ∂λ`(t, λ) + ∂λ ∫ S 2x2ρ(x; t, λ)dx+ λ∂λΩ(t, λ). (7.21) Proof. Consider the function H(x; t, λ) = −2 ∫ S log \|x− y\|ρ(y; t, λ)dy. By the Euler–Lagrange equality (4.7), we have H(x; t, λ) = `(t, λ)− 2x2, x ∈ (t, c), (7.22) H(x; t, λ) = `(t, λ)− 2x2 − λ, x ∈ (a, b). (7.23) Thus, by integrating it with respect to ρ(x; t, λ)dx, we obtain∫ S H(x; t, λ)ρ(x; t, λ)dx = `(t, λ)− λΩ(t, λ)− ∫ S 2x2ρ(x; t, λ)dx. We will evaluate ∂λ ∫ S H(x; t, λ)ρ(x; t, λ)dx in two different ways. From the above expression, it gives ∂λ ∫ S H(x; t, λ)ρ(x; t, λ)dx = ∂λ`(t, λ)− Ω(t, λ)− λ∂λΩ(t, λ)− ∂λ ∫ S 2x2ρ(x; t, λ)dx.(7.24) On the other hand, by Lebesgue’s dominated convergence theorem, and by the symmetry in x and y, we have ∂λ ∫ S ∫ S log \|x− y\|ρ(y; t, λ)ρ(x; t, λ)dydx = 2 ∫ S ∫ S log \|x− y\|∂λ(ρ(y; t, λ))ρ(x; t, λ)dydx. Therefore, by differentiating (7.22) and (7.23) with respect to λ, we obtain ∂λ ∫ S H(x; t, λ)ρ(x; t, λ)dx = 2 ∫ S ∂λ (H(x; t, λ)) ρ(x; t, λ)dx = 2∂λ`(t, λ)− 2Ω(t, λ). (7.25) Putting (7.24) and (7.25) together, we obtain (7.21). � Let us consider the function F (t, λ) = `(t, λ) + ∫ S 2x2ρ(x; t, λ)dx, Lemma 7.5. For t ∈ (−1, 1) and λ ∈ (0, λc(t)), we have the following relation Ω(t, λ) = ∂λ [ F (t, λc(t)) 2 ( λ λc(t) )2 + ∫ 1 λ λc(t) ξF ( t, λξ ) dξ ] . (7.26) Asymptotics for Hankel Determinants Associated to a Hermite Weight 39 Proof. From a direct calculation and a change of variables, the right-hand side of (7.26) is equal to∫ 1 λ λc(t) ξ∂λF ( t, λξ ) dξ = ∫ 1 λ λc(t) ∂uF (t, u) ∣∣ u=λ ξ dξ = λ ∫ λc(t) λ ∂uF (t, u) u2 du. (7.27) By using (7.21), which can be rewritten as ∂λF (t, λ) = Ω(t, λ)− λ∂λΩ(t, λ), the right-hand side of (7.27) becomes λ ∫ λc(t) λ Ω(t, u)− u∂uΩ(t, u) u2 du = Ω(t, λ), where the last equality is obtained via an integration by parts, and using the identity Ω(t, λc(t)) = 0. � Lemma 7.6. − ∫ λc(t) 0 Ω(t, λ)dλ = C1(t)− log 3 2 . (7.28) Proof. From Lemma 7.5, we directly obtain that − ∫ λc(t) 0 Ω(t, λ)dλ = −1 2 (F (t, λc(t))− F (t, 0)). (7.29) By (4.9) and (4.10), we obtain F (t, λc(t)) = 3 2 + 2 ( 4 3 t2 + 5 9 t √ 3 + t2 ) + 4t3 27 (√ 3 + t2 − t ) + 2 log ( 2 ( t+ √ 3 + t2 )) , (7.30) and by (4.16), we have F (t, 0) = 3 2 + 2 log 2. (7.31) By substituting (7.30) and (7.31) into (7.29), we obtain (7.28). � A Airy model RH problem We consider the following RH problem: (a) PAi : C \ ΣA → C2×2 is analytic, where ΣA is shown in Fig. 6. (b) PAi has the jump relations PAi,+(ζ) = PAi,−(ζ) ( 0 1 −1 0 ) , on R−, PAi,+(ζ) = PAi,−(ζ) ( 1 1 0 1 ) , on R+, PAi,+(ζ) = PAi,−(ζ) ( 1 0 1 1 ) , on e 2πi 3 R+ ∪ e− 2πi 3 R+. 40 C. Charlier and A. Deaño 2π 3 0 Figure 6. The jump contour ΣA for PAi(ζ). (c) As ζ →∞, z /∈ ΣA, we have PAi(ζ) = ζ− σ3 4 N ( I + ∞∑ k=1 Akζ −3k/2 ) e− 2 3 ζ3/2σ3 , (A.1) where N = 1√ 2 ( 1 i i 1 ) and A1 = 1 8 ( 1 6 i i − 1 6 ) . This model RH problem was introduced for the first time and solved in [18], and is now well- known. The unique solution of the above RH problem is given in terms of Airy functions, we have PAi(ζ) := MA ×  ( Ai(ζ) Ai(ω2ζ) Ai′(ζ) ω2Ai′(ω2ζ) ) e− πi 6 σ3 , for 0 < arg ζ < 2π 3 ,( Ai(ζ) Ai(ω2ζ) Ai′(ζ) ω2Ai′(ω2ζ) ) e− πi 6 σ3 ( 1 0 −1 1 ) , for 2π 3 < arg ζ < π,( Ai(ζ) −ω2Ai(ωζ) Ai′(ζ) −Ai′(ωζ) ) e− πi 6 σ3 ( 1 0 1 1 ) , for − π < arg ζ < −2π 3 ,( Ai(ζ) −ω2Ai(ωζ) Ai′(ζ) −Ai′(ωζ) ) e− πi 6 σ3 , for − 2π 3 < arg ζ < 0, with ω = e 2πi 3 , Ai the Airy function and MA := √ 2πe πi 6 ( 1 0 0 −i ) . B Bessel model RH problem We consider the following RH problem: (a) PBe : C \ ΣB → C2×2 is analytic, where ΣB is shown in Fig. 7. (b) PBe satisfies the jump conditions PBe,+(ζ) = PBe,−(ζ) ( 0 1 −1 0 ) , ζ ∈ R−, Asymptotics for Hankel Determinants Associated to a Hermite Weight 41 0 Figure 7. The jump contour ΣB for PBe(ζ). PBe,+(ζ) = PBe,−(ζ) ( 1 0 eπiα 1 ) , ζ ∈ e 2πi 3 R+, PBe,+(ζ) = PBe,−(ζ) ( 1 0 e−πiα 1 ) , ζ ∈ e− 2πi 3 R+. (B.1) (c) As ζ →∞, ζ /∈ ΣB, we have PBe(ζ) = ( 2πζ 1 2 )−σ3 2 N ( I + ∞∑ k=1 Bkζ −k/2 ) e2ζ 1 2 σ3 , (B.2) where N = 1√ 2 ( 1 i i 1 ) and B1 = 1 16 ( −(1+4α2) −2i −2i 1+4α2 ) . (d) As ζ tends to 0, the behaviour of PBe(ζ) is PBe(ζ) =  ( O(1) O(log ζ) O(1) O(log ζ) ) , \| arg ζ\| < 2π 3 ,( O(log ζ) O(log ζ) O(log ζ) O(log ζ) ) , 2π 3 < \| arg ζ\| < π, if α = 0, PBe(ζ) =  ( O(1) O(1) O(1) O(1) ) ζ α 2 σ3 , \| arg ζ\| < 2π 3 ,( O ( ζ− α 2 ) O ( ζ− α 2 ) O ( ζ− α 2 ) O ( ζ− α 2 )) , 2π 3 < \| arg ζ\| < π, if α > 0, PBe(ζ) = ( O ( ζ α 2 ) O ( ζ α 2 ) O ( ζ α 2 ) O ( ζ α 2 )) , if α < 0. (B.3) This RH problem was introduced and solved in [30]. Its unique solution is given by PBe(ζ) =  ( Iα ( 2ζ 1 2 ) i πKα ( 2ζ 1 2 ) 2πiζ 1 2 I ′α ( 2ζ 1 2 ) −2ζ 1 2K ′α ( 2ζ 1 2 )) , \| arg ζ\| < 2π 3 ,( 1 2H (1) α ( 2(−ζ) 1 2 ) 1 2H (2) α ( 2(−ζ) 1 2 ) πζ 1 2 ( H (1) α )′( 2(−ζ) 1 2 ) πζ 1 2 ( H (2) α )′( 2(−ζ) 1 2 ))eπiα2 σ3 , 2π 3 < arg ζ < π,( 1 2H (2) α ( 2(−ζ) 1 2 ) −1 2H (1) α ( 2(−ζ) 1 2 ) −πζ 1 2 ( H (2) α )′( 2(−ζ) 1 2 ) πζ 1 2 ( H (1) α )′( 2(−ζ) 1 2 ))e−πiα2 σ3 , −π < arg ζ < −2π 3 , (B.4) 42 C. Charlier and A. Deaño where H (1) α and H (2) α are the Hankel functions of the first and second kind, and Iα and Kα are the modified Bessel functions of the first and second kind. Acknowledgements C. Charlier was supported by the European Research Council under the European Union’s Seventh Framework Programme (FP/2007/2013)/ ERC Grant Agreement n. 307074. A. Deaño acknowledges financial support from projects MTM2012-36732-C03-01 and MTM2015-65888- C4-2-P from the Spanish Ministry of Economy and Competitivity. 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Asymptotics for Hankel Determinants Associated to a Hermite Weight with a Varying Discontinuity

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