Edge density expansions for the classical Gaussian and Laguerre ensembles

Edge densit y expansions for the classical Gaussian and Laguerre ensem bles P eter J. F orrester 1 , Anas A. Rahman 2 , and Bo-Jian Shen 1 1 Sc ho ol of Mathematics and Statistics, The Universit y of Melb ourne, Victoria 3010, Aus- tralia. Email: pjforr@unimelb.edu.au ; bojian.shen@unimelb.edu.au 2 Departmen t of Mathematics, The Univ ersity of Hong Kong, Hong Kong. Email: aarahman@hku.hk Abstract Recen t w ork of Bornemann has unco vered hitherto hidden in tegrable structures relating to the asymptotic expansion of quantities at the soft edge of Gaussian and Laguerre random matrix ensembles. These quantities are spacing distributions and the eigen v alue densit y , and the ﬁndings cov er the cases of the three symmetry classes orthogonal, unitary and symplectic. In this work we giv e a diﬀerent viewpoint on these results in the case of the soft edge scaled densit y , and in the Laguerre case w e initiate an analogous study at the hard edge. Our to ol is the scalar diﬀeren tial equation satisﬁed by the latter, kno wn from earlier work. Unlik e integral represen tations, these diﬀeren tial equations in soft edge scaling v ariables isolate the function of N whic h is the expansion v ariable. Moreov er, they give information on the correction terms whic h supplements the ﬁndings from the work of Bornemann. In the case of the Gaussian ensemble, w e can demonstrate analogous features for Dyson index β = 6, which suggests a broader class of mo dels, namely the classical β ensembles, with asymptotic expansions exhibiting integrable features. F or the Laguerre ensem bles at the hard edge, w e give the explicit form of the correction at second order for unitary symmetry , and at ﬁrst order in the orthogonal and symplectic cases. V arious diﬀeren tial relations are demonstrated. 1 In tro duction 1.1 The Gaussian and Laguerre random matrix ensembles F or G an N × N real or complex standard Gaussian matrix, let us form the Hermitian matrix H = 1 2 ( G + G † ). In the real case this is the construction of random matrices forming the Gaussian orthogonal ensemble (GOE), while in the complex case it sp eciﬁes matrices from the Gaussian unitary ensemble (GUE). An extensiv e account relating to these ensem bles is giv en in [ 24 , Ch. 1]. One key prop ert y is that their eigen v alue probability densit y functions hav e the functional form prop ortional to [ 24 , Prop. 1.3.4] N Y l =1 e − β x 2 l / 2 Y 1 ≤ j 0 Y 1 ≤ j − 1, this is said to sp ecify the Laguerre orthogonal ensemble (LOE) for β = 1, and the Laguerre unitary ensem ble (LUE) for β = 2. The constructions leading to ( 1.1 ) and ( 1.2 ) can be extended to give realisations of the case β = 4. This comes about by choosing G and X to b e N × N and n × N matrices with the elemen ts 2 × 2 complex matrices represen ting quaternions; see e.g. [ 24 , Eq. (1.20)]. Cho osing the tw o independent entries in the latter to b e standard complex Gaussians and forming H and W as in the ab ov e paragraphs gives Hermitian random matrices with a doubly degenerate sp ectrum. The probabilit y density function of the N independent eigen v alues in the spectrum is given by ( 1.1 ) for H , and ( 1.2 ) for W . 1.2 Outline of some ﬁndings of Bornemann Up to prop ortionality , the eigen v alue probability densit y functions ( 1.1 ) and ( 1.2 ) are of the form N Y l =1 w ( x l ) Y 1 ≤ j 0 resp ectiv ely . Let us denote b y E N ,β ( k ; ( s, ∞ ); w ( x )) the probabilit y that in an ensemble with eigenv alue probabilit y densit y function prop ortional to ( 1.3 ), there are k eigenv alues in the semi-inﬁnite in terv al ( s, ∞ ). Introduce to o the generating function E N ,β (( s, ∞ ); w ( x ); ξ ) := ∞ X k =0 (1 − ξ ) k E N ,β ( k ; ( s, ∞ ); w ( x )) . (1.4) A fundamen tal result in random matrix theory is that b y centering the v ariable s to the neigh- b ourho o d of the largest eigenv alue, and c ho osing a scale so that in that neighbourho o d the mean eigen v alue spacings are of order unity , the generating function ( 1.4 ) has a well deﬁned limiting form. Moreo v er, for a large class of w eights, this limiting form is indep enden t of the w eight w ( x ) (see [ 19 , 15 , 45 ]), an eﬀect referred to as universalit y . Explicit computation of the limiting form is simplest in the case β = 2. Then, the eigen v alues form a determinantal p oint pro cess (see, e.g., [ 24 , Ch. 5]), meaning that the k -p oint correlation ρ ( k ) ,N ( x 1 , . . . , x k ) is sp eciﬁed by a particular correlation kernel K N ( x, y ) according to ρ ( k ) ,N ( x 1 , . . . , x k ) = det[ K N ( x i , x j )] i,j =1 ,...,k . (1.5) As a consequence (see, e.g., [ 24 , § 9.1]), E N ,β (( s, ∞ ); w ( x ); ξ )    β =2 = det( I − ξ K ( s, ∞ ) ) , (1.6) where K ( s, ∞ ) is the integral op erator on ( s, ∞ ) with k ernel K N ( x, y ). In the case of the GUE, the latter is given in terms of Hermite p olynomials (see [ 24 , § 5.4]). F rom this starting p oint, it w as shown in [ 23 ] that E s β (( y , ∞ ); ξ )    β =2 := lim N →∞ E N ,β (( √ 2 N + y / ( √ 2 N 1 / 6 ) , ∞ ); e − x 2 ; ξ )    β =2 = det( I − ξ K s ( y , ∞ ) ) . (1.7) 2 Here, K s ( y , ∞ ) is the integral op erator on ( y , ∞ ) with k ernel K s ( x, y ) = Ai( x )Ai ′ ( y ) − Ai( y )Ai ′ ( x ) x − y , (1.8) no w kno wn as the Airy kernel. The sup erscript lab el “s” indicates that soft edge scaling has b een applied: Comparing with ( 1.6 ), we see that the end p oint of the semi-inﬁnite interv al has b een chosen to equal √ 2 N + y / ( √ 2 N 1 / 6 ). This is in keeping with the p osition of the largest eigen v alue in the GUE b eing to leading order lo cated at √ 2 N , and with 1 / N 1 / 6 b eing the scale of the spacing b et ween eigenv alues in this neighbourho o d. Note, though, that the limiting functional form E s β (( y , ∞ ); ξ ) | β =2 is v alid for a muc h larger class of mo dels than that of the GUE, in accordance with the universalit y phenomenon discussed in the previous paragraph. There is another prominent form of the limiting soft edge generating function E N ,β in the case β = 2 [ 51 ]. F or this, denote b y q ( x ; ξ ) the solution of the particular Painlev ´ e II equation with prescrib ed limiting b ehaviour q ′′ = xq + 2 q 3 , q ( x ; ξ ) ∼ x →∞ p ξ Ai( x ) . (1.9) Th us, it was sho wn in [ 51 ] that det( I − ξ K s ( y , ∞ ) ) = exp  − Z ∞ y ( x − y )( q ( x ; ξ )) 2 dx  . (1.10) This form makes it particularly clear that, from a mathematical viewp oint, E s β (( y , ∞ ); ξ ) | β =2 relates to integrable systems. Thirt y or so years after the results of [ 51 ], in a series of recen t works [ 9 , 10 , 11 ], Bornemann has uncov ered integrabilit y prop erties associated with the large N expansion of the N -dep enden t soft edge scaled generating function in the ﬁrst equality of ( 1.7 ). Thus, it has b een shown that this quantit y admits the large N asymptotic expansion, in p o wers of N − 2 / 3 , E N , 2 (( √ 2 N + y / ( √ 2 N 1 / 6 ) , ∞ ); e − x 2 ; ξ ) = E s 2 (( y , ∞ ); ξ ) + 1 4 N 2 / 3 E 1 , s , G 2 (( y , ∞ ); ξ ) + 1 16 N 4 / 3 E 2 , s , G 2 (( y , ∞ ); ξ ) + · · · , (1.11) with E 1 , s , G 2 (( y , ∞ ); ξ ) =  y 2 5 d dy − 3 10 d 2 dy 2  E s 2 (( y , ∞ ); ξ ) , E 2 , s , G 2 (( y , ∞ ); ξ ) =  −  141 350 + 8 y 3 175  d dy +  39 y 175 + y 4 50  d 2 dy 2 − 3 y 2 50 d 3 dy 3 + 9 200 d 4 dy 4  E s 2 (( y , ∞ ); ξ ) . (1.12) In the ﬁrst and second corrections, the sup erscript G indicates the Gaussian case. Unlike the leading term, these functional forms are not univ ersal. Nonetheless, there is a weak form of univ ersality in the sense that in the LUE case, the soft edge expansion is again in p ow ers of N − 2 / 3 . The expansion ( 1.11 ) con tinues, with the explicit form of E 3 , s , G 2 (( y , ∞ ); ξ ), the coeﬃcient of N − 2 / 64, given in [ 9 , Th. 2.1] with F ( t ) therein replaced b y E s 2 (( t, ∞ ); ξ ). One should be a ware, for a start, that the existence of an expansion in p ow ers of N − 2 / 3 is far from obvious when b eginning with the orthogonal polynomial expression of the correlation k ernel K N ( x, y ) 1 . In [ 9 ], this is c hec ked to high order b y establishing the corresp onding soft edge 1 W e know from, e.g., [ 12 , Lemma 2.1] that an expansion of the kernel can b e lifted to an expansion of the F redholm determinant. 3 expansion of the Hermite p olynomials to high order; it is exp ected that a general pro of using a Riemann-Hilb ert analysis can b e given analogous to that provided in [ 57 ], in a related setting. Less obvious still is that the successive correction terms should b e given by ξ -indep endent, p olynomial-w eigh ted diﬀeren tial op erators applied to the limiting distribution. This is read- ily appreciated if one recalls from [ 34 ] the complexit y of functional forms of E 1 , s , G 2 (( y , ∞ ); ξ ) a v ailable from earlier literature. In [ 10 ], it is noted that the structures ( 1.11 ) and ( 1.12 ) hav e consequences for the soft edge expansion of the density ρ GUE (1) ,N ( x ). This comes ab out from the fact that (see, e.g., [ 24 , Eq. (9.1)]) generally , E N ,β (( s, ∞ ); w ( x ); ξ ) = 1 + ξ Z ∞ s ρ (1) ,N ( x ) dx + . . . , (1.13) where the terms not shown on the right hand side are of higher p o wers in ξ 2 . Here, the dep endence of ρ (1) ,N ( x ) on w ( x ) and β in its notation hav e b een suppressed for notational clarit y . Let us no w sp ecialise to the GUE and expand the soft edge scaled density 1 √ 2 N 1 / 6 ρ GUE (1) ,N ( √ 2 N + y / ( √ 2 N 1 / 6 )) = ρ s (1) ( y ) + 1 4 N 2 / 3 ρ 1 , s , G (1) ( y ) + 1 16 N 4 / 3 ρ 2 , s , G (1) ( y ) + · · · , (1.14) as is consisten t with ( 1.11 ). On the righ t hand side, we hav e suppressed the fact that w e are sp ecialising to β = 2, although in the ﬁrst and second corrections we ha ve included a sup erscript G for the same reason as in ( 1.11 ). W e recall from [ 23 ] the explicit (univ ersal for β = 2) functional form of the limiting soft edge density , ρ s (1) ( y ) = (Ai ′ ( y )) 2 − y (Ai( y )) 2 . (1.15) With the diﬀeren tial op erators in ( 1.12 ) denoted D 1 , GUE , D 2 , GUE resp ectiv ely (and similarly at higher order although not made explicit in our presentation), it follows by taking the deriv ativ e in ξ at ξ = 0 on b oth sides of ( 1.11 ), as prescrib ed by ( 1.13 ), that for j = 1 , 2 , . . . , Z ∞ y ρ j, s , G (1) ( x ) dx = D j, GUE Z ∞ y ρ s (1) ( x ) dx. (1.16) Moreo ver, noting from ( 1.12 ) the factorisation D j, GUE = ˜ D j, GUE d dy , with ˜ D j, GUE a p olynomial diﬀeren tial op erator, v alid for j = 1 , 2 at least, it follo ws b y diﬀeren tiating both sides of ( 1.16 ) that ρ j, s , G (1) ( y ) = d dy ˜ D j, GUE ρ s (1) ( y ) . (1.17) That is, the correction terms of ( 1.14 ) are giv en b y polynomial-weigh ted diﬀeren tial operators applied to the limiting distribution. Most imp ortan t for our purp oses are tw o further asp ects of Bornemann’s ﬁndings. The ﬁrst is that, with an appropriate shifting of N by wa y of the v ariable N ′ s = N + ( β − 2) /β (this is equation ( 3.1 ) b elow), a soft edge expansion in p ow ers of N ′ s − 2 / 3 holds for the GOE and GSE density and, more generally , the generating function ( 1.11 ); the same holds in the LOE and LSE cases with the cav eat that the deﬁnitions of the soft edge scaling and N ′ s need to b e mo diﬁed (see § 4.1 – 4.3 b elow). The second, coming from [ 10 ], is that the successive terms in the asymptotic expansions of the soft edge scaled densities of the Gaussian and Laguerre ensem bles can b e expressed in terms of transcendental basis functions — of count three for the GUE and LUE, and ﬁve for the GOE, GSE, LOE, LSE — with co eﬃcients that are p olynomials. In the case of the GUE, these are a 1 ( y ) = (Ai( y )) 2 , a 2 ( y ) = (Ai ′ ( y )) 2 , a 3 ( y ) = Ai( y ))Ai ′ ( y ); (1.18) th us for example from ( 1.15 ), we hav e ρ s (1) ( y ) = − y a 1 ( y ) + a 2 ( y ). 2 In distinction to ( 1.11 ), the expansion ( 1.13 ) is conv ergent for all | ξ | < 1. 4 1.3 Scalar diﬀeren tial equation c haracterisation of the densit y and summary of our ﬁndings The to ol for our study of expansions of the form ( 1.14 ) are linear third (GUE and LUE) and ﬁf th (GOE, GSE, LOE, LSE) order diﬀeren tial equations satisﬁed b y the densit y . A uniﬁed approac h co vering each of the Gaussian, Laguerre and Jacobi weigh ts, v alid for all ev en β and their duals 4 /β (the diﬀerential equation is of degree 2 β + 1), has b een given in [ 48 ]. In App endix A, for β = 2, we give a diﬀeren t uniﬁed approach applying to all the classical w eights (the three listed ab o v e as well as the Cauch y weigh t) using orthogonal p olynomial theory . F or example, for the global scaled GUE densit y ρ GUE , g (1) ,N ( x ) := √ 2 N ρ GUE (1) ,N ( √ 2 N x ) (the eﬀect of global scaling is to map the leading order supp ort to the interv al ( − 1 , 1)), w e hav e the diﬀerential equation   1 4 N  2 d 3 dx 3 − ( x 2 − 1) d dx + x   ρ GUE , g (1) ,N ( x ) = 0 . (1.19) With a diﬀerent interpretation (density of spinless free F ermions on a line, conﬁned by a har- monic trap — for a recen t review on the relation of suc h quan tum many b o dy systems to random matrices, see [ 18 ]), an equiv alent third order diﬀeren tial equation c haracterisation ﬁrst appeared in the 1979 work [ 43 ]. Suppose no w we know the existence of an expansion in inv erse (p ossibly fractional) p ow ers of N 3 . Then, it follows from the diﬀerential equation that the expansion in fact only in volv es ev en p ow ers of N . Moreo ver, the individual terms in the expansion are related by a sequence of nested inhomogeneous linear diﬀerential equations which all share the same homogeneous part. In the con text of the Stieltjes transform introduced in fo otnote 3 , this observ ation (and its consequences), can b e found in [ 41 ]. It w as recen tly noted in [ 31 ] that the soft edge scaling of ( 1.19 ) naturally manifests the expansion ( 1.14 ). In Section 2 , w e study said expansion using the general approac h outlined in the previous paragraph. W e ﬁnd that exactly the particular solutions of the aforementioned nested inhomogeneous linear diﬀeren tial equations (with no con tribution from the homogeneous solution) sp ecify b oth ρ 1 , s , G (1) ( y ) and ρ 2 , s , G (1) ( y ), but not ρ 3 , s , G (1) ( y ). This same prop erty is found at the soft edge for the GOE, GSE (see § 3.2 ) and the Laguerre ensem bles LUE (see § 4.1 – 4.3 ), LOE, LSE (see § 5.1 ). Likewise at the hard edge for the LUE (see § 4.4 ), but not the LOE (see § 5.2 ), where we instead ﬁnd that the ﬁrst correction when so characterised does inv olv e a m ultiple of the limiting density . Here, the hard edge refers to a rescaling of the form x 7→ x/ (4 N ′ h ) with N ′ h sp eciﬁed in ( 4.25 ). An imp ortant p oin t is that the solutions of both the homogeneous and inhomogeneous equations in all cases considered can b e sought among a set of transcendental basis functions, as iden tiﬁed in [ 10 ] at the soft edge (recall the ﬁnal paragraph of the previous subsection), and in the presen t w ork at the Laguerre hard edge. It w as noted in [ 31 ] that the homogeneous part of the inhomogeneous linear diﬀerential equation for the bilateral Laplace transform of the soft edge scaled GUE density is ﬁrst order. This allo ws for a simpler analysis, with eac h correction term ha ving the structure ( 2.15 ) b elow. Moreo v er, in equation ( B.9 ) of App endix B, w e give an integral form ula based on a saddle p oint analysis which can (when aided by computer algebra, and sub ject to asso ciated ov erhead constraints) pro duce the terms in the asymptotic expansion to any order. Our metho ds of analysing linear diﬀeren tial equations satisﬁed b y the densit y are applied to the GOE and GSE in Section 3 . As already men tioned, this equation is no w of degree ﬁv e. Nonetheless, the shifted v ariable N ′ s (recall the b eginning of the paragraph con taining ( 1.18 )) is clearly identiﬁable, and leads to a ﬁnite (three terms in total) decomp osition of this op erator at the soft edge, with comp onen ts prop ortional to ( N ′ s ) − 2 j / 3 for j = 0 , 1 , 2. Moreov er, consisten t 3 This is only for illustrativ e purp oses: F or the global densit y , all but the leading term contain oscillatory functions of N ; see, e.g., [ 39 ]. In tro ducing the Stieltjes transform W GUE N ( x ) := R ∞ −∞ ρ GUE , g (1) ,N ( y )( x − y ) − 1 dy , x / ∈ R , it is kno wn from [ 40 ], [ 41 ], [ 56 ] that this smoothed quantit y satisﬁes the hypothesis and also the diﬀerential equation ( 1.19 ) but with the zero on the right hand side replaced b y 2 N . 5 with ﬁndings in [ 9 ], up on the use of a simple scaling of the indep endent v ariable, each op e rator in this decomp osition is the same for b oth the GOE and GSE. W e can no w pro ceed to c hec k that the ﬁrst and second corrections in the analogue of ( 1.14 ) can be determined as particular solutions of the implied inhomogeneous linear equations, with no additive comp onent from the solution of the homogeneous equation as commen ted on in the previous paragraph. The bilateral Laplace transform, for whic h the leading term and the ﬁrst t wo corrections can b e mad e explicit, also exhibits this prop erty . In Remark 3.5 we commen t on an analysis of the soft edge scaled densit y in the β = 6 case of the Gaussian ensembles, from the viewp oint of the known sev en th order linear diﬀerential equation for this quantit y [ 48 ]. The study of soft edge scaling for the Laguerre ensem bles, undertak en in sections 4 and 5 , has from Bornemann’s ﬁndings man y similarities with their Gaussian coun terparts (in particular sharing the same transcenden tal bases for the same symmetry class), although the expansion v ariable N ′ s needs to be mo diﬁed. In fact, the required mo diﬁcation dep ends on whether the soft edge results from the case of the Laguerre parameter a ﬁxed, or a scaled as is natural in applications to multiv ariable statistics ( n, N as appearing in ( 1.2 ) sim ultaneously going to inﬁnit y , with their ratio ﬁxed). W e note that the latter has tw o further sub cases that must b e distinguished — the left and righ t soft edges. Also considered in these sections is the case of hard edge scaling, whereby eigenv alues in the neigh b ourho o d of the origin are scaled to b e of order unit y apart. While an earlier study [ 35 ] has iden tiﬁed the appropriate c hoice of N ′ h , and another has noted asymptotic expansions in pow ers of ( N ′ h ) − 2 [ 26 , Remark 4.3.1], missing from the early literature is the transenden tal bases replacing ( 1.18 ). W e give such bases in terms of Bessel functions. Asymptotic analysis, based on the relation b etw een Laguerre and h yp ergeometric p olynomials, is p erformed to compute the second order correction term in the analogue of ( 1.14 ) for the hard edge scaled LUE (the ﬁrst order term is kno wn [ 34 ]), and the ﬁrst order correction term for the LOE. Surprisingly (k eeping in mind experience of early calculations in our pap er), for the latter it is found that from the viewp oint of the corresp onding coupled linear diﬀerential equations, there is an additiv e m ultiple of the solution of the homogeneous equation. 2 The GUE edge densit y 2.1 Previously established results Let ρ GUE , s (1) ,N ( y ) denote the left hand side of ( 1.14 ), and th us the GUE density with soft edge scaling. Starting with ( 1.19 ) w e hav e shown in [ 31 , Eq. (12)] that this scaled densit y satisﬁes the linear third order diﬀeren tial equation  d 3 dy 3 − 4 y d dy + 2  ρ GUE , s (1) ,N ( y ) = 1 N 2 / 3  y 2 d dy − y  ρ GUE , s (1) ,N ( y ) . (2.1) In k eeping with the recen t ﬁnding in [ 10 ] as revised in § 1.2 , it follows immediately that ρ GUE , s (1) ,N ( y ) p ermits a large N expansion in p o wers of N − 2 / 3 , ρ GUE , s (1) ,N ( y ) = r G 0 ( y ) + 4 − 1 N − 2 / 3 r G 1 ( y ) + 4 − 2 N − 4 / 3 r G 2 ( y ) + · · · ; (2.2) cf. ( 1.14 ). A consequence (see [ 31 , Eq. (13)] with the scaling r G j ( y ) 7→ 4 − j r G j ( y )) is that the suc- cessiv e r G j ( y ) are related b y the nested inhomogeneous, linear, third order diﬀerential equations  d 3 dy 3 − 4 y d dy + 2  r G j ( y ) = 4  y 2 d dy − y  r G j − 1 ( y ) , j = 0 , 1 , . . . , (2.3) with r G − 1 ( y ) = 0 (note that the homogeneous parts of these diﬀerential equations are identical for all j = 0 , 1 , . . . ). The case j = 1 was ﬁrst recorded in [ 48 , Eq. (4.10)]. 6 R emark 2.1 . F rom ( 1.17 ), w e kno w that r G j ( y ) for j = 1 , 2 can be expressed as a j -dep endent diﬀeren tial op erator acting on r G 0 ( y ), a fact whic h moreov er is exp ected to be true for general j . Using the ﬁrst equation in ( 1.12 ), in the case j = 1, this relation reads d dy  − 3 10 d dy + y 2 5  r G 0 ( y ) = r G 1 ( y ) . (2.4) This diﬀerential relation sp eciﬁes the solution of ( 2.3 ) with j = 1 (note that the solution of the latter is not unique without further information). Another ﬁnding of [ 10 ] (whic h can b e used to giv e a veriﬁcation pro of of ( 2.4 )) is that the r G j ( y ) ha v e the structure r G j ( y ) = α G j ( y )(Ai( y )) 2 + β G j ( y )(Ai ′ ( y )) 2 + γ G j ( y )Ai( y )Ai ′ ( y ) (2.5) for p olynomials α G j ( y ) , β G j ( y ) , γ G j ( y ) of degrees that dep end on j . Note that since d dy (Ai( y )) 2 = 2Ai( y )Ai ′ ( y ) , d dy (Ai ′ ( y )) 2 = 2 y Ai( y )Ai ′ ( y ) , d dy Ai( y )Ai ′ ( y ) = (Ai ′ ( y )) 2 + y (Ai( y )) 2 , (2.6) as follo ws from the rules of diﬀeren tiation and the Airy diﬀeren tial equation, this structure is closed with resp ect to diﬀeren tiation. W e list in T able 1 low order cases as known from [ 23 ] ( j = 0), [ 39 ] ( j = 1) and [ 10 ] ( j = 2). j α G j ( y ) β G j ( y ) γ G j ( y ) 0 − y 1 0 1 − 3 5 y 2 2 5 y 3 5 2 39 175 y 3 + 9 100 − 3 175 y 2 − 1 25 y 4 − 99 175 y T able 1: Explicit form of low order cases of the co eﬃcients in ( 2.5 ). In the usual notation asso ciated with the Airy diﬀerential equation, it was noted in [ 31 ] that the third order homogeneous diﬀeren tial equation obtained b y setting j = 0 in ( 2.3 ) has the three linearly indep endent solutions (Ai ′ ( y )) 2 − y (Ai( y )) 2 , (Bi ′ ( y )) 2 − y (Bi( y )) 2 , Ai ′ ( y )Bi ′ ( y ) − y Ai( y )Bi( y ) . (2.7) Due to their asymptotic b ehaviours 4 , only the ﬁrst of these is relev ant to ( 2.1 ), whic h from T able 1 and ( 2.5 ) is precisely r G 0 ( y ) (recall to o ( 1.15 )). One can chec k that the equations ( 2.3 ) for j = 0 , 1 , 2 are satisﬁed b y the explicit functional forms giv en b y ( 2.5 ) and T able 1 . Moreov er, for j = 1 , 2, r G j ( y ) is the particular solution whic h is linearly indep endent of the homogeneous solution. How ever, for j = 3, it turns out that the particular solution con tains an additive factor that is prop ortional to the ﬁrst of the functional forms in ( 2.7 ) and so is not uniquely determined. It was also noted in [ 31 ] that introducing the bilateral Laplace transform 5 u j ( γ ) := Z ∞ −∞ e γ y r G j ( y ) dy , Re( γ ) > 0 (2.8) 4 The ﬁrst con verges to zero as y → ∞ , while the latter tw o div erge to −∞ in this limit. 5 This quan tity pla yed a k ey role in the analysis of [ 46 ] relating the GUE to particular intersection num b ers. 7 leads to a simpiﬁcation of ( 2.3 ). Thus, by multiplying b oth sides of ( 2.3 ) by e γ y (Re( γ ) > 0), in tegrating ov er the range y ∈ R , and simplifying using in tegration by parts assuming no con tribution from the endp oints, it follows that 4 γ u ′ j ( γ ) + (6 − γ 3 ) u j ( γ ) = − 4( γ u ′′ j − 1 ( γ ) + 3 u ′ j − 1 ( γ )) , u − 1 ( γ ) := 0 . (2.9) In the case j = 0 w e see that the third order homogeneous equation has reduced to a ﬁrst order homogeneous equation. This is consistent with the transform ( 2.8 ) only b eing w ell deﬁned for the ﬁrst of the three linearly indep endent solutions ( 2.7 ). 2.2 Results relating to { u j ( γ ) } W e consider ﬁrst a direct ev aluation, using ( 2.8 ) and our knowledge of r G j ( y ) ( j = 0 , 1 , 2). As a preliminary , w e ev aluate the bilateral Laplace transform of the Airy function pro ducts app earing in ( 2.5 ). Prop osition 2.2. L et Re( γ ) > 0 . We have h 1 ( γ ) := Z ∞ −∞ e γ y (Ai( y )) 2 dy = e γ 3 / 12 2 √ π γ , h 2 ( γ ) := Z ∞ −∞ e γ y (Ai ′ ( y )) 2 dy =  1 γ + d dγ  h 1 ( y ) , h 3 ( γ ) := Z ∞ −∞ e γ y Ai( y )Ai ′ ( y ) dy = − γ 2 h 1 ( γ ) . (2.10) Pr o of. The ﬁrst of these is returned by Mathematica, but this pac k age fails on the remaining t wo. T o obtain the second ev aluation, we make use of in tegration by parts and the Airy diﬀerential equation. The third only requires in tegration by parts. Corollary 2.3. With Re( γ ) > 0 , let u j ( γ ) b e sp e ciﬁe d by ( 2.8 ). We have u 0 ( γ ) = e γ 3 / 12 2 √ π γ 3 / 2 , u 1 ( γ ) = − e γ 3 / 12 160 √ π γ 5 / 2  60 + 20 γ 3 + γ 6  , u 2 ( γ ) = e γ 3 / 12 179200 √ π γ 7 / 2  − 42000 + 28000 γ 3 + 14840 γ 6 + 680 γ 9 + 7 γ 12  . (2.11) Pr o of. Substituting ( 2.5 ) in ( 2.8 ) and integrating by parts shows that u j ( γ ) = α G j  d dγ  h 0 ( γ ) + β G j  d dγ  h 1 ( γ ) + γ G j  d dγ  h 2 ( γ ) . (2.12) F or j = 0 , 1 , 2, the p olynomials { α G j , β G j , γ G j } are kno wn from T able 1 , while { h 0 , h 1 , h 2 } are kno wn from Prop osition 2.2 . The result now follows from explicit computation. R emark 2.4 . 1. In ( 2.8 ), let us write γ = γ r + iγ i with γ r , γ i ∈ R and γ r > 0. Then, we see that u j ( γ r + iγ i ) is the F ourier transform of e γ r y r G j ( y ), with F ourier v ariable γ i . Hence, by taking the inv erse F ourier transform, it follo ws e γ r y r G j ( y ) = 1 2 π Z ∞ −∞ e − iy x u ( γ r + ix ) dx. (2.13) One use of this formula is for the purpose of pro viding a plot of r G j ( y ) giv en u j ( γ ). F or j = 0 , 1 , 2, this can b e compared against the same plots calculated from the exact functional forms kno wn from ( 2.5 ) and T able 1 — agreement is obtained. 2. T aking the bilateral Laplace transform of b oth sides of ( 2.4 ) shows that −  3 10 γ 2 + 1 5 γ d 2 dγ 2  u 0 ( γ ) = u 1 ( γ ) , (2.14) as can b e veriﬁed from the explicit formulas in ( 2.11 ). 8 W e can c heck that the results of Corollary 2.3 are consisten t with the coupled diﬀeren tial equations ( 2.9 ). Moreov er, pro ceeding inductively , w e see for general p ositive in teger j that these equations p ermit solutions u j ( γ ) = e γ 3 / 12 √ π γ (2 j +3) / 2 2 j X l =0 c l,j γ 3 l , (2.15) where the c l,j are rationals. F or j not a m ultiple of three, these solutions are unique as then the solution to the homogeneous part of ( 2.9 ) (which is prop ortional to u 0 ( γ )) is not consisten t with the functional form ( 2.15 ), i.e., there is no 0 ≤ l ≤ 2 j such that the factor γ 3 l − (2 j +3) / 2 in the l -th term of ( 2.15 ) matches the equiv alent factor of γ − 3 / 2 in u 0 ( γ ). In particular, this allo ws us to indep endently derive u 1 ( γ ) , u 2 ( γ ) in ( 2.11 ). In the case j = 3, making use of kno wledge of u 2 ( γ ) from ( 2.11 ) and substitution of ( 2.15 ) gives that u 3 ( γ ) = − 4 3 e γ 3 / 12 √ π γ 9 / 2  105 16384 + bγ 3 + 1099 196600 γ 6 + 223 229376 γ 9 + 17089 412876800 γ 12 + 27 45875200 γ 15 + 1 393216000 γ 18  , (2.16) for some undetermined b . Note that the latter relates to the solution of the homogeneous part of ( 2.9 ). Using a direct method based on a saddle point analysis sp eciﬁed in Appendix B, w e compute that b = − 35 16384 . (2.17) With u 3 ( γ ) no w sp eciﬁed, taking the functional form ( 2.15 ) as ansatz in ( 2.9 ) then gives u 4 ( γ ) = 4 4 e γ 3 / 12 √ π γ 11 / 2  − 4725 1048576 + 315 262144 γ 3 + 3759 1048576 γ 6 + 43471 22020096 γ 9 + 61483 293601280 γ 12 + 113941 14533263360 γ 15 + 114691 924844032000 γ 18 + 37 44040192000 γ 21 + 1 503316480000 γ 24  . (2.18) Next, with knowledge of u 4 ( γ ), this pro cess can b e rep eated to sp ecify u 5 ( γ ). As already noted b elo w ( 2.15 ), and explictly demonstrated in ( 2.16 ), con tinuing this approach to the calculation of u 6 ( γ ) there will b e an additive factor prop ortional to u 0 ( γ ) whic h is undetermined. In this case, the direct metho d of App endix B could b e applied to deduce the analogue of ( 2.17 ). 3 The GOE and GSE edge densit y 3.1 Previously established results In order to present the results in a compact form, here w e adopt the con ven tion used in [ 10 ] for the w eight function: Instead of using w G ( x ) = e − β x 2 / 2 , w e use e − x 2 / 2 for the GOE case and e − x 2 for the GSE case. It has b een demonstrated in [ 10 ] that for β = 1 , 4, by in tro ducing a shifted v ariable 6 N ′ s := N + ( β − 2) / (2 β ) , (3.1) under the soft edge scaling ρ G β E ,s (1) ,N ( y ) := 1 √ 2( N ′ s ) 1 / 6 ρ (1) ,N ( p 2 N ′ s + y / ( √ 2( N ′ s ) 1 / 6 ); β ) , (3.2) 6 Here, we in tro duce a further subscript lab el “s” denoting soft to distinguish from other deﬁnitions of N ′ whic h o ccur at, for example, the hard edge to be considered b elow. The expansion parameter h n ′ , ∞ in [ 10 ] relates to N ′ s b y h n ′ , ∞ = 1 4 β − 1 3 ( N ′ s ) − 2 3 . 9 the densit y of the GOE/GSE admits an asymptotic expansion in p o wers of ( N ′ s ) − 2 / 3 . T o presen t the results in a uniﬁed wa y , w e write the expansion as 7 β 1 6 ρ G β E ,s (1) ,N ( y ) = r G ,β 0 ( β 1 3 y ) + (8 p β N ′ s ) − 2 / 3 r G ,β 1 ( β 1 3 y ) + (8 p β N ′ s ) − 4 / 3 r G ,β 2 ( β 1 3 y ) + · · · . (3.3) Analogous to the GUE case, the functions r G ,β j ( y ) are found to hav e the form of a linear com bination inv olving the Airy function [ 10 ], r G ,β j ( y ) = α G ,β j ( y )(Ai( y )) 2 + β G ,β j ( y )(Ai ′ ( y )) 2 + γ G ,β j ( y )Ai( y )Ai ′ ( y ) + ξ G ,β j ( y )Ai( y )AI ν ( y ) + η G ,β j ( y )Ai ′ ( y )AI ν ( y ) . (3.4) Here, the p olynomial co eﬃcients α G ,β j ( y ) , β G ,β j ( y ) , γ G ,β j ( y ) , ξ G ,β j ( y ) , η G j ( y ) are indep endent of β and AI ν is the anti-deriv ative of the Airy function deﬁned as AI ν ( y ) := ν − Z ∞ y Ai( t ) dt, (3.5) with ν = ν β :=  1 , β = 1 , 0 , β = 4 . (3.6) Note that we hav e lim y →−∞ AI 1 ( y ) = lim y →∞ AI 0 ( y ) = 0. F or the low order cases, the co eﬃ- cien ts are given in T able 2 . j α G ,β j ( y ) β G ,β j ( y ) γ G ,β j ( y ) ξ G ,β j ( y ) η G ,β j ( y ) 0 − y 1 0 1 2 0 1 − 1 2 y 2 2 5 y 3 10 − y 10 y 2 10 2 3 y 3 25 + 279 700 − 27 y 2 350 − y 4 100 − 27 y 140 y 5 100 + 9 y 2 140 − 3 y 3 70 − 9 70 T able 2: Explicit form of low order cases of the co eﬃcients in ( 3.4 ). R emark 3.1 . The analogues of ( 1.12 ) for the GOE and GSE are kno wn from [ 9 , Eqns. (5.2a) and (5.2b)]. Moreov er, these op erators p ermit a factorisation of the same form of that noted b elo w ( 1.16 ). F rom ( 1.17 ) w e then read oﬀ that  − 3 5 d 2 dy 2 + y 2 5 d dy + 2 5 y  r G ,β 0 ( y ) = r G ,β 1 ( y ) (3.7) for b oth β = 1 and 4. This result can b e veriﬁed using T able 2 and the form ulas ( 2.6 ) supple- men ted by d dy  Ai( y )AI ν ( y )  = Ai ′ ( y )AI ν ( y ) + (Ai( y )) 2 , d dy  Ai ′ ( y )AI ν ( y )  = y Ai( y )AI ν ( y ) + Ai ′ ( y )Ai( y ) . (3.8) Consider now the general β Gaussian ensem ble (w eight e − β x 2 / 2 ) and write the soft edge expansion of the density as ρ G β E ,s (1) ,N ( y ) = R G ,β 0 ( y ) + (8 N ′ s ) − 2 / 3 R G ,β 1 ( y ) + · · · . (3.9) 7 Note that if ρ G β E ,s (1) ,N ( y ) in ( 3.2 ) was deﬁned with the N ′ s in the argument replaced b y N , the leading correction w ould be of order N − 1 / 3 , with a functional form in y prop ortional to the deriv ative of r G ,β 0 ( β 1 3 y ); see [ 17 , Remark 1.7], [ 10 , § 1.3]. 10 In comparison to ( 2.2 ) and ( 3.3 ), there is agreemen t for β = 1 , 2 and th us R G ,β j ( y ) = r G j ( y ) for β = 2, and R G ,β j ( y ) = r G ,β j ( y ) for β = 1. How ever for β = 4, since the weigh t has b een tak en as e − x 2 in ( 3.3 ) and as e − 2 x 2 in ( 3.9 ), the expansions diﬀer by a scaling. Comparison of the results ( 2.4 ) and ( 3.7 ) suggest that for general β ,  − 3 5 β d 2 dy 2 + y 2 5 d dy + 2 5 y  R G ,β 0 ( y ) = R G ,β 1 ( y ) . (3.10) F or β ev en, the density of the G β E has a β -dimensional integral form [ 4 ]. This was used in [ 20 ] to establish an ev aluation of R G ,β 0 ( y ) as a β -dimensional in tegral, and in [ 36 ] to similarly express R G ,β 1 ( y ) in terms of a β -dimensional in tegral. W e exp ect that an integration by parts strategy as is common in studies of Selb erg type integrals (see [ 24 , § 4.6]), can b e used to establish that these forms are related b y ( 3.10 ) (such a strategy was used in the recent work [ 33 ] to establish an analogous identit y for the circular β ensemble). 3.2 Coupled diﬀeren tial equations and bilateral Laplace transform Without soft edge scaling, ﬁfth order linear diﬀeren tial equations satisﬁed b y ρ GOE (1) ,N ( x ), ρ GSE (1) ,N ( x ) ha ve b een derived in [ 56 ]. Changing to soft edge v ariables as sp eciﬁed in ( 3.2 ) giv es a terminating large N ′ s form of the corresp onding diﬀerential op erators in p ow ers of ( N ′ s ) − 2 / 3 . Prop osition 3.2. Intr o duc e the diﬀer ential op er ators D G ,β 0 := 4 β d 5 dy 5 − 20 y d 3 dy 3 + 12 d 2 dy 2 + 16 β y 2 d dy − 8 β y , D G ,β 1 := − 5 y 2 d 3 dy 3 + 6 y d 2 dy 2 +  8 β y 3 + 14 − 4 β − 16 β  d dy − 6 β y 2 , D G ,β 2 := β y 3  y d dy − 1  . (3.11) Then, for β = 1 and β = 4 , the soft-e dge sc ale d density satisﬁes  D G ,β 0 + 1 ( N ′ s ) 2 / 3 D G ,β 1 + 1 ( N ′ s ) 4 / 3 D G ,β 2  ρ G β E ,s (1) ,N ( y ) = 0 . (3.12) Corollary 3.3. The terms in the exp ansion ( 3.3 ) satisfy the neste d inhomo gene ous diﬀer ential e quations D G ,β 0 r G ,β j ( β 1 3 y ) = −  4 β 1 3 D G ,β 1 r G ,β j − 1 ( β 1 3 y ) + 16 β 2 3 D G ,β 2 r G ,β j − 2 ( β 1 3 y )  , r G ,β − 1 ( y ) = r G ,β − 2 ( y ) = 0 , (3.13) or e quivalently, ˜ D G ,β 0 r G ,β j ( y ) + ˜ D G ,β 1 r G ,β j − 1 ( y ) + ˜ D G ,β 2 r G ,β j − 2 ( y ) = 0 , (3.14) wher e D G ,β 0 := d 5 dy 5 − 5 y d 3 dy 3 + 3 d 2 dy 2 + 4 y 2 d dy − 2 y , D G ,β 1 := − 5 y 2 d 3 dy 3 + 6 y d 2 dy 2 +  8 y 3 − 6  d dy − 6 y 2 , D G ,β 2 := 4 y 3  y d dy − 1  . (Her e, we have use d the fact that 4 β + 16 β = 20 , β = 1 , 4 to simplify the formula.) 11 W e see that the scaling of the v ariables has remo ved the β dep endence in the op erators, making the corrections r G ,β j ( y ) for the GOE/GSE describ ed b y the same system of equations. W e can c heck from the diﬀerentiation form ulas ( 2.6 ) and ( 3.8 ) that making an ansatz consistent with the polynomials in T able 2 for j = 1 , 2 but with unkno wn co eﬃcien ts , the latter is uniquely determined by the appropriate case of ( 3.14 ). Th us, as for the GUE, r G ,β j ( y ) for j = 1 , 2 are particular solutions distinct from the case j = 0 whic h solves the corresp onding homogeneous equation. Similar to the GUE case, we introduce the bilateral Laplace transform u β j ( γ ) := Z ∞ −∞ e γ y r G ,β j ( y ) dy , Re( γ ) > 0 . (3.15) Then from ( 3.14 ), we compute that u β j ( γ ) satisﬁes L G ,β 0 u β j ( γ ) + L G ,β 1 u β j − 1 ( γ ) + L G ,β 2 u β j − 2 ( γ ) = 0 , u β − 1 ( γ ) = u β − 2 ( γ ) = 0 , (3.16) where L G ,β 0 = 18 γ 2 − γ 5 + 5( γ 3 − 2) d dγ − 4 γ d 2 dγ 2 , L G ,β 1 = 48 γ + 36 γ 2 d dγ − 5(6 − γ 3 ) d 2 dγ 2 − 8 γ d 3 dγ 3 , L G ,β 2 = − 20 d 3 dγ 3 − 4 γ d 4 dγ 4 . (3.17) Esp ecially , the leading term satisﬁes a second order homogeneous equation  18 γ 2 − γ 5  u β 0 ( γ ) + 5  γ 3 − 2  d dγ u β 0 ( γ ) − 4 γ d 2 dγ 2 u β 0 ( γ ) = 0 . (3.18) F or this, one can verify that the t wo fundamental solutions are given by 8 exp  γ 3 3  , exp  γ 3 12  3 γ 3 / 2 2 + exp  γ 3 4  √ π γ 3 / 2 Erf γ 3 / 2 2 !! . (3.19) Supplemen tary to Proposition 2.2 , w e ev aluate the bilateral Laplace transform of the relev ant basis functions AiAI ν and Ai ′ AI ν . Prop osition 3.4. L et Re( γ ) > 0 . F or ν = 0 , 1 w e have h 4 ( γ ) := Z ∞ −∞ e γ y Ai( y )AI ν ( y ) dy = 1 2 e γ 3 3 Erf γ 3 / 2 2 ! − 1 ! + ν e γ 3 3 , (3.20) h 5 ( γ ) := Z ∞ −∞ e γ y Ai ′ ( y )AI ν ( y ) dy = − h 1 ( γ ) − γ h 4 ( γ ) . (3.21) Pr o of. The ﬁrst ev aluation is obtained by in tegration by parts and solving a ﬁrst order diﬀeren- tial equation. The constant is ﬁxed b y the equation R ∞ −∞ Ai( y )AI 1 ( y ) dy = 1 2 . The second only requires integration by parts. As an immediate result, we can deduce explicit forms for u β j ( γ ) with small j from that of 8 These are returned by the computer algebra pack age Mathematica. 12 r G ,β j ( y ): u β 0 ( γ ) = 1 4 (2 ν β − 1) e γ 3 / 3 + e γ 3 / 12 4 √ π γ 3 / 2 2 + √ π e γ 3 4 γ 3 2 Erf γ 3 2 2 !! , u β 1 ( γ ) = − e γ 3 / 12 80 √ π γ 5 / 2  30 + 25 γ 3 + 8 γ 6  − e γ 3 / 3 20 γ 2  5 + γ 3  − 1 + 2 ν β + Erf γ 3 / 2 2 !! , u β 2 ( γ ) = e γ 3 / 12 89600 √ π γ 7 / 2  − 21000 + 37100 γ 3 + 79870 γ 6 + 19975 γ 9 + 896 γ 12  + e γ 3 / 3 1400 γ  700 + 875 γ 3 + 170 γ 6 + 7 γ 9  − 1 + 2 ν β + Erf γ 3 / 2 2 !! . (3.22) Here, one sees that u β 0 ( γ ) is a linear com bination of the t wo fundamen tal solutions ( 3.19 ) of ( 3.18 ), and that u β 1 ( γ ) , u β 2 ( γ ) are particular solutions of the corresp onding cases of the inho- mogeneous diﬀerential equation ( 3.16 ), whic h do not contain an additive term prop ortional to either of the fundamen tal solutions. This prop ert y of u β 1 ( γ ) , u β 2 ( γ ) is then the same as that observ ed for β = 2 (GUE case). R emark 3.5 . F rom [ 48 , § 2.3] w e hav e a v ailable a sev enth order linear diﬀeren tial equations for the densit y of the Gaussian β ensem ble with β = 6 and its dual β = 2 / 3. Changing v ariables to a soft edge scaling, these op erators can b e shown to p ermit a form analogous to ( 2.1 ) and ( 3.12 ), and th us to p ermit the solution ( 3.9 ) whic h expands in p ow ers of 1 / ( N ′ s ) 2 / 3 . As an explicit example, for β = 6 and j = 0 , 1, we ha ve D G ,β 0 r G ,β j ( y ) = −D G ,β 1 r G ,β j − 1 ( y ) , r G ,β − 1 ( y ) = 0 , (3.23) where D G ,β 0 is as rep orted in [ 48 , Th. 4.1]. A computer algebra aided calculation gives D G ,β 1 = − 42 y 2 d 5 dy 5 + 42 y d 4 dy 4 + 12(98 y 3 − 9) d 3 dy 3 − 1404 y 2 d 2 dy 2 − 324 y (16 y 3 − 5) d dy + 3456 y 3 − 234 . Ho wev er, for these β there is no evidence that the limiting density can be expressed in terms of Airy functions as known for β = 1 , 2 and 4. Instead, from [ 20 ], there is a six-dimensional in tegral form for β = 6, and no kno wn explicit functional form for β = 2 / 3. Thus, there is no scop e to express the functional forms of corrections in terms of a basis of explicit transcenden tal functions. 4 The LUE edge densities W e kno w from [ 40 ] and [ 48 ] that the LUE density ρ LUE (1) ,N ( x ) (note that in this notation, the de- p endence on the Laguerre parameter a has b een suppressed) satisﬁes a third order homogeneous linear diﬀerential equation D LUE ρ LUE (1) ,N ( x ) = 0, where D LUE := x 3 d 3 dx 3 + 4 x 2 d 2 dx 2 −  x 2 − 2( a + 2 N ) x + a 2 − 2  x d dx +  ( a + 2 N ) x − a 2  . (4.1) The analysis of this at the soft edge dep ends on the parameter a b eing ﬁxed, or being prop or- tional to N . 13 4.1 Soft edge — the case a ﬁxed Considering ﬁrst the case that a is ﬁxed relative to N , let us now introduce the particular soft edge scaling v ariable y b y x = 4 N + 2 a + 2(2 N + a ) 1 / 3 y ; (4.2) cf. [ 24 , § 7.2.2] with the shift N 7→ N + a/ 2 motiv ated by a ﬁnding of the study [ 35 ]. With N ′ s ,a := N + a/ 2, after dividing through by (2 N ′ s ,a ) 2 , a (computer algebra aided) cal- culation gives that the diﬀeren tial op erator ( 4.1 ) then has the terminating large N ′ s ,a expansion 1 (2 N ′ s ,a ) 2 D LUE    x 7→ 4 N ′ s ,a +2(2 N ′ s ,a ) 1 / 3 y = D LUE , s ,a 0 + 1 (2 N ′ s ,a ) 2 / 3 D LUE , s ,a 1 + 1 (2 N ′ s ,a ) 4 / 3 D LUE , s ,a 2 + 1 (2 N ′ s ,a ) 2 D LUE , s ,a 3 , (4.3) where D LUE , s ,a 0 := d 3 dy 3 − 4 y d dy + 2 , D LUE , s ,a 1 := 3 y d 3 dy 3 + 4 d 2 dy 2 − 8 y 2 d dy + 2 y , D LUE , s ,a 2 := 3 y 2 d 3 dy 3 + 8 y d 2 dy 2 − (4 y 3 + ( a 2 − 2)) d dy , D LUE , s ,a 3 := y 3 d 3 dy 3 + 4 y 2 d 2 dy 2 − ( a 2 − 2) y d dy − a 2 . (4.4) Note that D LUE , s ,a 0 is the same op erator as app earing on the left hand side of ( 2.1 ). Setting ρ LUE , s ,a (1) ,n ( y ) := (2 N ′ s ,a ) 1 / 3 ρ LUE (1) ,N (4 N ′ s ,a + 2(2 N ′ s ,a ) 1 / 3 y ), the decomp osition ( 4.3 ) sho ws that the diﬀeren tial equation admits solutions with a large N ′ s ,a asymptotic expansion of the form ρ LUE , s ,a (1) ,N ( y ) = r L , s ,a 0 ( y ) + (2 N ′ s ,a ) − 2 / 3 r L , s ,a 1 ( y ) + (2 N ′ s ,a ) − 4 / 3 r L , s ,a 2 ( y ) + · · · , (4.5) since ( 4.3 ) dep ends on N ′ s ,a only through in teger p ow ers of ( N ′ s ,a ) − 2 / 3 ; cf. ( 2.2 ). F rom the earlier w ork [ 39 ], we kno w that r L , s ,a 0 ( y ) = r G 0 ( y ) as sp eciﬁed by ( 1.15 ), in keeping with the univ ersality of the soft edge limit for matrix ensem bles with unitary symmetry . In particular, this quan tity is indep endent of a . Analogous to ( 2.3 ), we hav e that successiv e correction terms are linked by coupled inhomogeneous diﬀerential equations. Prop osition 4.1. F or k = 0 , 1 , . . . , we have D LUE , s ,a 0 r L , s ,a k ( y ) = −  D LUE , s ,a 1 r L , s ,a k − 1 ( y ) + D LUE , s ,a 2 r L , s ,a k − 2 ( y ) + D LUE , s ,a 3 r L , s ,a k − 3 ( y )  , (4.6) with r L , s ,a − 3 ( y ) = r L , s ,a − 2 ( y ) = r L , s ,a − 1 ( y ) = 0 . F rom [ 10 ], w e kno w that the r L , s ,a j ( y ) admit the same structural expansion ( 2.5 ) as for the r G j ( y ). With regards to r L , s ,a 1 ( y ), w e observ e from ( 4.4 ) that the diﬀerential op erator D LUE , s ,a 1 is indep enden t of the parameter a . It then follows from ( 4.6 ) with k = 1 that r L , s ,a 1 ( y ) is similarly indep enden t of a . It turns out that the case a = 0 of r L , s ,a 1 ( y ) is known from [ 9 , Eq. (3.10a) with τ = 1], [ 10 , Th. 4.3 with τ = 1] (see also T able 3 b elow). Hence, for all a > − 1, r L , s ,a 1 ( y ) = 1 5  3 y 2 (Ai( y )) 2 − 2 y (Ai ′ ( y )) 2 + 2Ai( y )Ai ′ ( y )  . (4.7) W e can c heck that this functional form is consistent with the inhomogeneous diﬀerential equation ( 4.6 ) with k = 1. No w that w e ha ve kno wledge of r L , s ,a 0 ( y ) , r L , s ,a 1 ( y ), we can furthermore use 14 ( 4.6 ) with k = 2 to sp ecify the inhomogeneous diﬀerential equation satisﬁed by r L , s ,a 2 ( y ). W e seek a solution of this equation with the structure ( 2.5 ), and for the p olynomials therein ha ving the same form as those in T able 1 with j = 2, α L 2 ( y ) = a 3 y 3 + a 0 , β L 2 ( y ) = b 2 y 2 , γ L 2 ( y ) = c 4 y 4 + c 1 y . (4.8) Suc h a solution is unique as it do es not con tain an additiv e m ultiple of r L , s 0 , whic h is the solution of the homogeneous p ortion of the diﬀerential equation. Carrying out the calculation gives r L , s ,a 2 ( y ) =  − 96 175 y 3 + 4 − 2 a 2 100  (Ai( y )) 2 + 37 175 y 2 (Ai ′ ( y )) 2 −  1 25 y 4 + 74 175 y  Ai( y )Ai ′ ( y ) . (4.9) As with r L , s ,a 1 ( y ), the case a = 0 of r L , s ,a 2 ( y ) is av ailable in [ 10 , Th. 4.3 with τ = 1] (see also T able 3 b elow), and agreement is found with this sp ecialisation of ( 4.9 ). Imp ortan t in relation to the analogue of the ﬁrst equation in ( 1.12 ) for the ﬁxed- a Laguerre soft edge generating function expansion E N , 2 ((4 N ′ s ,a +2(2 N ′ s ,a ) 1 / 2 y , ∞ ); x a e − x ; ξ ) = E s 2 (( y , ∞ ); ξ ) + 1 (2 N ′ s ,a ) 2 / 3 E 1 , s ,a 2 (( y , ∞ ); ξ ) + · · · (4.10) is a second order diﬀeren tial operator mapping r L , s ,a 0 ( y ) to r L , s ,a 1 ( y ). Th us, w e seek the ﬁxed- a Laguerre analogue of ( 2.4 ) for the GUE. Prop osition 4.2. We have d 2 dy 2 r L , s ,a 0 ( y ) = − 2Ai( y )Ai ′ ( y ) , d dy r L , s ,a 0 ( y ) = − (Ai( y )) 2 , (4.11) and c onse quently, − 1 5 d dy  d dy + y 2  r L , s ,a 0 ( y ) = r L , s ,a 1 ( y ) . (4.12) Pr o of. T o deriv e ( 4.11 ), we mak e use of ( 1.15 ) and ( 2.6 ). With this established, ( 4.12 ) can b e v eriﬁed after recalling that r L , s ,a 1 ( y ) is giv en by ( 4.7 ). F rom the assumption that E s 2 (( y , ∞ ); ξ ) and E 1 , s ,a 2 (( y , ∞ ); ξ ) are related by a ξ -indep endent second order diﬀerential op erator, it follows from ( 4.12 ) that − 1 5  d dy + y 2  d dy E s 2 (( y , ∞ ); ξ ) = E 1 , s ,a 2 (( y , ∞ ); ξ ) . (4.13) This can b e compared against the result of [ 9 , Eq. (3.7a) with τ = 1], which corresp onds to the case a = 0 (note that ( 4.13 ) is indep endent of a ). Agreemen t is found. 4.2 Righ t soft edge — the case a prop ortional to N F or β = 2, w e hav e from ( 1.2 ) that a = n − N . Here, w e set n = γ N , γ ≥ 1 so that a = ( γ − 1) N , and is th us prop ortional to N , as is consisten t with earlier work [ 44 , 9 ]. W e moreov er in tro duce the soft edge scaling v ariable y according to x = (1 + √ γ ) 2 N ′ s + γ − 1 / 6 (1 + √ γ ) 4 / 3 ( N ′ s ) 1 / 3 y . (4.14) (F or β = 2, N ′ s = N ; using N ′ s in ( 4.14 ) is required later when the cases β = 1 and 4 are considered.) It is further conv enien t to introduce the mo diﬁcation of N ′ s 9 ˆ N ′ s = (4 γ ) 1 / 2 τ N ′ s , τ = 4 √ γ + 1 / √ γ + 2 . (4.15) 9 In the w ork [ 10 ], the expansion parameter h n,p , when speciﬁed at right soft edge, relates to our ˆ N ′ s b y h n,p = ( ˆ N ′ s ) − 2 / 3 for β = 2 and h n,p = ( √ β ˆ N ′ s ) − 2 / 3 for β = 1 , 4. 15 After a change of v ariables in ( 4.1 ), the LUE densit y scaled at the right soft edge with a prop ortional to N , denoted by ρ LUE ,sr (1) ,N ( y ), can b e chec ked to satisfy the diﬀerential equation ( ˆ N ′ s ) 2 D LUE ,sr 0 + ( ˆ N ′ s ) 4 / 3 D LUE ,sr 1 + ( ˆ N ′ s ) 2 / 3 D LUE ,sr 2 + D LUE ,sr 3 ! ρ LUE ,sr (1) ,N ( y ) = 0 , (4.16) where D LUE ,sr 0 = d 3 dy 3 − 4 y d dy + 2 , D LUE ,sr 1 = τ  3 y d 3 dy 3 + 4 d 2 dy 2 − 4 y 2 d dy − 2 y  − 4  y 2 d dy − y  , D LUE ,sr 2 = τ 2  3 y 2 d 3 dy 3 + 8 y d 2 dy 2 + 2 d dy  − 4 τ y 3 d dy , D LUE ,sr 3 = τ 3  y 3 d 3 dy 3 + 4 y 2 d 2 dy 2 + 2 y d dy  . (4.17) As exp ected by the universalit y of the limiting soft edge density , the diﬀeren tial op erator D LUE ,sr 0 is iden tical to that on the left hand side of ( 2.1 ). Now in tro ducing the asymptotic expansion in p o w ers of ˆ N ′− 2 / 3 s established in [ 10 ], ρ LUE ,sr (1) ,N ( y ) = r L , sr 0 ( y ) + r L , sr 1 ( y ) ˆ N ′− 2 / 3 s + r L , sr 2 ( y ) ˆ N ′− 4 / 3 s + . . . , (4.18) coupled diﬀerential equations for { r L , sr k ( y ) } are immediate. Prop osition 4.3. We have D LUE ,sr 0 r L , sr k ( y ) + D LUE ,sr 1 r L , sr k − 1 ( y ) + D LUE ,sr 2 r L , sr k − 2 ( y ) + D LUE ,sr 3 r L , sr k − 3 ( y ) = 0 , (4.19) wher e r L , sr − 3 ( y ) = r L , sr − 2 ( y ) = r L , sr − 1 ( y ) = 0 . F rom ( 4.19 ), it can b e seen that unlik e the ﬁxed parameter case, the ﬁrst correction term r L , sr 1 ( y ) no w dep ends on the Laguerre parameter a (through dep endence on τ ). As for the Gaussian case, it is sho wn in [ 10 ] that the r L , sr j ( y ) ha v e the structure r L , sr j ( y ) = α L , sr j ( y )(Ai( y )) 2 + β L , sr j ( y )(Ai ′ ( y )) 2 + γ L , sr j ( y )Ai( y )Ai ′ ( y ) . (4.20) The low order co eﬃcients, as calculated in [ 10 ], are listed in T able 3 . j α L , sr j ( y ) β L , sr j ( y ) γ L , sr j ( y ) 0 − y 1 0 1 3(2 τ − 1) 5 y 2 − 2(2 τ − 1) 5 y 3 − τ 5 2 − 214 τ 2 − 79 τ − 39 175 y 3 + ( τ − 3) 2 100 143 τ 2 − 103 τ − 3 175 y 2 − (2 τ − 1) 2 25 y 4 + 29 τ 2 − 4 τ − 99 175 y T able 3: Explicit form of low order cases of the co eﬃcients in ( 4.20 ). Setting τ = 0 reclaims the GUE results of T able 1 (see [ 10 , Remark 4.1]). Our p oint in relation to these explict functional forms and Prop osition 4.3 is that, as for the case of a ﬁxed considered in the previous subsection, b oth r L , sr 1 ( y ) and r L , sr 2 ( y ) are particular solutions of the resp ective inhomogeneous diﬀerential equations ( 4.19 ). R emark 4.4 . Analogous to ( 4.12 ), as noted in [ 10 ], the corrections r L , sr 1 ( y ), r L , sr 2 ( y ) can b e related to r L , sr 0 ( y ) by particular diﬀerential op erators (of degree t wo and four resp ectively). Explicitly , for r L , sr 1 ( y ), as can b e chec ked using T able 3 and ( 1.15 ), we hav e 1 5 d dy  τ − 3 2 d dy − (2 τ − 1) y 2  r L , sr 0 ( y ) = r L , sr 1 ( y ) . (4.21) In keeping with [ 10 , Remark 4.1], setting τ = 0 reclaims ( 2.4 ). 16 4.3 Left soft edge — the case a prop ortional to N In the con text of the LUE, the left soft edge was ﬁrst considered in [ 5 ]. With the same notation as in the last subsection, it is realized by the scaling v ariable x = (1 − √ γ ) 2 N ′ s − γ − 1 / 6 ( √ γ − 1) 4 / 3 ( N ′ s ) 1 / 3 y . T o ﬁt the left soft edge case, w e also mo dify the parameters ˆ N ′ s , l := (4 γ ) 1 / 2 τ l N ′ s , τ l := 4 √ γ + 1 / √ γ − 2 . (4.22) Denote b y ρ LUE , sl (1) ,N ( y ) the left soft edge scaled LUE densit y . Starting with ( 4.1 ), and pro ceeding as in the calculation of ( 4.16 ), we ﬁnd that ( ˆ N ′ s , l ) 2 D LUE , sl 0 + ( ˆ N ′ s , l ) 4 / 3 D LUE , sl 1 + ( ˆ N ′ s , l ) 2 / 3 D LUE , sl 2 + D LUE , sl 3 ! ρ LUE , sl (1) ,N ( y ) = 0 , (4.23) where D LUE ,sl 0 = d 3 dy 3 − 4 y d dy + 2 , D LUE , sl 1 = − τ l  3 y d 3 dy 3 + 4 d 2 dy 2 − 4 y 2 d dy − 2 y  − 4 y 2 d dy + 4 y , D LUE , sl 2 = τ 2 l  3 y 2 d 3 dy 3 + 8 y d 2 dy 2 + 2 d dy  + 4 τ l y 3 d dy , D LUE , sl 3 = − τ 3 l  y 3 d 3 dy 3 + 4 y 2 d 2 dy 2 + 2 y d dy  . (4.24) W e see that ( 4.23 ) is iden tical in structure to ( 4.16 ), while ( 4.24 ) is iden tical to ( 4.17 ) with τ 7→ − τ l . Hence, deﬁning { r L , sl j ( y ) } in analogy with ( 4.18 ), the coupled diﬀeren tial equations of Prop osition 4.3 remain v alid with this replacement. Assuming r L , sl j ( y ) for j = 1 , 2 are determined as the particular solutions with no additiv e contribution from r L 0 ( y ) (as is the case for the right soft edge), it follo ws that these functions ha ve the form ( 4.20 ) with co eﬃcients as in T able 3 , no w setting τ = − τ l . 4.4 The hard edge Generally in the Laguerre ensembles, the hard edge refers to the neigh b ourho o d of the origin with the parameter a ﬁxed. In the case of the densit y with β even, and the probability densit y function of the smallest eigen v alue for general β > 0 with a ev en, or for β = 2 with general a > − 1, it was demonstrated in [ 35 ] 10 that an optimal rate of con vergence is obtained through the hard edge scaling x = y 4 N ′ h , N ′ h := N + a β . (4.25) Our interest here is sp eciﬁc to the LUE, which corresp onds to β = 2. W e kno w from [ 35 , Eqns. (2.15)–(2.17)] that the hard edge scaled density ρ LUE , h (1) ,N ( y ) = (1 / 4 N ′ h ) ρ LUE , h (1) ,N ( y / 4 N ′ h ) has the large N expansion ρ LUE , h (1) ,N ( y ) = r L , h 0 ( y ) + 1 ( N ′ h ) 2 r L , h 1 ( y ) + O  1 ( N ′ h ) 3  , (4.26) where r L , h 0 ( y ) , r L , h 1 ( y ) are giv en in terms of Bessel functions according to r L , h 0 ( y ) = 1 4  ( J a ( √ y )) 2 − J a +1 ( √ y ) J a − 1 ( √ y )  , r L , h 1 ( y ) = − 1 192  (2 y + a 2 ) J a ( √ y ) 2 + 4 √ y J a ( √ y ) J ′ a ( √ y ) + y J ′ a ( √ y )) 2  . (4.27) 10 In the latter setting see also [ 22 , 6 , 47 , 42 ]. 17 Here, the notation J ′ a ( √ y ) is for J ′ a ( u ) | u = √ y . W e remark to o that the limit law as sp eciﬁed by r L , h 0 ( y ) w as already known from the earlier work [ 23 ]. Our general metho d of analysing the linear diﬀerential equation satisﬁed by the densit y allo ws us to easily demonstrate that the expansion ( 4.26 ) contains only even p o wers of ( N ′ h ) − 1 . Prop osition 4.5. With D LUE sp e ciﬁe d by ( 4.1 ), we have D LUE    x = y / (4 N ′ h ) = y 3 d 3 dy 3 + 4 y 2 d 2 dy 2 + ( y − a 2 + 2) y d dy + y 2 − a 2 − y 3 (4 N ′ h ) 2 d dy . (4.28) Henc e, the har d e dge density p ermits the lar ge N ′ h exp ansion ρ LUE , h (1) ,N ( y ) = r L , h 0 ( y ) + ( N ′ h ) − 2 r L , h 1 ( y ) + ( N ′ h ) − 4 r L , h 2 ( y ) + · · · (4.29) (cf. ( 4.26 )). F urther, the suc c essive terms ar e r elate d by the neste d inhomo ge ous diﬀer ential e quations D LUE , h 0 r L , h j ( y ) = −D LUE , h 1 r L , h j − 1 ( y ) , r L , h − 1 ( y ) = 0 ( j = 0 , 1 , 2 , . . . ) , (4.30) wher e we have use d the notation D LUE , h 0 + ( N ′ h ) − 2 D LUE , h 1 for the right hand side of ( 4.28 ). Pr o of. The form ( 4.28 ) is immediate from ( 4.1 ). The existence of a well deﬁned large N expan- sion of the hard edge scaled density in inv erse p o wers of N is assured b y ( 4.44 ) and the working of App endix D below. Since the op erator ( 4.28 ) acting on the hard edge density ρ LUE , h (1) ,N ( y ) gives zero, ( 4.29 ) follows. Substituting this in the rewritten diﬀerential equation giv es ( 4.30 ). Using the Bessel function iden tities 2 a u J a ( u ) = J a − 1 ( u ) + J a +1 ( u ) , 2 J ′ a ( u ) = J a − 1 ( u ) − J a +1 ( u ) , (4.31) w e see that the r L , h 0 ( y ) as sp eciﬁed in ( 4.27 ) can b e rewritten as r L , h 0 ( y ) = 1 4   1 − a 2 y  ( J a ( √ y )) 2 + ( J ′ a ( √ y )) 2  . (4.32) Comparing this with the functional form of r L , h 1 ( y ) in ( 4.27 ) suggests that we seek solutions of the coupled equations ( 4.30 ) of the form r L , h j ( y ) = α L , h j ( y ) 1 y ( J a ( √ y )) 2 + β L , h j ( y )( J ′ a ( √ y )) 2 + γ L , h j ( y ) 1 √ y J a ( √ y ) J ′ a ( √ y ) , (4.33) where α L , h j ( y ) , β L , h j ( y ) , γ L , h j ( y ) are p olynomials; cf. ( 2.5 ). The explicit forms of the latter can b e read oﬀ from ( 4.32 ) and the second equation in ( 4.27 ); we record their v alues in T able 3 . j α L , h j ( y ) β L , h j ( y ) γ L , h j ( y ) 0 1 4 ( y − a 2 ) 1 4 0 1 − 1 192 (2 y + a 2 ) y − 1 192 y − 1 48 y T able 4: Explicit forms of some low order cases of the co eﬃcients in ( 4.33 ). The functional form ( 4.33 ) is compatible with the nested diﬀerential equations ( 4.30 ) since it remains closed under the deriv ativ e op eration y d dy , or equiv alen tly D y := d dy y . The relev ance of this property , to no w b e veriﬁed, is that the diﬀeren tial operator ( 4.28 ) can b e written as a p olynomial of either of these op erations. 18 Prop osition 4.6. L et D y b e as deﬁne d ab ove. L et b 1 ( y ) = 1 y ( J a ( √ y )) 2 , b 2 ( y ) = ( J ′ a ( √ y )) 2 , b 3 ( y ) = 1 √ y J a ( √ y ) J ′ a ( √ y ) . (4.34) We have D y b 1 ( y ) = b 3 ( y ) , D y b 2 ( y ) = − ( y − a 2 ) b 3 ( y ) , D y b 3 ( y ) = 1 2 ( − ( y − a 2 ) b 1 ( y ) + b 2 ( y )) . (4.35) Pr o of. These follow from the deﬁnitions and the rules for diﬀerentiation, making use to o of the second order diﬀerential equation satisﬁed by the Bessel function, written in the form J ′′ a ( u ) = − 1 u 2  uJ ′ a ( u ) + ( u 2 − a 2 ) J a ( u )  . (4.36) Corollary 4.7. In addition to the diﬀer ential r elation ( 4.30 ) with j = 1 , we have the explicit diﬀer ential r elation r L , h 1 ( y ) = −  1 12 D 2 y r L , h 0 ( y ) + 1 48 D y ( y r L , h 0 ( y )) + 1 24 ( a 2 − 2) D y r L , h 0 ( y )  . (4.37) L et the gener ating function of ther e b eing k ( k = 0 , 1 , 2 , . . . ) eigenvalues in the interval (0 , s ) in the LUE b e exp ande d for lar ge N ′ h ac c or ding to 11 E N , 2 ((0 , y / (4 N ′ h )); x a e − x ; ξ ) = E h 2 ((0 , y ); ξ ) + 1 ( N ′ h ) 2 E 1 , L , h 2 ((0 , y ); ξ ) + 1 ( N ′ h ) 4 E 2 , L , h 2 ((0 , y ); ξ ) + · · · . (4.38) Under the assumption that E h 2 ((0 , y ); ξ ) and E 1 , L , h 2 ((0 , y ); ξ ) ar e r elate d by a ξ -indep endent se c ond or der diﬀer ential op er ator applie d to the former, we have E 1 , L , h 2 ((0 , y ); ξ ) = −  1 12 y 2 d dy + 1 48 y 2 + 1 24 a 2 y  d dy E h 2 ((0 , y ); ξ ) . (4.39) Pr o of. Using Prop osition 4.6 and the form of r L , h 0 ( y ) as implied by T able 3 , we compute D y r L , h 0 ( y ) = 1 4 ( J a ( √ y )) 2 , D 2 y r L , h 0 ( y ) = 1 4  √ y J a ( √ y ) J ′ a ( √ y ) + ( J a ( √ y )) 2  . (4.40) These allow ( 4.37 ) to b e v eriﬁed. The form ula ( 4.39 ) then follo ws from the same argument as that leading to ( 1.17 ). R emark 4.8 . 1. F rom [ 23 ] w e ha ve the simple exact result E N ,β ((0 , y / (4 N )); e − β x/ 2 ; ξ ) | ξ =1 = e − β y / 8 . Th us, all correction terms in a large N expansion are iden tically zero. Indeed, substituting E h 2 ((0 , y ); ξ ) | ξ =1 = e − y / 4 (4.41) in the righ t hand side of ( 4.39 ) with a = 0 (in k eeping with the weigh t b eing e − β x/ 2 | β =2 ) returns zero. Another simple exact result is that [ 35 , Prop. 10] E N ,β ((0 , y / (4 N ′ h )); xe − β x/ 2 ; ξ ) | ξ =1 = e − β y / 8 0 F 1  2 β ; y 4  + 1 ( N ′ h ) 2 y 48 e − β y / 8  − (1 − 1 /β ) 0 F 1  2 β ; y 4  + ((1 − 1 /β ) + y β / 8) 0 F 1  2 β + 1; y 4   + · · · . (4.42) 11 Existence of an expansion of this form follows from Prop osition D.2 b elow. 19 In the case β = 2, we can c hec k that the functional forms of the leading term and the ﬁrst correction are consistent with ( 4.39 ) when setting a = 1 (here, the weigh t is xe − x ). 2. Characterisations of b oth E h 2 ((0 , y ); ξ ) and E 1 , L , h 2 ((0 , y ); ξ ) in terms of a particular σ -Painlev ´ e I I I ′ transcenden t are known [ 51 , 38 , 35 ]. This allows us to v erify ( 4.39 ), and thus establish its v alidity without any assumptions. The required working is sk etched in App endix C. 3. The generating function E h 2 ((0 , y ); ξ ) with ξ = 1 and the Laguerre parameter a a p ositive in teger p ermits an expression as a particular matrix a verage o ver Haar distributed unitary matrices of size a (see, e.g., [ 37 , Eq. (2.1)]). This matrix in turn has an interpretation in terms of the length of the longest increasing subsequence of a random p erm utation [ 49 ]. In this context, the scaled a → ∞ limit of E h 2 ((0 , y ) with y 7→ a 2 − 2 a ( a/ 2) 1 / 3 y is of interest [ 13 ]. The optimal expansion parameter and the resulting functional forms in the corresp onding asymptotic expansion are given in [ 3 , 27 , 7 , 8 ]. The former for β = 2 is a − 2 / 3 . F rom the presen t viewp oin t, this can b e understo o d from the op erator decomp osition D LUE , h    y 7→ a 2 − 2 a ( a/ 2) 1 / 3 y = a 2 D LUE , hs 0 + a 4 / 3 D LUE , hs 1 + a 2 / 3 D LUE , hs 2 + D LUE , hs 3 . (4.43) Here, D LUE , h is the diﬀerential op erator sp eciﬁed in Prop osition 4.5 . F or economy of space, w e do not list the explicit forms of D LUE , hs i , except to sa y that for i = 0, this is equal to the op erator on the left hand side of ( 2.1 ). Not a v ailable in earlier literature is the ev aluation of r L , h 2 ( y ). W e w ould like to compute this here. The starting p oin t is to use the form of the correlation k ernel K LUE N ( x, y ) for the LUE sp eciﬁed in [ 35 , Eq. (2.4)]. The LUE densit y is the limit x → y of this. Making use of L’Hˆ opital’s rule and a Laguerre p olynomial identit y , w e obtain ρ LUE (1) ,N ( x ) = N ! Γ( a + N ) x a e − x  L ( a ) N − 1 ( x ) L ( a ) N ( x ) − L ( a +1) N − 1 ( x ) L ( a − 1) N ( x )  = Γ( N + a + 1) (Γ( a + 1)) 2 Γ( N ) x a e − x  1 F 1 ( − N + 1; a + 1; x ) 1 F 1 ( − N ; a + 1; x ) − a a + 1 1 F 1 ( − N + 1; a + 2; x ) 1 F 1 ( − N ; a ; x )  , (4.44) where the second equality mak es use of the hypergeometric p olynomial form of Laguerre p oly- nomials given in ( D.1 ) b elo w. In relation to computing r L , h 2 ( y ), we hav e from ( 4.25 ) with β = 2 and ( 4.26 ) that w e require the large N expansion of ( 4.44 ) with x = y / (4 N ′ h ). F rom Prop osition D.1 of App endix D, we hav e a v ailable the large N form of the h yp ergeometric p olynomials in ( 4.44 ) with x 7→ x/ N . An appropriate N -dep endent scaling of x allows for the latter to b e used in ( 4.44 ), leading then to the sought ev aluation. Prop osition 4.9. In the notation of ( 4.33 ), r L , h 2 ( y ) is sp e ciﬁe d by α L , h 2 ( y ) = y 92160  14 a 2 ( − 4 + a 2 ) + 2(48 − 9 a 2 ) y − 26 y 2  , β L , h 2 ( y ) = y 92160  192 − 128 a 2 + 20 a 4 + ( − 104 + 20 a 2 ) y + 5 y 2  , γ L , h 2 ( y ) = y 92160  14( − 4 + a 2 ) + 6 y  . (4.45) Pr o of. According to ( 4.44 ) and ( D.1 ), we require use of Proposition D.1 with α = a, a ± 1. This is immediate, since the op erators in ( D.2 ) and ( D.3 ) are independent of α . In the latter form ulas it is con v enien t to make use of the deriv ativ e formula D k x 0 F 1 ( α + 1; − x ) = ( − x ) k Q k l =1 ( l + a ) 0 F 1 ( α + k + 1; − x ) , (4.46) 20 whic h is implemented b y default in the computer algebra system Mathematica. With x 7→ x/ N , w e use the results of Proposition D.1 to expand the diﬀerence of h yp ergeometric p olynomials in ( 4.44 ) in p ow ers of N − 1 . After this, w e m ust replace x b y y 4 (1 − ϵ ), ϵ := a/ (2 N ′ h ), and no w expand in p ow ers of 1 / N ′ h . In theory , this is carried out using a T aylor series, f ((1 − ϵ ) y / 4) = f ( y / 4) − ϵ D x f ( x )    x 7→ y / 4 + ϵ 2 2! D 2 x f ( x )    x 7→ y / 4 + · · · . (4.47) In practice, this step can b e carried out internally in Mathematica, whic h automatically ev al- uates the deriv ativ es according to ( 4.46 ), as previously remarked. Pro ceeding in this w ay , w e arriv e at the expansion in p o w ers of 1 / N ′ h of the com bination of hypergeometric p olynomials in ( 4.44 ) with x replaced as sp eciﬁed. Again in ternally in Mathematica, w e up date this expan- sion b y m ultiplying it b y the expansion in pow ers of 1 / N ′ h of e − x/ (4 N ′ h ) . One observ es that an expansion in terms of 1 / ( N ′ h ) 2 results. With regards to the prefactor Γ( N + a + 1) / Γ( N ), we ﬁrst assume a ∈ Z ≥ 0 . Under this circumstance, Γ( N + a + 1) Γ( N ) = N a +1 a Y l =0  1 + l N  . (4.48) Making use of ( D.4 ), terms in the 1 / N expansion are seen to b e p olynomials in a , thus allowing us to remov e the restriction a ∈ Z ≥ 0 (see also [ 53 ]). This latter expansion, multiplied b y (4 N ′ h ) − ( a +1) , then expanded in p ow ers of 1 / N ′ h , shows (4 N ′ h ) − ( a +1) Γ( N + a + 1) Γ( N ) = 4 − ( a +1)  1 − a 3 + 3 a 2 + 2 a 24( N ′ h ) 2 + 5 a 6 + 12 a 5 − 25 a 4 − 60 a 3 + 20 a 2 + 48 a 5760( N ′ h ) 4 + · · ·  . (4.49) (The working in the pro of of Prop osition D.2 b elow assures that this expansion is in p o wers of ( N ′ h ) − 2 .) Multiplying ( 4.49 ) b y the expansion obtained in the paragraph b efore, and multiplying by the factor of 1 / (Γ( a + 1)) 2 as also app ears in ( 4.44 ), then using the FullSimplify command in Mathematica, we obtain a form equiv alen t to ( 4.33 ) (speciﬁcally , it inv olv es J a − 1 ( √ y ) instead of J ′ a ( √ y ), which can b e eliminated b y appropriate use of ( 4.31 )) with the co eﬃcients as sp eciﬁed in ( 4.45 ). R emark 4.10 . 1. Ab ov e ( 4.44 ) men tion is made of the LUE correlation k ernel K LUE N ( x, y ). In subsection D.2 of App endix D w e make of its form in terms of h yp ergeometric p olynomials. Moreov er, w e use that form to exhibit that the hard edge scaled large N ′ h expansion is of the form 1 4 N ′ h K LUE N ( x/ 4 N ′ h , y / 4 N ′ h ) = K h ∞ , 0 ( x, y ) + 1 ( N ′ h ) 2 K h ∞ , 1 ( x, y ) + 1 ( N ′ h ) 4 K h ∞ , 2 ( x, y ) + · · · (4.50) Since the correlation k ernel fully determines the general k -p oint correlation functions, whic h in turn fully determines the generating function in ( 4.38 ) (see [ 24 , Eq.(9.1)]), this establishes the v alidity of the expansion in ( 4.38 ). An analogous argumen t has b een used in the recen t work [ 33 ] in relation to establishing an expansion in p o wers of 1 / N 2 for the spacing distributions of the bulk scaled circular ensem bles. 2. F rom the viewp oin t of the inhomogeneous equation ( 4.30 ), the solution for j = 2 giv en in Prop osition 4.9 do es not contain an additive multiple of r L , h 0 ( y ), and so can b e c haracterised as the particular solution with this prop erty . Moreo ver, a consistency chec k on Prop osition 4.9 is that ( 4.30 ) with j = 2 is indeed satisﬁed. This can b e veriﬁed with the aid of computer algebra. 21 5 The LOE and LSE edge densities Denote by ρ L β E (1) ,β ,N ( x ) the densit y of the Laguerre β ensemble; for even β , it is known from [ 48 ] to satisfy a diﬀerential equation: D L β E N ρ L β E (1) ,β ,N ( x ) = 0 , (5.1) where D L β E N is a linear diﬀerential op erator of order β + 1. By a duality satisﬁed by the sp ectral momen ts [ 21 , 32 ], the same equation c haracterises the density with β → 4 /β , up to mappings of N and a . Explicit forms of the op erators are giv en for β = 1 , 2 , 4 in [ 48 ]; recall ( 4.1 ) for the case of β = 2 (LUE) and see [ 48 , Eq. (2.29)], with the proviso that the Laguerre weigh t is taken as x a e − x instead of as x a e − β x/ 2 as in ( 1.2 ), for β = 1 , 4. W e now study the cases β = 1 (LOE) and β = 4 (LSE). In the interest of eﬃciency of presen tation, atten tion will be fo cused on tw o classes of edge setting (for the LUE, we considered four): the righ t soft edge with a = β 2 ( γ − 1) N + ( β 2 − 1), as is relev an t to Wishart matrices in multidimensional statistics and considered in the presen t context in [ 9 , 10 ], and the hard edge. 5.1 Righ t soft edge — the case a = β 2 ( γ − 1) N + ( β 2 − 1) Let ˆ N ′ s b e as in ( 4.15 ). F or this choice of a , changing v ariables in the diﬀerential op erators [ 48 , Eq. (2.29) with x 7→ β x/ 2] according to ( 4.14 ) to give a soft edge scaling gives the form 5 X k =0 1 ( ˆ N ′ s ) 2 k/ 3 D L ,β ,sr k ! ρ L β E ,sr (1) ,N ( y ) = 0 . (5.2) The op erators D L ,β ,sr k , determined with the aid of computer algebra, are listed as a prop osition in the following. Prop osition 5.1. With r efer enc e to ( 5.2 ), and with τ as in ( 4.15 ) , we have D L ,β ,sr 0 = 4 β d 5 dy 5 − 20 y d 3 dy 3 + 12 d 2 dy 2 + 16 β y 2 d dy − 8 β y , (5.3) D L ,β ,sr 1 = 20 τ β y d 5 dy 5 + 40 τ β d 4 dy 4 − 5(4 + 12 τ ) y 2 d 3 dy 3 + (24 − 52 τ ) y d 2 dy 2 . + 16 β (2 + τ ) y 3 d dy + 2(16 τ − 12) d dy − 8 β (3 − τ ) y 2 , (5.4) D L ,β ,sr 2 = 40 τ 2 y 2 β d 5 dy 5 + 160 τ 2 y β d 4 dy 4 +  93 τ 2 β − 60 τ y 3 − 60 τ 2 y 3  d 3 dy 3 −  16 τ + 140 τ 2  y 2 d 2 dy 2 +  16 β y 3 + 32 β τ y 3 − 8 τ  y d dy −  16 β y 3 − 8 β τ y 3 + 16 τ − 10 τ 2  , (5.5) 22 D L ,β ,sr 3 = 40 τ 3 y 3 β d 5 dy 5 + 240 τ 3 y 2 β d 4 dy 4 +  279 τ 3 β − 60 τ 2 y 3 − 20 τ 3 y 3  y d 3 dy 3 +  38 τ 3 β − 104 τ 2 y 3 − 76 τ 3 y 3  d 2 dy 2 +  16 β τ y 3 − 8 τ 2 − 32 τ 3  y 2 d dy −  12 τ 2 − 2 τ 3  y , (5.6) D L ,β ,sr 4 = 20 τ 4 y 4 β d 5 dy 5 + 160 τ 4 y 3 β d 4 dy 4 +  279 τ 4 β − 20 τ 3 y 3  y 2 d 3 dy 3 +  76 τ 4 β − 64 τ 3 y 3  y d 2 dy 2 −  τ 4 β + 24 τ 3 y 3  d dy − 4 τ 3 y 2 , (5.7) D L ,β ,sr 5 = τ 5  4 y 5 β d 5 dy 5 + 40 y 4 β d 4 dy 4 + 93 y 3 β d 3 dy 3 + 38 y 2 β d 2 dy 2 − y β d dy + 1 β  . (5.8) Comparing ( 5.3 ) with the ﬁrst op erator in ( 3.11 ) sho ws that they are equal, D L ,β ,sr 0 = D G ,β 0 . This is in keeping with the limiting soft edge density b eing the same for the Gaussian and Laguerre ensembles, dep ending only on the symmetry class ( β v alue). A feature of the explicit forms of D L ,β ,sr k for k = 1 , 2 is that for τ → 0, D L ,β ,sr k → D G ,β k , where the D G ,β k are sp eciﬁed in Prop osition 3.2 (for k > 2, the latter v anish). This prop ert y is consistent with [ 10 , Remark 4.1]. One observes to o that analogous to a property of the diﬀerential op erators ( 3.11 ), we hav e that the scaled op erators ˜ D L ,β k := β ( k − 2) / 3 D L ,β ,sr k    y 7→ β − 1 / 3 y (5.9) are independent of β . This suggests pro ceeding analogous to ( 3.3 ) and introducing the scaled expansion β 1 6 ρ L ,β ,sr (1) ,N ( y ) = r L ,β , sr 0 ( β 1 3 y ) + ( p β ˆ N ′ s ) − 2 / 3 r L ,β , sr 1 ( β 1 3 y ) + ( p β ˆ N ′ s ) − 4 / 3 r L ,β , sr 2 ( β 1 3 y ) + · · · . (5.10) F rom this expansion, the analogue of ( 3.14 ) in Corollary 3.3 can b e form ulated. Corollary 5.2. F or j = 0 , 1 , . . . we have 5 X k =0 ˜ D L ,β k r L ,β , sr j − k ( y ) = 0 , (5.11) wher e r L ,β , sr l ( y ) = 0 for l < 0 . As is consisten t with the fact that ˜ D L ,β 0 = ˜ D G ,β 0 , one has for b oth β = 1 and 4 that r L ,β , sr 0 ( y ) = r G ,β 0 ( y ), with the latter as sp eciﬁed by ( 3.4 ) and T able 2 . Moreo ver, we kno w from [ 10 ] that the analogue to ( 3.4 ) holds: r L ,β ,sr j ( y ) = α L ,β , sr j ( y )(Ai( y )) 2 + β L ,β , sr j ( y )(Ai ′ ( y )) 2 + γ L ,β , sr j ( y )Ai( y )Ai ′ ( y ) + ξ L ,β , sr j ( y )Ai( y )AI ν ( y ) + η L ,β , sr j ( y )Ai ′ ( y )AI ν ( y ) (5.12) for certain p olynomial coeﬃcients α L ,β , sr j ( y ) , . . . , η L ,β , sr j ( y ). F or j = 0 , 1 , 2, these are sp eciﬁed explicitly in [ 10 , Th. 4.1]. F or j = 0, we read these p olynomials oﬀ from the ﬁrst ro w of T able 2 . F or j = 1, they are sp eciﬁed by α L ,β , sr 1 ( y ) = 2 τ − 1 2 y 2 , β L ,β , sr 1 ( y ) = − 2(2 τ − 1) 5 y , γ L ,β , sr 1 ( y ) = 3 − τ 10 , ξ L ,β , sr 1 ( y ) = − 3 τ + 1 10 y , η L ,β , sr j ( y ) = − 2 τ − 1 10 y 2 . (5.13) 23 Setting τ = 0 reclaims the v alues of these co eﬃcients for β = 1 , 4 as sp eciﬁed in T able 2 ; recall the discussion in the paragraph b elow ( 5.8 ). They are lengthier again for j = 2 and we refer to [ 10 ] for their explicit form. Results from [ 9 , § 6] and [ 10 ] also relate r L ,β ,sr j ( y ) for j = 1 , 2 (and j = 3 to o) to r L ,β ,sr 0 ( y ) via a diﬀerential op erator. F or example, d dy  −  2 τ − 1 5  y 2 + τ − 3 5 d dy  r L ,β ,sr 0 ( y ) = r L ,β ,sr 1 ( y ) , (5.14) whic h for τ = 0 reduces to ( 3.7 ). 5.2 The hard edge In this subsection, w e consider the hard edge scaled densit y for the LOE/LSE. T o present the results in a uniﬁed wa y , we adopt a conv en tion similar to that used in [ 10 ] for the weigh t functions: w β ( x ) := ( x a − 1 2 e − x , β = 1 , x a +1 e − x , β = 4 , (5.15) where a is ﬁxed at the hard edge. Denote b y ρ L β E , h (1) ,N ( y ) the corresp onding density scaled in the hard edge v ariable ( 4.25 ). In tro duce to o the mo diﬁcation of ( 4.25 ), ˆ N ′ h = ( N + ( a − 1) / 2 , β = 1 , N + ( a + 1) / 4 , β = 4 . (5.16) Then, as a consequence of kno wledge of the explicit form of the op erator D L β E N in ( 5.1 ) for β = 1 and 4, we hav e the following result for the expansion of the density with resp ect to ( 5.16 ). Prop osition 5.3. L et ˜ a = a 2 − 1 , (5.17) and sp e cify ˆ N ′ h by ( 5.16 ) . F or the diﬀer ential op er ator D L β E N in ( 5.1 ) with β = 1 , 4 , after a tr ansformation of the L aguerr e weight function x a e − x → w β ( x ) as sp e ciﬁe d by ( 5.15 ) and intr o duction of the har d e dge sc ale d variable, we have D L β E N    x = y / (4 ˆ N ′ h ) w → w β = ˜ a 2 − (4 + 3˜ a ) y + 2 y 2 +  4 y 2 − 4( − 3 + ˜ a ) y − 16 − 14˜ a + ˜ a 2  y d dy +  38 y + 16 − 22˜ a  y 2 d 2 dy 2 +  10 y + 88 − 5˜ a  y 3 d 3 dy 3 + 40 y 4 d 4 dy 4 + 4 y 5 d 5 dy 5 ! + y 2 (2 √ β ˆ N ′ h ) 2 ˜ a − y + 2( − 2 + ˜ a − 2 y ) y d dy − 16 y 2 d 2 dy 2 − 5 y 3 d 3 dy 3 ! + y 5 (2 √ β ˆ N ′ h ) 4 d dy := D L β E , h 0 + D L β E , h 1 (2 p β ˆ N ′ h ) − 2 + D L β E , h 2 (2 p β ˆ N ′ h ) − 4 . (5.18) Thus, the har d e dge density for the LOE/LSE satisﬁes  D L β E , h 0 + D L β E , h 1 (2 p β ˆ N ′ h ) − 2 + D L β E , h 2 (2 p β ˆ N ′ h ) − 4  ρ L β E , h (1) ,N ( y ) = 0 (5.19) and so p ermits the lar ge ˆ N ′ h exp ansion ρ L β E , h (1) ,N ( y ) = r L , h ,β 0 ( y ) + 1 (2 √ β ˆ N ′ h ) 2 r L , h ,β 1 ( y ) + 1 (2 √ β ˆ N ′ h ) 4 r L , h ,β 2 ( y ) + · · · . (5.20) 24 With r L , h ,β − 1 ( y ) = r L , h ,β − 2 ( y ) = 0 , the suc c essive terms ar e r elate d by D L β E , h 0 r L , h ,β j ( y ) = −D L β E , h 1 r L , h ,β j − 1 ( y ) − D L β E , h 2 r L , h .β j − 2 ( y ) , ( j = 0 , 1 , 2 , . . . ) . (5.21) W e remark that the op erator D L β E , h 0 w as rep orted earlier in [ 48 , Th. 4.2] with κx = y and a mo diﬁed meaning of ˜ a . It is known from [ 28 ], [ 24 , Eq. (7.110) and (7.155)] that for β = 1 , 4, r L , h ,β 0  y 2  = 1 2  J a ( √ y ) 2 − J a +1 ( √ y ) J a − 1 ( √ y )  + J a ( √ y ) 2 √ y JI a ( √ y ) , (5.22) where JI a ( y ) := Z ∞ y J a ( t ) dt. (5.23) One notes that the right hand side of ( 5.22 ) is indep enden t of β ; compare the β -indep enden t deﬁnition of JI a ( y ) in ( 5.23 ) with the β -dep endent deﬁnition of AI ν ( y ) in ( 3.5 ). There is nonetheless indirect β -dep endence through the parameter a due to our redeﬁnition of the weigh t w β ( x ) given in ( 5.15 ). The functional form ( 5.22 ) suggests introducing the further basis functions b 4 ( y ) = 1 √ y J a ( √ y )JI a ( √ y ) , b 5 ( y ) = J ′ a ( √ y )JI a ( √ y ) (5.24) (cf. the basis functions on the second line of ( 3.4 )), for whic h the analogue of ( 4.35 ) is given by D y b 4 ( y ) = 1 2 ( b 4 ( y ) + b 5 ( y ) − y b 1 ( y )) , D y b 5 ( y ) = 1 2 ( b 5 ( y ) − ( y − a 2 ) b 4 ( y ) − y b 3 ( y )) . (5.25) The facts that the span of { b 1 ( y ) , . . . , b 5 ( y ) } with polynomial co eﬃcien ts is closed under the op eration D y = d dy y and that r L , h ,β 0 ( y / 2) is expressible as such a linear combination encourages us to seek solutions of ( 5.21 ) of a form extending ( 4.33 ), r L , h ,β j  y 2  = α L , h ,β j ( y ) b 1 ( y ) + β L , h ,β j ( y ) b 2 ( y ) + γ L , h ,β j ( y ) b 3 ( y ) + ξ L , h ,β j ( y ) b 4 ( y ) + η L , h ,β j ( y ) b 5 ( y ) , (5.26) where α L , h ,β j ( y ) , . . . , η L , h ,β j ( y ) are p olynomials, the same for b oth β = 1 , 4. Comparing with ( 5.22 ) shows that for the case j = 0, α L , h ,β 0 ( y ) = y − a 2 2 , β L , h ,β 0 ( y ) = 1 2 , γ L , h ,β 0 ( y ) = 0 , ξ L , h ,β 0 ( y ) = 1 2 , η L , h ,β 0 ( y ) = 0 . (5.27) With knowledge of ( 5.27 ), and aided by computer algebra, we can compute a particular solution of the ﬁfth order inhomogeneous equation D L β E , h 0 Y ( y ) = −D L β E , h 1 r L , h ,β 0 ( y ) (5.28) corresp onding to the case j = 1 of ( 5.21 ), and we can further identify the solution of the homogeneous equation in this class. Prop osition 5.4. Solutions of ( 5.28 ) with Y ( y / 2) of the form ( 5.26 ) have the structur e Y ( y / 2) = C r L , h ,β 0 ( y / 2) + P ( y / 2) (5.29) wher e C is a c onstant and P ( y / 2) is such that the c o eﬃcients in ( 5.26 ) do not c ontain an additive multiple of ( 5.27 ). Mor e over, P ( y / 2) = −  ( y − a 2 ) 2 + a 4 − 2 a 2 16  b 1 ( y ) − y − 3 a 2 + 3 24 b 2 ( y ) − y 12 b 3 ( y ) + y 48 b 4 ( y ) − y + 2 a 2 − 2 48 b 5 ( y ) . (5.30) 25 In the previous cases considered, w e ha ve found that b oth the ﬁrst and second corrections are solutions of the corresp onding inhomogeneous equation with no contribution from the solution of the homogeneous equation. Sp ecialising to β = 1 and making use of the kno wn ﬁnite N expression [ 2 ], [ 24 , Eq. (7.154)], [ 10 ] 1 2 ρ L β E , h (1) ,N  y 2     β =1 = ρ LUE , h (1) ,N − 1 ( y ) − 1 4 ˆ N ′ h ( N − 1)! 4Γ( a − 1 + N ) x ( a − 1) / 2 e − x/ 2 L ( a ) N − 1 ( x ) ×  2 Z x 0 L ( a ) N − 2 ( u ) u ( a − 1) / 2 e − u/ 2 du − Γ(( N + 1) / 2)Γ( a − 1 + N ) 2 a/ 2 − 3 / 2 Γ( N )Γ(( N + a ) / 2)      x = y / (4 ˆ N ′ h ) , (5.31) w e will inv estigate the v alidity of that prop erty in the present setting. Thus, our task is to compute the large ˆ N ′ h = N + ( a − 1) / 2 expansion of ( 5.31 ), making explicit the O(( ˆ N ′ h ) − 2 ) ﬁrst order correction, which sub ject to the structural hypothesis ( 5.26 ) m ust b e of the form ( 5.29 ). Prop osition 5.5. Consider the exp ansion ( 5.20 ) for β = 1 . As with the le ading functional form ( 5.22 ), after r eplacing the ar gument y by y / 2 , the ﬁrst c orr e ction has the functional form ( 5.26 ). With r L , h ,β 0 ( y / 2) sp e ciﬁe d by the c o eﬃcients ( 5.27 ), and P ( y / 2) sp e ciﬁe d by ( 5.30 ), we have that r L , h ,β 1 ( y / 2) is sp e ciﬁe d by the right hand side of ( 5.29 ) with C = (1 − a 2 ) / 4 . Thus, fr om the viewp oint of the underlying inhomo gene ous diﬀer ential e quation, ther e is a c ontribution fr om the solution of the homo gene ous p art. Sp e ciﬁc al ly, r L , h ,β 1 ( y / 2) is given by ( 5.26 ) with c o eﬃcients α L , h ,β 1 ( y ) = 2 y − y 2 16 , β L , h ,β 1 ( y ) = − y 24 , γ L , h ,β 1 ( y ) = − y 12 , ξ L , h ,β 1 ( y ) = y − 6 a 2 + 6 48 , η L , h ,β 1 ( y ) = − y + 2 a 2 − 2 48 . (5.32) Pr o of. Starting from the explicit functional form ( 5.31 ), this requires an asymptotic analysis similar to that used for the pro of of Prop osition 4.9 . While the latter was to one order higher, in the presen t problem we are faced with the task of simplifying v arying integrals. F or this, as is seen in our detailed working of App endix E, essential use is made of the second order diﬀeren tial equation satisﬁed b y J a ( u ) (this gives a mechanism explaining wh y the asymptotic expansion has the structure ( 5.26 )). W e conclude with an analogue of Corollary 4.7 . Corollary 5.6. In addition to the diﬀer ential r elation ( 5.21 ) with j = 1 , we have the explicit diﬀer ential r elation r L , h ,β 1  y 2  = − 1 12  8 D 2 y r L , h ,β 0  y 2  + D y  y r L , h ,β 0  y 2  + 2( a 2 − 5) D y r L , h ,β 0  y 2  . (5.33) F or β = 1 , 4 and w β ( x ) sp e ciﬁe d by ( 5.15 ), let the gener ating function of ther e b eing k ( k = 0 , 1 , 2 , . . . ) eigenvalues in the interval (0 , s ) b e exp ande d for lar ge ˆ N ′ h ac c or ding to E N ,β ((0 , y / (4 ˆ N ′ h )); w β ( x ); ξ ) = E h β ((0 , y ); ξ ) + 1 (2 √ β ˆ N ′ h ) 2 E 1 , L , h β ((0 , y ); ξ ) + 1 (2 √ β ˆ N ′ h ) 4 E 2 , L , h β ((0 , y ); ξ ) + · · · . (5.34) Under the assumption that E h β ((0 , y ); ξ ) and E 1 , L , h β ((0 , y ); ξ ) ar e r elate d by a ξ -indep endent se c ond or der diﬀer ential op er ator applie d to the former, we have E 1 , L , h β ((0 , y ); ξ ) = − 1 6  4 y 2 d dy + y 2 + ( a 2 − 1) y  d dy E h β ((0 , y ); ξ ) . (5.35) 26 Pr o of. Using Prop osition 4.6 and its extension ( 5.25 ), along with the form of r L , h ,β 0 ( y / 2) given in ( 5.22 ), we compute D y r L , h ,β 0  y 2  = 1 4  ( J a ( √ y )) 2 + 1 √ y J a ( √ y )JI a ( √ y ) + J ′ a ( √ y )JI a ( √ y )  , D 2 y r L , h ,β 0  y 2  = 1 8  ( J a ( √ y )) 2 + √ y J a ( √ y ) J ′ a ( √ y ) + a 2 + 1 − y √ y J a ( √ y )JI a ( √ y ) + 2 J ′ a ( √ y )JI a ( √ y )  . Th us, we v erify ( 5.33 ). Changing v ariables by a factor of 2 produces a relation b et ween r L , h ,β 0 ( y ) and r L , h ,β 1 ( y ), which gives ( 5.35 ) upon applying the same reasoning as that leading to ( 1.17 ). R emark 5.7 . Recall from Remark 4.8 the exact results ( 4.41 ) and ( 4.42 ). With our w eight w β ( x ) b eing sp eciﬁed b y ( 5.15 ), the ﬁrst of these gives that when β = 1 and a = 1, or β = 4 and a = − 1, we ha v e E h β ((0 , y ); ξ ) | ξ =1 = e − y / 4 , E j, L , h β ((0 , y ); ξ ) | ξ =1 = 0 , ( j = 1 , 2 , · · · ) . This is indeed consistent with ( 5.35 ). On the other hand, setting β = 1 in the second of these results shows that when a = 3, we ha v e E h 1 ((0 , y ); ξ ) | ξ =1 = e − y / 4 0 F 1  2; y 2  , E 1 , L , h 1 ((0 , y ); ξ ) | ξ =1 = y 2 24 e − y / 4 0 F 1  3; y 2  . Calculation sho ws that this is again consistent with ( 5.35 ). A similar c heck with β = 4 and a = 0 also holds. Ac kno wledgemen ts The w ork of PJF is supp orted b y a gran t from the Australian Research Council, Discov ery Pro ject DP250102552. AAR is supp orted by Hong Kong R GC grants GRF 16304724 and GRF 17304225. BJS is supp orted by the Shanghai Jiao T ong Universit y Overseas Joint P ostdo c- toral F ellowship Program. Corresp ondence on the topic of this w ork from F. Bornemann is appreciated. App endix A: An orthogonal p olynomial approac h to the third order diﬀerential equations Here, we consider the classical unitary ensembles, with eigenv alue probabilit y density function prop ortional to N Y l =1 w ( x l ) 1 x l ∈ I Y 1 ≤ j − 1. Adopting the notation in [ 50 ], we denote Q (2 q , 1 − p ) N ( x ) := R ( η , ¯ η ) N ( x ) from now on. A.1 The raising and lo w ering op erators It is fundamental that orthogonal p olynomials satisfy the three term recurrence relation xp N ( x ) = p N +1 ( x ) + d N p N ( x ) + e N p N − 1 ( x ) , (A.5) and with classical weigh ts, the structure relation σ ( x ) p ′ N ( x ) = a N p N +1 ( x ) + b N p N ( x ) + c N p N − 1 ( x ); (A.6) with regards to the latter see, e.g., [ 2 ]. In these equations, a N , b N , c N , d N and e N are certain constan ts that can b e determined by comparing co eﬃcients on both sides. Eliminating one of the three terms on the right hand side results in the following equations  σ ( x ) d dx + a N ( d N − x ) − b N  p N ( x ) = ( c N − a N e N ) p N − 1 ( x ) , (A.7)  σ ( x ) d dx + c N e N ( d N − x ) − b N  p N ( x ) =  a N − c N e N  p N +1 ( x ) . (A.8) 12 In the Gaussian case, our deriv ation reduces to that given in [ 56 ]. 28 The op erators in the brack ets on the left hand side are usually called the lo wering and raising op erators, resp ectiv ely . F or the cases w e are considering, these relations can b e sp eciﬁed b y the follo wing: • Hermite P olynomials H n ( x ) H ′ n ( x ) = 2 nH n − 1 ( x ) , H ′ n ( x ) = 2 xH n ( x ) − H n +1 ( x ) . • Laguerre P olynomials L ( α ) n ( x ) xL ( α ) ′ n ( x ) = nL ( α ) n ( x ) − ( n + α ) L ( α ) n − 1 ( x ) , xL ( α ) ′ n ( x ) = ( n + 1) L ( α ) n +1 ( x ) − ( n + α + 1 − x ) L ( α ) n ( x ) . • Jacobi P olynomials P ( α,β ) n ( x ) (2 n + α + β )(1 − x 2 ) P ( α,β ) ′ n ( x ) = − n [(2 n + α + β ) x − ( β − α )] P ( α,β ) n ( x ) + 2( n + α )( n + β ) P ( α,β ) n − 1 ( x ) , (2 n + α + β )(1 − x 2 ) P ( α,β ) ′ n ( x ) = ( n + α + β ) [(2 n + α + β ) x + ( α − β )] P ( α,β ) n ( x ) − 2( n + 1)( n + α + β + 1) P ( α,β ) n +1 ( x ) . • R.-R. p olynomials Q ( α,β ) n ( x ) Q ( α,β ) ′ n ( x ) = n (2 β + n − 1) Q ( α,β ) n − 1 ( x ) , (1 + x 2 ) Q ( α,β ) ′ n ( x ) = Q ( α,β ) n +1 ( x ) − (2( β − n − 1) x + α ) Q ( α,β ) n ( x ) . A.2 Diﬀeren tial equations satisﬁed by the eigenv alue densities of classical ensem bles Let K N ( x, x ) b e sp eciﬁed b y ( A.2 ). F or simplicit y , w e deﬁne the operator θ : f ( x ) 7→ d dx ( σ ( x ) f ( x )), whic h will b e used extensiv ely . As a simple consequence of equation ( A.4 ), w e hav e θ K N ( x, x ) = ( λ N − λ N − 1 ) w ( x ) h N − 1 p N ( x ) p N − 1 ( x ) . (A.9) F urther applications of θ to K N ( x, x ) giv e θ 2 K N ( x, x ) = τ ( x ) θ K N ( x, x ) + ( λ N − λ N − 1 ) σ ( x ) w ( x ) h N − 1 ( p ′ N ( x ) p N − 1 ( x ) + p N ( x ) p ′ N − 1 ( x )) = τ ( x ) θ K N ( x, x ) + ( λ N − λ N − 1 ) σ ( x ) K N ( x, x ) + 2( λ N − λ N − 1 ) σ ( x ) w ( x ) h N − 1 p N ( x ) p ′ N − 1 ( x ) , (A.10) θ 3 K N = τ ′ σ θ K N + τ θ 2 K N + ( λ N − λ N − 1 ) w h N − 1 ( σ ′ ( p N p N − 1 ) ′ + ( λ N + λ N − 1 ) σ p N p N − 1 + 2 σ 2 p ′ N p ′ N − 1 ) = τ ′ σ θ K N + τ θ 2 K N + σ ′ ( θ 2 K N − τ θ K N ) + ( λ N + λ N − 1 ) σ θ K N + 2( λ N − λ N − 1 ) σ 2 w h N − 1 p ′ N p ′ N − 1 , (A.11) 29 where τ and σ are the p olynomials in Pearson’s equation ( A.3 ). Equations ( A.2 ),( A.9 ),( A.10 ) and ( A.11 ) are linear equations of ﬁve unkno wn quan tities p ′ N p ′ N − 1 , p N p ′ N − 1 , p ′ N p N − 1 , p N p N − 1 , K N . In order to obtain an diﬀerential equation for K N , w e need an extra equation to form a closed system. This can b e accomplished through the raising and low ering operators for the classical orthogonal p olynomials. By m ultiplying together b oth sides of equation ( A.7 ) and equation ( A.8 ), with the latter ha ving N replaced by N − 1, we obtain ( c N − a N e N )( a N − 1 e N − 1 − c N − 1 ) p N − 1 p N =( c N ( d N − x ) − b N e N ) σ p N p ′ N − 1 + ( a N − 1 ( d N − 1 − x ) − b N − 1 ) e N σ p ′ N p N − 1 + ( c N ( d N − x ) − b N e N )( a N − 1 ( d N − 1 − x ) − b N − 1 ) p N p N − 1 + e N σ 2 p ′ N p ′ N − 1 . (A.12) This can b e further sp eciﬁed according to the following: • Hermite P olynomials H N ( x ) H ′ N H ′ N − 1 − 2 xH ′ N H N − 1 = − 2 N H N H N − 1 . • Laguerre P olynomials L ( α ) N ( x ) xL ( α ) ′ N L ( α ) ′ N − 1 + ( N + α − x ) L ( α ) ′ N L ( α ) N − 1 − N L ( α ) N L ( α ) ′ N − 1 = − N L ( α ) N L ( α ) N − 1 . • Jacobi P olynomials P ( α,β ) N ( x ) Let D N = 2 N + α + β , we ha ve D N ( D N − 2)(1 − x 2 ) P ( α,β ) ′ N P ( α,β ) ′ N − 1 + N ( D N − 2) [ D N x + ( α − β )] P ( α,β ) N P ( α,β ) ′ N − 1 + 4 N ( N + α )( N + β )( N + α + β − 1) P ( α,β ) N P ( α,β ) N − 1 = D N ( N + α + β − 1) [( D N − 2) x + ( α − β )] P ( α,β ) ′ N P ( α,β ) N − 1 . • R.-R. p olynomials Q ( α,β ) N ( x ) ( x 2 + 1) Q ( α,β ) ′ N − 1 Q ( α,β ) ′ N + (2( β − N ) x + α ) Q ( α,β ) N − 1 Q ( α,β ) ′ N = 2(2 β + N − 1) Q ( α,β ) N − 1 Q ( α,β ) N . Therefore, eliminating the terms from equation ( A.2 ),( A.9 ),( A.10 ),( A.11 ) and ( A.12 ) leads to a third order diﬀerential equation for the density ρ (1) ,N ( x ) of the classical unitary ensembles. App endix B: Direct determination of { u j ( γ ) } The bilateral Laplace transform of the soft edge scaled GUE density is deﬁned b y F N ( γ ) := Z ∞ −∞ e γ x ρ GUE , s (1) ,N ( x ) dx, ρ GUE , s (1) ,N ( x ) := 1 √ 2 N 1 / 6 ρ GUE (1) ,N ( √ 2 N + x/ ( √ 2 N 1 / 6 )) . (B.1) Substituting in ( 2.2 ) and recalling ( 2.15 ) we see that for large N , F N ( γ ) = u 0 ( γ ) + 1 N 2 / 3 u 1 ( γ ) + 1 N 4 / 3 u 2 ( γ ) + · · · . (B.2) Here, we would lik e to undertake a direct computation of the large N expansion of F N ( γ ), and th us of { u j ( γ ) } (at least for small j ). W e b egin by a simple change of v ariables to obtain from ( B.1 ) that F N ( γ ) = e − 2 γ N 2 / 3 Z ∞ −∞ e γ √ 2 N 1 / 6 y ρ GUE (1) ,N ( y ) dy . (B.3) 30 Using now a result of Ullah from 1985 [ 54 ] for the F ourier transform of the GUE densit y w e can ev aluate the integral to obtain F N ( γ ) = e − 2 γ N 2 / 3 e γ 2 N 1 / 3 / 2 L (1) N − 1 ( − γ 2 N 1 / 3 ) , (B.4) where L ( α ) N ( x ) denotes the Laguerre p olynomial. The task now is to obtain the large N expansion of the particular Laguerre p olynomial in ( B.4 ). F or this we in tro duce the contour in tegral form of the Laguerre p olynomial L (1) N − 1 ( − z ) = e − z 2 π i 1 c I | ( s/c ) − 1 | = R  1 − c s  − N e z s/c ds, (B.5) where R > 0 , c > 0 are arbitrary; see [ 14 , § 2]. This with c = γ / N 1 / 3 and z = γ 2 N 1 / 3 substituted in ( B.4 ) gives F N ( γ ) = e − 2 γ N 2 / 3 e γ 2 N 1 / 3 / 2 N 1 / 3 γ 1 2 π i I | ( N 1 / 3 s/γ ) − 1 | = R  1 − γ N 1 / 3 s  − N e γ N 2 / 3 s ds. (B.6) W e remark that from this integral representation, the large N expansion ( B.2 ) is far from eviden t. Nonetheless, we will sho w that at least for small orders it provides a systematic computation scheme. With a view to applying the saddle p oint metho d, w e expand  1 − γ N 1 / 3 s  − N = exp  N ∞ X l =1 1 l  γ N 1 / 3 s  l  . (B.7) Substituting in ( B.6 ) shows that the leading order exp onen tial factors in the integrand are e γ N 2 / 3 ( s +1 /s ) . This has critical points s = ± 1, of whic h s = 1 maximises these factors. Hence, according to the saddle p oint method, the contour of integration should b e deformed to pass through this point. Since the second deriv ative of s + 1 /s is p ositiv e, the direction of steepest descen t is parallel to the imaginary axis, which tells us to introduce the v ariable t according to ( s − 1) = it/ ( N 1 / 3 γ 1 / 2 ). Doing this, the leading order exp onential factors reduce to e 2 γ N 2 / 3 e − t 2 times terms lo w er order in N . An asymptotic expansion to all algebraic orders in N requires that the full expression ( B.7 ) b e similarly expanded ab out s = 1. With α k,p := ∞ X l = p +1 ( l ) k l  γ N 1 / 3  l , (B.8) a straightforw ard calculation gives from the expansion of the integrand about s = 1, and the in tro duction of the v ariable t , the asymptotic expression F N ( γ ) ∼ e γ 3 / 12 2 π γ 3 / 2 Z ∞ −∞ e − ( t + iγ 3 / 2 / 2) 2 e N ( α 0 , 3 + α 1 , 2 ( − it/ ( N 1 / 3 γ 1 / 2 ))+ α 2 , 1 ( − it/ ( N 1 / 3 γ 1 / 2 )) 2 / 2!) × ∞ Y j =3 e N α j, 0 ( − it/ ( N 1 / 3 γ 1 / 2 )) j /j ! dt. (B.9) All factors in the in tegrand, apart from the ﬁrst, go to zero lik e p ow ers of N − 1 / 3 . Expliciltly , to ﬁrst order in N − 1 / 3 , we calculate F N ( γ ) ∼ e γ 3 / 12 2 π γ 3 / 2 Z ∞ −∞ e − ( t + iγ 3 / 2 / 2) 2  1 + N − 1 / 3  γ 4 4 − iγ 5 / 2 t − 3 2 γ t 2 + i γ 1 / 2 t 3  dt. (B.10) 31 Changing v ariables t 7→ t − iγ 3 / 2 / 2, and noting that all o dd monomials integrate to zero against e − t 2 , we are left with only the even monomials to consider. F or the term of order N − 1 / 3 , as can b e read oﬀ from ( B.10 ), these terms are seen to add up to zero. Thus, we conclude that there is no term in the asymptotic expansion at order N − 1 / 3 , in agreement with ( 2.2 ). The calculation of the asymptotic expansion to higher orders in N − 1 / 3 starting from ( B.9 ) is w ell suited to the use of computer algebra. In particular, expanding up to and including order N − 2 , w e reclaim u 1 ( γ ), u 2 ( γ ) as presen ted in ( 2.11 ), and further u 3 ( γ ) as sp eciﬁed by ( 2.16 ) with b given by ( 2.17 ). R emark B.1 . F or large N but ﬁxed z it is known [ 55 ], [ 14 ] that L ( − a ) N ( − z ) ∼ e − z / 2 2 √ π e 2 √ N z z 1 / 4 − a/ 2 N 1 / 4+ a/ 2  1 +  3 − 12 a 2 + 24(1 − a ) z + 4 z 2 48 √ z  1 N 1 / 2 + O  1 N   . (B.11) In comparison, from ( B.4 ) and ( B.10 ) we hav e L (1) N ( − γ 2 N 1 / 3 ) ∼ e 2 γ N 2 / 3 e − γ 2 N 1 / 3 / 2 e γ 3 / 12 2 √ π γ 3 / 2  1 + O( N − 2 / 3 )  . (B.12) F ormally substituting z = γ 2 N 1 / 3 in ( B.11 ) with a = − 1 gives L (1) N ( − γ 2 N 1 / 3 ) ∼ e 2 γ N 2 / 3 e − γ 2 N 1 / 3 / 2 1 2 √ π γ 3 / 2  1 + γ 3 12 + · · ·  . (B.13) This is consisten t with ( B.12 ) if it should b e that each successiv e term in the N − 1 / 2 expansion inside the main brac kets in ( B.11 ) should give successiv e terms in the p o wer series expansion of e γ 3 / 12 , as is the case with the display ed term. App endix C: A σ -P ainlev ´ e I I I ′ transcenden t v eriﬁcation of ( 4.39 ) Let σ 0 ( x ; ξ ) b e the σ -Painlev ´ e I I I ′ transcenden t sp eciﬁed as the solution of the particular σ - P ainlev´ e I I I ′ equation ( xσ ′′ ) 2 + σ ′ (1 + 4 σ ′ )( xσ ′ − σ ) = ( aσ ′ ) 2 (C.1) sub ject to the b oundary condition σ 0 ( x ; ξ ) ∼ x → 0 + − ξ xr L , h 0 ( x ) . (C.2) Let σ 1 ( x ; ξ ) b e characterised as satisfying a particular second order linear diﬀerential equation, A ( x ) σ ′′ 1 + B ( x ) σ ′ 1 + C ( x ) σ 1 = D ( x ) , (C.3) sub ject to the b oundary condition σ 1 ( x ; ξ ) ∼ x → 0 + − ξ xr L , h 1 ( x ) . (C.4) In ( C.3 ), the functions A ( x ) , . . . , D ( x ) dep end on σ 0 and/or its ﬁrst and second deriv ative; their explicit forms are given in [ 35 , Eqns. (2.27) and (2.31)]. W e hav e from [ 52 , 38 ] that E h 2 ((0 , y ); ξ ) = exp  Z y 0 σ 0 ( x ; ξ ) dx x  , (C.5) and from [ 35 ] that E 1 , L , h 2 ((0 , y ); ξ ) = exp  Z y 0 σ 1 ( x ; ξ ) dx x  (C.6) 32 (in [ 35 ] σ 1 ( x ; ξ ) is denoted ˆ σ 2 ( x ; ξ )). Substituting in ( 4.39 ) shows that the latter is v alid provided  1 48 y + a 2 − 2 24  σ 0 + 1 12  y σ ′ 0 + σ 2 0  = Z y 0 σ 1 ( x ; ξ ) dx x . (C.7) Diﬀeren tiating this expresses σ 1 ( y ; ξ ) in terms of { y , σ 0 , σ ′ 0 , σ ′′ 0 } . Substituting ( C.2 ) and ( C.4 ) regarded as functional forms for small ξ gives precisely the diﬀeren tial relation ( 4.37 ) express- ing r L , h 1 ( y ) in terms of r L , h 0 ( y ). It then remains to establish ( C.3 ) with σ 1 ( y ; ξ ) so expressed. Analogous tasks ha ve b een met in [ 29 , 27 , 33 ]. The new form of ( C.3 ) is an identit y inv olving y , σ 0 and its ﬁrst four deriv atives. The third and fourth deriv atives can b e eliminated by dif- feren tiating ( C.7 ) an appropriate num b er of times. This lea ves a diﬀerential identit y inv olving { y , σ 0 , σ ′ 0 , σ ′′ 0 } . The use of computer algebra allo ws for the latter to b e factorised, with one of the factors v anishing due to ( C.1 ). App endix D: Hard edge scaling and h yp ergeometric p olynomials D.1 Scaled large N expansion of L ( α ) N ( x/ N ) The Laguerre p olynomials are expressed as particular hypergeometric p olynomials according to L ( α ) N ( x ) =  N + α N  1 F 1 ( − N ; α + 1; x ) . (D.1) Here, we consider the large N expansion of the hypergeometric polynomial in ( D.1 ) with x 7→ x/ N , up to and including terms of order 1 / N 4 . Prop osition D.1. L et D p x := x p d p dx p . F or lar ge N , we have 1 F 1 ( − N ; α + 1; x/ N ) =  1 − 1 N 1 2 D 2 x + 1 N 2  1 8 D 4 x + 1 3 D 3 x  − 1 N 3  1 48 D 6 x + 1 6 D 5 x + 1 4 D 4 x  + 1 N 4  1 384 D 8 x + 1 24 D 7 x + 13 72 D 6 x + 1 5 D 5 x  + · · ·  0 F 1 ( α + 1; − x ) (D.2) and 1 F 1 ( − N + 1; α + 1; x/ N ) =  1 − 1 N  1 2 D 2 x + D x  + 1 N 2  1 8 D 4 x + 5 6 D 3 x + D 2 x  − 1 N 3  1 48 D 6 x + 7 24 D 5 x + 13 12 D 4 x + D 3 x  + 1 N 4  1 384 D 8 x + 1 16 D 7 x + 17 36 D 6 x + 77 60 D 5 x + D 4 x  + · · ·  0 F 1 ( α + 1; − x ) (D.3) Pr o of. Consider ﬁrst ( D.2 ). W e use the facts that 1 F 1 ( − N ; α + 1; x/ N ) = ∞ X p =0 ( − N ) p p !( α + 1) p  x N  p , ( − N ) p N p = ( − 1) p p − 1 Y l =1  1 − l N  , and then the expansion p − 1 Y l =1  1 − l N  = 1 − 1 N p − 1 X l =1 l + 1 N 2 X 1 ≤ l 1

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