Strong asymptotics for the Pollaczek multiple orthogonal polynomials ensembles
A. I. Aptekarev, G. Lopez Lagomasino, A. Martinez-Finkelshtein
SStrong asymptotics for the Pollaczek multiple orthogonalpolynomials ensembles
A. I. Aptekarev, G. L´opez Lagomasino, and A. Mart´ınez–FinkelshteinAugust 24, 2018
Abstract
We study the asymptotic properties of a class of multiple orthogonal polynomials withrespect to a Nikishin system generated by two measures ( σ , σ ) with unbounded supports(supp( σ ) ⊂ R + , supp( σ ) ⊂ R − ), and such that the second measure σ is discrete. Theweak asymptotics for these polynomials was obtained by Sorokin in [40]. We use his resultand the Riemann–Hilbert analysis to derive the strong asymptotics of these polynomialsand of the reproducing kernel. If we are given p weight functions w , . . . , w p : R → R with finite moments and a multi-index (cid:126)n = ( n , . . . , n p ) ∈ Z o + \ { (cid:126) } , the polynomials satisfying the orthogonality relations (cid:90) ∞−∞ P (cid:126)n ( x ) x k w j ( x ) dx = 0 for k = 0 , . . . , n j − , j = 1 , . . . , p, (1)are known as multiple orthogonal polynomials (or MOP) of type II. These polynomials appearin a natural way in certain models of random matrices and non-intersecting paths, fact thatwas observed first in [10] for the Hermitian random matrix model with external source. Thegeneral notion of a multiple orthogonal polynomial ensemble (generalizing in a certain sensethe well-known concept of a biorthogonal ensemble of A. Borodin [11]) was introduced recentlyin [28] (see also [3, 29]): Definition 1. A multiple orthogonal polynomial ensemble is a probability density functionon R n , with n = | (cid:126)n | = n + · · · + n p , of the form P ( x , . . . , x n ) = 1 Z n det (cid:104) x i − j (cid:105) i,j =1 ,...,n det [ ϕ i ( x j )] i,j =1 ,...,n (2)for certain functions ϕ , . . . , ϕ n : R → R whose linear span is equal tospan { x k w j ( x ) | k = 0 , . . . , n j − , j = 1 , . . . , p } . (3)We say that the MOP ensemble (2) is generated by the weight functions w , . . . , w p and themulti-index (cid:126)n = ( n , . . . , n p ). 1 a r X i v : . [ m a t h . C A ] O c t bviously, the necessary condition for the consistency of this definition is that the productin the right hand side of (2) has fixed sign for all ( x , . . . , x n ) ∈ R n , and that the normalizingconstant (“partition function”, chosen such that the integral of P over R n equals 1) satisfies Z n = (cid:90) R n det (cid:104) x i − j (cid:105) i,j =1 ,...,n det [ ϕ i ( x j )] i,j =1 ,...,n dx · · · dx n ∈ R \ { } . (4)This expression can be transformed (see, for example, in [3]) into a block Hankel determinant Z n = c n det (cid:2) H · · · H p (cid:3) with p rectangular blocks, where H j = (cid:20)(cid:90) ∞−∞ x i + k − w j ( x ) dx (cid:21) i =1 ,...,n, k =1 ,...,n j is of size n × n j and contains the moments of the weight w j .Being a multiple orthogonal polynomial ensemble a determinantal point process, there isa kernel K n such that (2) can be written as P ( x , . . . , x n ) = 1 n ! det [ K n ( x i , x j )] i,j =1 ,...,n . (5)In fact, K n ( x, y ) = n (cid:88) i =1 n (cid:88) j =1 (cid:2) A − n (cid:3) j,i x i − ϕ j ( y ) , (6)where (cid:2) A − n (cid:3) j,i denotes the ( ji )th entry of the inverse of the matrix A n = (cid:20)(cid:90) x i − ϕ j ( x ) dx (cid:21) i,j =1 ,...,n . With this notation, Z n = n ! det A n . Since Z n (cid:54) = 0, we see that the matrix A n is invertible,and the kernel (6) is well-defined.It is well known that in this context, for every k = 1 , . . . , n , R ( x , . . . , x k ) := n !( n − k )! (cid:90) R n − k P ( x , . . . , x n ) dx k +1 · · · dx n = det [ K n ( x i , x j )] i,j =1 ,...,k . (7)Also the (monic) multiple orthogonal polynomial of type II, P (cid:126)n , defined by (1), exists, isuniquely determined, and has the probabilistic interpretation of being the “average charac-teristic polynomial” of the ensemble (2): P (cid:126)n ( z ) = E n (cid:89) j =1 ( z − x j ) . (8)2his conclusion is based on the integral representation P (cid:126)n ( z ) = 1 Z n (cid:90) R n n (cid:89) j =1 ( z − x j ) (cid:89) i 2, form a Nikishin system N ( σ , σ ) generated by σ = µ , supported on R + , and the discrete measure σ whose support is contained in( −∞ , P (cid:126)n is uniquely determined up to a constantfactor and deg P (cid:126)n = n + n ; in other words, in our problem all indices (cid:126)n ∈ Z \ { (cid:126) } are normal . Using [22, Theorem 1.2] we also know that all zeros of P (cid:126)n are simple and lie in(0 , + ∞ ). In the sequel we normalize P (cid:126)n to be monic.Here we are interested in the re-scaled asymptotic behavior of the diagonal sequence ofpolynomials ( P (cid:126)n ), (cid:126)n = ( n, n ), n ∈ N . For simplicity, we adopt the notation P n = P (cid:126)n , withdeg P n = 2 n . The monic rescaled polynomials Q n ( x ) = c n P n (4 n x ) = x n + lower degree terms , c n = (4 n ) − n , (15)are characterized by the orthogonality conditions (cid:90) + ∞ x k Q n ( x ) w j,n ( x ) dx = 0 , k = 0 , . . . , n − , j = 1 , , (16)where w ,n ( z ) := 1sinh (cid:0) πnz / (cid:1) , w ,n ( z ) := 1 z / cosh (cid:0) πnz / (cid:1) . (17)Given a smooth oriented curve on the plane, we use the subindex + (resp., − ) to denotethe left (resp., the right) side of the curve and the boundary values of any function fromthe corresponding side induced by the given orientation. In the case of R we use standardorientation, so the +-side is reached from the upper half plane and the − -side from the lowerone. Also, unless we explicitly say otherwise, we adopt the convention that z / denotes themain branch of the square root in C \ R − , positive on R + , while √ x = x / , is its restrictionto x ≥ 0. In particular, the w j,n are holomorphic and non-vanishing in C \ R + . We wish to remark that σ is not finite and the two cited theorems from [22] require this assumption.However, it is easy to see that they are true also in our context: all what is really needed from the measure σ for their proof is the existence of (cid:82) ( z − x ) − dσ ( x ) for all z outside the support of σ , which is clearly thecase. emark 1. It might be convenient to point out the following relation with the multipleorthogonality considered in [30]. There, the orthogonality weights were (after an appropriaterescaling) w ,n ( x ) = x α/ exp (cid:18) − nxt (1 − t ) (cid:19) I α (cid:18) n √ axt (cid:19) , w ,n ( x ) = x ( α +1) / exp (cid:18) − nxt (1 − t ) (cid:19) I α +1 (cid:18) n √ axt (cid:19) . Using the definition of the modified Bessel function and setting here α = − / √ a = πt ,we get w ,n ( x ) = 1 π √ n exp (cid:18) − nxt (1 − t ) (cid:19) x w ,n ( x ) , w ,n ( x ) = 1 π √ n exp (cid:18) − nxt (1 − t ) (cid:19) w ,n ( x ) . In particular, with this choice of the parameters, w ,n ( x ) w ,n ( x ) = x w ,n ( x ) w ,n ( x ) , which explains the connections of our analysis in the following sections with that in [30].The strong asymptotics of the MOP Q n is described in the following result: Theorem 1. Let H ( ζ ) = ζ √ (cid:18) ζζ + ζ − (cid:19) / denote the holomorphic branch in C \ ( −∞ , ( − √ / , normalized by H (1) = 1 . Then:(i) for z ∈ C \ [0 , p + ] , with p + = (cid:16) √ − (cid:17) , Q n ( z ) = e ng ( z ) H ( ζ ( z )) (cid:18) O (cid:18) n ( | z | + 1) (cid:19)(cid:19) , (18) locally uniformly away from the interval [0 , p + ] .(ii) in a small neighborhood of (0 , p + ) in the upper half plane, Q n ( z ) = e ng ( z ) (cid:18) H ( ζ ( z )) + e nψ ( z ) H ( ζ ( z )) + O (cid:18) n (cid:19)(cid:19) . (19) In particular, on compact subsets of (0 , p + ) , Q n ( x ) = (cid:18) e ng ( x ) H ( ζ ( x )) + e ng − ( x ) H ( ζ − ( x )) + O (cid:18) n (cid:19)(cid:19) . (20)5 ere ζ , ζ are the holomorphic branches of the algebraic function ζ ( z ) defined by the equation z = 1 + ζζ (1 − ζ ) , normalized by ζ ( z ) = 1 + O (cid:18) z (cid:19) , ζ ( z ) = 1 z / + O (cid:18) z (cid:19) , z → ∞ , and g ( z ) = (cid:82) log( z − t ) dλ ( t ) is given in terms of the first component λ of the uniquesolution of a vector equilibrium problem described in Proposition 3 below. Remark 2. Taking into account the definition of g , formula (18) obviously implies the n -throot asymptotic result from [40]:lim n n log | Q n ( z ) | = −P λ ( z ) , where P λ is the logarithmic potential of λ defined in (24).Regarding the CD kernel K n , introduced in (6), we have Theorem 2. For the rescaled weights w j,n defined in (17) , the limiting mean density of thepositions of the particles from the corresponding multiple polynomial ensemble exists and issupported on [0 , p + ] : lim n →∞ n K n ( x, x ) = λ (cid:48) ( x ) , x ∈ (0 , p + ) , where λ has the same meaning as in Theorem 1.Moreover, for x ∗ ∈ (0 , p + ) , lim n →∞ nλ (cid:48) ( x ∗ ) K n (cid:18) x ∗ + xnλ (cid:48) ( x ∗ ) , x ∗ + ynλ (cid:48) ( x ∗ ) (cid:19) = sin π ( x − y ) π ( x − y ) , uniformly for x and y on compact subsets of R . The non-linear steepest descent analysis based on the Riemann-Hilbert characterizationof MOP (see Section 2) allows also to obtain the limit formulas for Q n and K n close to theendpoints of [0 , p + ]. We are not writing these formulas explicitly here, but an interestedreader can easily assemble them from the expressions appearing in Section 4. The starting point for our analysis is the Riemann-Hilbert interpretation of multiple orthog-onality (1), valid for the arbitrary multi-index (cid:126)n = ( n , n ).Consider the following Riemann-Hilbert problem (RHP). Given (cid:126)n ∈ Z \ { (cid:126) } find a 3 × (cid:98) Y , analytic in C \ R + , such that:6RH- Y (cid:98) Y has continuous boundary values on R + related by the jump condition (cid:98) Y + ( x ) = (cid:98) Y − ( x ) w ( x ) w ( x )0 1 00 0 1 , x ∈ (0 , + ∞ ) . (RH- Y (cid:98) Y ( z ) = (cid:0) I + O ( z − ) (cid:1) diag ( z n + n , z − n , z − n ), as z → ∞ , z ∈ C \ R + , where I stands for the 3 × Y (cid:98) Y ( z ) = O (cid:0) | z | − / | z | − / (cid:1) , z → z ∈ C \ R + . Proposition 1. For each (cid:126)n ∈ Z \{ (cid:126) } , the problem (RH- Y Y 3) has a unique solutionwhich is given by the matrix P (cid:126)n ( z ) = P (cid:126)n ( z ) πi (cid:82) P (cid:126)n ( x ) w ( x ) dxx − z πi (cid:82) P (cid:126)n ( x ) w ( x ) dxx − z d P (cid:126)n − (cid:126)e ( z ) d πi (cid:82) P (cid:126)n − (cid:126)e ( x ) w ( x ) dxx − z d πi (cid:82) P (cid:126)n − (cid:126)e ( x ) w ( x ) dxx − z d P (cid:126)n − (cid:126)e ( z ) d πi (cid:82) P (cid:126)n − (cid:126)e ( x ) w ( x ) dxx − z d πi (cid:82) P (cid:126)n − (cid:126)e ( x ) w ( x ) dxx − z , where (cid:126)e = (1 , , (cid:126)e = (0 , , and d − j = d − (cid:126)n,j = − πi (cid:90) x n j − P (cid:126)n − e j ( x ) w j ( x ) dx, j = 1 , . (21)Notice that the normality of the multi-indices (cid:126)n = ( n , n ) guarantees that the integralsin (21) are non-vanishing. Proof. The proof is basically contained in [45]; however, the measures there are supportedon the whole real line, which slightly simplifies the analysis. We will sketch a proof here forconvenience of the reader.Using the Sokhotski-Plemelj formula and the orthogonality conditions it is easy to verifythat P (cid:126)n satisfies (RH- Y Y Y 3) is trivially satisfied by thefirst column, while for the second and third columns it follows from the fact that w j ( x ) = O (1 / √ x ) , x → , j = 1 , Y Y 3) it is easy to deduce thatthe first column of (cid:98) Y has to be made up of multiple orthogonal polynomials with respect tothe weights w j with the multi-indices (cid:126)n , (cid:126)n − (cid:126)e , and (cid:126)n − (cid:126)e , respectively, and that the secondand third columns must be the corresponding second type functions of these polynomialswith respect to w and w normalized appropriately. The constants appearing in the secondand third row are needed to guarantee (RH- Y (cid:126)n , (cid:126)n − (cid:126)e , and (cid:126)n − (cid:126)e are normal, thus the corresponding monic multiple orthogonalpolynomials are uniquely determined. Remark 3. Using the expression for P (cid:126)n it is not difficult to verify that det P (cid:126)n ≡ z ∈ C (fact established for C \ [0 , + ∞ ) and extended by analyticity to the whole plane), which isthe standard tool for proving the uniqueness of P (cid:126)n . With this approach, only the normalityof the multi-index (cid:126)n is needed; however, the normality of the other two multi-indices is usefulin order to give an explicit description of the second and third rows of P (cid:126)n .7or the rescaled polynomials (15) we can write an analogous RHP using the connectionbetween Q n and P n and Proposition 1: given n ∈ N , find a 3 × Y analyticin C \ R + such that:(RH-Y1) For x ∈ (0 , + ∞ ) there is the jump condition Y + ( x ) = Y − ( x ) w ,n ( x ) w ,n ( x )0 1 00 0 1 , (RH-Y2) Y ( z ) = (cid:0) I + O ( z − ) (cid:1) diag (cid:0) z n , z − n , z − n (cid:1) , as z → ∞ , z ∈ C \ R − ,(RH-Y3) Y ( z ) = O (cid:0) | z | − / | z | − / (cid:1) , z → z ∈ C \ R + .Indeed, using the change of variables z → n z , x → n x it is easy to see that theRHP (RH-Y1)–(RH-Y3) and (RH- Y Y 3) (with (cid:126)n = ( n, n )) reduce to one another.From Proposition 1 it follows that (RH-Y1)–(RH-Y3) has a unique solution which may beexpressed in terms of P (cid:126)n , (cid:126)n = ( n, n ). In particular, the first row of Y is (cid:18) Q n ( z ) , πi (cid:90) + ∞ Q n ( x ) w ,n ( x ) dxx − z , πi (cid:90) + ∞ Q n ( x ) w ,n ( x ) dxx − z (cid:19) . The proof of Theorem 2 is based on the following characterization of the kernel K n in termsof the solution Y of (RH-Y1)–(RH-Y3), see [10, 14]: K n ( x, y ) = 12 πi ( x − y ) (cid:0) w ,n ( y ) w n ( y ) (cid:1) Y − ( y ) Y + ( x ) . (22) In the asymptotic analysis of multiple orthogonal polynomials with respect to a generalNikishin system N ( σ , σ ) in which σ is discrete, the associated model vector equilibriumproblem exhibits an external field acting on supp( σ ) plus a constraint on σ . This situationis encountered, for example, in [30], as well as for Pollaczek weights w j in [40]; see [4] for theanalysis of a general case.Let µ be a positive Borel measure with support contained in R and satisfying (cid:90) log(1 + | x | ) dµ ( x ) < ∞ , (23)(or the equivalent condition (cid:82) log(1 + | x | ) dµ ( x ) < ∞ , as used in [8]). Its potential andlogarithmic energy are defined as P µ ( x ) := (cid:90) log 1 | x − y | dµ ( y ) , I ( µ ) := (cid:90) (cid:90) log 1 | x − y | dµ ( x ) dµ ( y ) , (24)8espectively. From (23) it follows that I ( µ ) > −∞ . Let M e be the collection of all measures µ satisfying (23) and for which I ( µ ) < ∞ . If, additionally, | µ | = (cid:90) R dµ ( x ) = c, c > , we write µ ∈ M e ( c ). When µ , µ ∈ M e , their mutual energy is defined as I ( µ , µ ) = (cid:90) (cid:90) log 1 | x − y | dµ ( x ) dµ ( y ) , which is finite and I ( µ − µ ) = I ( µ ) + I ( µ ) − I ( µ , µ ) . Moreover for µ , µ ∈ M e ( c ), we have I ( µ − µ ) ≥ µ = µ (see [13, Theorem 2.5], [39, Theorem 2.1], and also [42,Chapter I] if the measures have bounded support).Let σ, supp( σ ) = R − , | σ | > 1, be a positive Borel measure such that for every compactsubset K ⊂ R − we have that P σ | K is continuous on C . As usual, σ | K denotes the restrictionof σ to K . We define M ( σ ) = { (cid:126)µ = ( µ , µ ) T ∈ M e (2) × M e (1) : supp( µ ) ⊂ R + , supp( µ ) ⊂ R − , µ ≤ σ } , where ( · ) T stands for transpose. By µ ≤ σ we mean that σ − µ is a positive measure. Sincewe have assumed that P σ | K is continuous on C for every compact K it readily follows that P µ is continuous on C .Let ϕ be a bounded from below continuous function on R + . Define A = (cid:18) − − (cid:19) , f = (cid:18) ϕ (cid:19) . For (cid:126)µ = ( µ , µ ) T ∈ M ( σ ) we introduce the vector function W (cid:126)µ ( x ) = (cid:90) log 1 | x − y | dA(cid:126)µ ( y ) + f ( x ) = ( W (cid:126)µ ( x ) , W (cid:126)µ ( x )) T and the functional J ϕ ( (cid:126)µ ) = (cid:90) ( W (cid:126)µ ( x ) + f ( x )) · d(cid:126)µ ( x ) = (cid:90) ( W (cid:126)µ ( x ) + ϕ ( x )) dµ ( x ) + (cid:90) W (cid:126)µ ( x ) dµ ( x ) . (26)Set J ϕ = inf { J ϕ ( (cid:126)µ ) : (cid:126)µ ∈ M ( σ ) } . Considering measures with compact support, it is easy to show that there exists (cid:126)µ ∈ M ( σ )such that J ϕ ( (cid:126)µ ) < ∞ ; therefore, −∞ ≤ J ϕ < + ∞ .9 efinition 2. A vector measure (cid:126)λ ∈ M ( σ ) is extremal if J ϕ ( (cid:126)λ ) = J ϕ > −∞ .In the study of the existence and uniqueness of an extremal measure one can combinethe techniques employed in [8] and [27] (see also [4]). In [8, Definition 1.6] growth conditionsat infinity are imposed on the vector external field f which we cannot require here (in factthe second component of f is identically zero). The growth condition is used in [8, Theorem1.7] to prove the lower semi-continuity of the functional J ϕ ( · ) and from there deduce theexistence of an extremal measure. However, [8, Theorem 1.8(b)] remains valid assumingthat J ϕ > −∞ and that a minimizer of the functional exists. From [27] one can use themore relaxed assumption of weak admissibility of the extremal problem (see Assumption 2.1therein), sufficient to prove the lower semi-continuity of a certain modified functional whichwe introduce promptly (see (28) below). For a detailed discussion see [4, Section 4]. Proposition 2. Assume that J ϕ > −∞ . The following statements are equivalent: ( A ) There exists (cid:126)λ ∈ M ( σ ) such that J ϕ ( (cid:126)λ ) = J ϕ . ( B ) There exists (cid:126)λ ∈ M ( σ ) such that (cid:82) W (cid:126)λ · d ( (cid:126)ν − (cid:126)λ ) ≥ for all (cid:126)ν ∈ M ( σ ) . ( C ) There exist (cid:126)λ = ( λ , λ ) T ∈ M ( σ ) and constants γ , γ such that ( i ) W (cid:126)λ ( x ) = 2 P λ ( x ) − P λ ( x ) + ϕ ( x ) (cid:26) = γ , x ∈ supp( λ ) , ≥ γ , x ∈ R + , ( ii ) W (cid:126)λ ( x ) = 2 P λ ( x ) − P λ ( x ) (cid:26) ≤ γ , x ∈ supp( λ ) , ≥ γ , x ∈ supp( σ − λ ) . If either condition is satisfied, they all have the same unique solution and the constants γ , γ are unique as well. Some additional properties are contained in (see [4, Lemma 4.2]) Lemma 1. Let (cid:126)λ be extremal in the sense of Definition 2. Then, P λ , P λ are continuous in C , supp( λ ) is connected, and ∈ supp( λ ) . If xϕ (cid:48) ( x ) is an increasing function on R + then supp( λ ) is connected. Should ϕ be increasing on R + then ∈ supp( λ ) . Finally, if lim x → + ∞ ( ϕ ( x ) − x )) = + ∞ (27) then supp( λ ) is a compact set, supp( λ ) = R − , and the λ , λ verify (23) . Following [27] we introduce a modified logarithmic energy of a measure µ as follows I ∗ ( µ ) := (cid:90) (cid:90) log (cid:112) | x | (cid:112) | y | | x − y | dµ ( x ) dµ ( y ) . µ, ν is given by I ∗ ( µ, ν ) := (cid:90) (cid:90) log (cid:112) | x | (cid:112) | y | | x − y | dµ ( x ) dν ( y ) . The advantage of this definition comes from the fact that (see (2.10)-(2.11) in [27]), | x − y | (cid:112) | x | (cid:112) | y | ≤ , x, y ∈ C . Therefore, the kernel in the previous integrals is uniformly bounded from below.Let us introduce M ∗ ( σ ) = { ( µ , µ ) t ∈ M (2) × M (1) : supp( µ ) ⊂ R + , supp( µ ) ⊂ R − , µ ≤ σ } , where M ( c ) denotes the class of all positive Borel measures with total mass c > 0. Ob-serve that unlike in the definition of M ( σ ) we neither assume (23) nor the finiteness of thelogarithmic energy of the measures. However, if µ, ν verify (23) then I ∗ ( µ, ν ) = I ( µ, ν ) + 12 | ν | (cid:90) log(1 + | x | ) dµ ( x ) + 12 | µ | (cid:90) log(1 + | x | ) dν ( x ) . Having this in mind, we introduce the following functional on M ∗ ( σ ): J ∗ ϕ ( (cid:126)µ ) = 2( I ∗ ( µ ) − I ∗ ( µ , µ ) + I ∗ ( µ )) + (cid:90) (2 ϕ − | x | )) dµ , (28)assuming that ϕ satisfies lim inf x →∞ ( ϕ ( x ) − x ) > −∞ , (29)so that J ∗ ϕ ( (cid:126)µ ) > −∞ for all (cid:126)µ ∈ M ∗ ( σ ). Moreover, inf { J ∗ ϕ ( (cid:126)µ ) : (cid:126)µ ∈ M ( σ ) } > −∞ . Itis understood that J ∗ ϕ ( (cid:126)µ ) = + ∞ when I ∗ ( µ ) = + ∞ or I ∗ ( µ ) = + ∞ . Assumption (29)ensures the weak admissibility of the extremal problem. Straightforward calculations yieldthat J ∗ ϕ ( (cid:126)µ ) = J ϕ ( (cid:126)µ ), (cid:126)µ ∈ M ( σ ).The following lemma is a direct consequence of [27, Corollary 2.7]: Lemma 2. Assume that ϕ verifies (29) , then J ∗ ϕ ( · ) is strictly convex on the set where it isfinite and admits a unique minimizer. If the components of the minimizer verify (23) then itminimizes J ϕ ( · ) as well. Summarizing we have that if ϕ satisfies (27) then J ϕ > −∞ and there exists (cid:126)λ ∈ M ( σ )such that J ϕ ( (cid:126)λ ) = J ϕ which allows us to use Proposition 2.Let us return to the polynomials Q n satisfying (16). Fix n ∈ N . We have ( w ,n dx, w ,n dx ) = N ( w ,n dx, σ ,n ), where σ ,n = 4 π (cid:88) k ≥ δ − [(2 k +1) / (2 n )] . Q n, , deg Q n, = n, whose zeros are simple and contained in theconvex hull of supp( σ ,n ), such that (cid:90) x ν Q n ( x ) Q n, ( x ) dx sinh( πn √ x ) = 0 , ν = 0 , . . . , n − , (30)and (cid:90) t ν Q n, ( t ) Q n ( t ) (cid:90) Q n ( x ) Q n, ( x ) dx ( x − t ) sinh( πn √ x ) dσ ,n ( t ) = 0 , ν = 0 , . . . , n − . (31)That is, Q n and Q n, satisfy full orthogonality relations with respect to certain varyingmeasures. From (30)-(31) one can establish a connection (see [4, Section 3.3] between theasymptotic zero distribution of the sequences of polynomials { Q n } n ≥ , { Q n, } n ≥ and thesolution of a vector equilibrium problem of the type discussed above in which ϕ ( x ) = π √ x, dσ ( x ) = dx (cid:112) | x | . Obviously, ϕ ( x ) and xϕ (cid:48) ( x ) are increasing on R + and (27) takes place. An explicit solutionfor the corresponding equilibrium problema is given in [4, Proposition 3.1].In fact, in [40] V.N. Sorokin proved for this very interesting case the following result whichwe will use. Proposition 3. Let p − = − (cid:32) √ − (cid:33) ≈ − . , p + = (cid:18) √ − (cid:19) ≈ . . There exists a unique pair of measures λ and λ , which satisfy the following equilibriumconditions: • supp( λ ) = [0 , p + ] ⊂ R + , | λ | = 2 , and supp( λ ) = R − , | λ | = 1 . • λ is absolutely continuous, and λ (cid:48) ( x ) (cid:40) = 1 / (2 (cid:112) | x | ) , x ∈ [ p − , ,< / (2 (cid:112) | x | ) , x < p − . (32) In other words, with dσ ( x ) = dx (cid:112) | x | = i dx x / , x ∈ R − , (33) the measure σ − λ is non-negative and supported on ( −∞ , p − ] . With the external field ϕ ( x ) = π √ x > on R + , (34) there exists a unique constant ω ∈ R such that P λ ( x ) − P λ ( x ) + ϕ ( x ) (cid:40) = ω, x ∈ [0 , p + ] ,> ω, x > p + ; (35)2 P λ ( x ) − P λ ( x ) (cid:40) = 0 , x ≤ p − ,< , x ∈ ( p − , . (36) Moreover. lim n →∞ µ Q n = λ / , lim n →∞ µ Q n, = λ . (37)lim n →∞ (cid:18)(cid:90) | Q n ( x ) | | Q n, ( x ) | dx sinh( nπ √ x ) (cid:19) /n = e − ω . (38) and lim n →∞ (cid:32)(cid:90) Q n, ( t ) | Q n ( t ) | (cid:90) Q n ( x ) | Q n, ( x ) | dx | x − t | sinh( πn √ x ) dσ ,n ( t ) (cid:33) /n = e − ω . (39)Using the pair of equilibrium measures λ j described in Proposition 3 we define as usualthe g -functions g j ( z ) = (cid:90) log( z − t ) dλ j ( t ) , j = 1 , . (40)In this definition we understand by log( z − · ) its principal branch in C \ ( −∞ , p + ]. Wesummarize next some of their properties needed for out steepest descent analysis.For the sake of brevity we use the notation υ = υ ( z ) := exp (cid:16) πz / (cid:17) , z ∈ C \ R − , (41)so that | υ | > C \ R − , υ + υ − = 1 on R − , and w ,n ( z ) = 2 υ n − υ − n , w ,n ( z ) = 2 z / ( υ n + υ − n ) . We have also the following straightforward identities: w ,n ( z ) ± z / w ,n ( z ) = 4 υ ± n υ n − υ − n , w ,n ( z ) ± z / w ,n ( z ) = ± υ ± n , z ∈ C \ R − , (42)as well as w j,n + ( x ) = − w j,n − ( x ) , x < , j = 1 , . (43)13 roposition 4. The g -functions defined in (40) satisfy the following properties:(i) exp ( g + g − − g + ω ) ( x ) (cid:40) = υ, on [0 , p + ] , < υ, for x > p + .(ii) exp ( g + g − − g ) ( x ) (cid:40) = 1 , for x < p − , > , on ( p − , .(iii) exp ( g − g − ) ( x ) = υ on [ p − , .(iv) For x ∈ [0 , p + ] , g ( x ) − g − ( x ) = 2 πi (cid:90) p + x dλ ( t ) . (44) (v) With z = x + iy , ∂∂y (cid:60) (cid:16) g ( z ) − g ( z ) − πz / (cid:17) (cid:12)(cid:12)(cid:12)(cid:12) z = x + i = 2 π (cid:32) λ (cid:48) ( x ) − (cid:112) | x | (cid:33) , x < . In particular, this derivative is ≤ on R − , and < for x < p − .(vi) There is an open sector with its vertex at p − and containing ( −∞ , p − ) where (cid:12)(cid:12) e g − g ( z ) υ − (cid:12)(cid:12) < . (45) Moreover, there exists an ε > such that for |(cid:61) z | < ε , p − + ε < (cid:60) z < − ε , (cid:12)(cid:12) e g − g ( z ) υ − (cid:12)(cid:12) > (the relations on R − hold in the sense of the boundary values of the left hand sides).Proof. All these identities are direct consequence of Proposition 3. Indeed, ( i ), ( ii ) and ( iii )follow directly from (35), (36) and (32), respectively. For ( iv ) we use the definition of g .In order to prove ( v ) we use the equilibrium conditions and the Cauchy–Riemann formulas.Inequality (45) is a consequence of ( v ), while (46) follows from (36).We introduce finally two other auxiliary functions. From the analyticity of the densityof λ it follows that the right hand side in (44) can be extended as a multivalued analyticfunction to a neighborhood U of the interval [0 , p + ]. Hence, we can define the holomorphicbranch ψ ( z ) = − πi (cid:90) p + z dλ ( t ) , z ∈ U \ ( −∞ , p + ] . (47)By (44), ψ ± ( x ) = ∓ ( g ( x ) − g − ) ( x ) , x ∈ (0 , p + ) and ψ ± (0) = ± πi. (48)Since ddx ψ ( x ) = 2 πiλ (cid:48) ( x ) ∈ i R + , 14t implies that ∂∂y (cid:60) ψ ( x + iy ) (cid:12)(cid:12)(cid:12)(cid:12) y =+0 = − πλ (cid:48) ( x ) < , x ∈ (0 , p + ) . (49)Furthermore, function exp (2 g − g + ω + ψ ) ( z ) /υ is holomorphic B δ \ (0 , p + ), where B δ = { z : | z − p + | < δ } , with δ < p + / 2. Considering itsboundary values on B δ ∩ (0 , p + ) and using ( i ) of Proposition 4 we conclude thatexp (2 g − g + ω ) ( z ) = υ exp ( − ψ ( z )) , z ∈ B δ \ (0 , p + ) . (50)Observe that this identity has a holomorphic continuation to C \ ( −∞ , p + ], so we can actuallyuse it to extend the definition of exp( ψ ) there: e ψ ( z ) = υ exp ( − g + g − ω ) ( z ) , z ∈ C \ ( −∞ , p + ] . (51)With this definition, and taking into account (48), we conclude that e ψ + ( x ) = e ψ − ( x ) , x ∈ ( p − , . (52)We can apply analogous arguments when defining (cid:98) ψ ( z ) = − πi (cid:90) p − z d ( σ − λ ) ( t ) (53)in a neighborhood of p − , cut along ( −∞ , p − ]. The starting point of the steepest descent asymptotic analysis is the RHP (RH-Y1)–(RH-Y3)for the matrix Y . Using the notation (41) we define the following matrix-valued functions in C \ R − : A L ( z ) := (cid:18) − / (2 z / ) z / / (cid:19) , A R ( z ) = (cid:32) − z / υ n w z / z / υ n w (cid:33) . We have that A R ( z ) = A L ( z ) B ( z ) , with B ( z ) := (cid:18) z / υ n (cid:19) . (54)In particular, det A L ( z ) = det A R ( z ) ≡ 1, and A R ( z ) = A L ( z ) (cid:0) I + O (cid:0) | υ | − n (cid:1)(cid:1) , z → ∞ , z ∈ C \ R − . (55)15oreover, by (42), ( w , w ) A R ( x ) = (cid:18) υ n υ n − υ − n , (cid:19) , x > , (56)and for x < A − L − ( x ) A L + ( x ) = (cid:32) − / (2 x / )2 x / (cid:33) , A − R − ( x ) A R + ( x ) = (cid:32) υ n + x / υ − n + (cid:33) . (57)Now we open lenses as in the Figure 1, and define the new matrix (written block-wise): X ( z ) = Y ( z ) (cid:18) A R ( z ) (cid:19) in the domains limited by ∆ ± and ( p − , + ∞ ) , (58)and X ( z ) = Y ( z ) (cid:18) A L ( z ) (cid:19) in the domains limited by ∆ ± and ( −∞ , p − ) . (59)The newly defined matrix X is the unique solution of the following RHP:(RH-X1) X = X n is holomorphic in C \ ( R ∪ ∆ + ∪ ∆ − ), has continuous boundary valueson all contours, and these satisfy X + ( z ) = X − ( z ) J X ( z ) , with J X ( z ) = υ n υ n − υ − n 01 1 , x > , J X ( z ) = υ n + x / υ − n + , x ∈ ( p − , , J X ( z ) = (cid:18) B ± ( z ) (cid:19) , z ∈ ∆ ± , J X ( z ) = − / (2 x / )2 x / , x ∈ ( −∞ , p − ) . (RH-X2) X ( z ) = (cid:0) I + O ( z − ) (cid:1) (cid:18) A L ( z ) (cid:19) diag (cid:0) z n , z − n , z − n (cid:1) as z → ∞ , z ∈ C \ R .(RH-X3) X ( z ) = O (cid:0) | z | − / | z | − / (cid:1) as z → 0, and X ( z ) = O (1) as z → p − .16 ∆ + ∆ − p − X ( z ) = Y ( z ) (cid:32) A L ( z ) (cid:33) X ( z ) = Y ( z ) (cid:32) A L ( z ) (cid:33) X ( z ) = Y ( z ) (cid:32) A R ( z ) (cid:33) X ( z ) = Y ( z ) (cid:32) A R ( z ) (cid:33) Figure 1: Global lens opening.Indeed, jump relations (RH-X1) are obtained by direct calculations, while (RH-X2) is aconsequence of the obvious identitydiag (cid:0) z n , z − n , z − n (cid:1) (cid:18) C (cid:19) = (cid:18) C (cid:19) diag (cid:0) z n , z − n , z − n (cid:1) , valid for any 2 × C .Finally, (RH-X3) is a result of a direct combination of (RH-Y3) and of the fact that A R ( z ) = O (cid:18) | z | / (cid:19) as z → . Now we use the functions g j defined in (40) in order to normalize the behavior at infinity.Set U ( z ) = diag ( e nω , , X ( z ) diag (cid:16) e − n ( g ( z )+ ω ) , e n ( g ( z ) − g ( z )) , e ng ( z ) (cid:17) . (60)Then U is analytic in C \ ( R ∪ ∆ + ∪ ∆ − ), and U + ( z ) = U − ( z ) J U ( z ) , with J U ( z ) = e − n ( g − g − )( x ) 4 υ n υ n − υ − n e n ( g + g − − g + ω )( x ) e n ( g − g − )( x ) , x > , J U ( z ) = υ n + e − n ( g − g − ) x / e − n ( g + g − − g ) υ − n + e n ( g − g − ) , x ∈ ( p − , , J U ( z ) = ± z / υ n e n (2 g − g ) , z ∈ ∆ ± , J U ( z ) = − / (2 x − / ) e n ( g + g − − g ) x / e − n ( g + g − − g ) , x ∈ ( −∞ , p − ) . g -functions summarized in Proposition 4 wehave that: • The jump matrix J U on [0 , p + ] has the form J U ( z ) = e nψ + ( x ) 41 − υ − n e nψ − ( x ) , while on ( p + , + ∞ ), J U ( z ) = υ n υ n − υ − n e n ( g + g − − g + ω )( x ) 01 1 = υ n υ n − υ − n e n (2 g − g + ω )( x ) 01 1 , and the entry (1 , 2) of the jump matrix J U is exponentially decaying for x > p + . • The jump matrix J U on ( p − , 0) has the form J U ( z ) = x / e − n ( g + g − − g ) , and the (3 , 2) entry is exponentially decaying. • We can choose the contours ∆ ± in such a way that the entry (2 , 3) of the jump matrix J U on ∆ ± is also exponentially decaying. • The jump matrix J U for x < p − has the form J U ( z ) = − / (2 x / )2 x / . In summary, J U is exponentially close to the identity matrix I on all contours, except onsupp( λ ) ∪ supp( σ − λ ). Furthermore, U ( z ) = (cid:0) I + O ( z − ) (cid:1) (cid:18) A L ( z ) (cid:19) as z → ∞ , z ∈ C \ R . Clearly, U has the same behavior at z = p − and at the origin as X , see (RH-X3). We fix the jump on [0 , p + ] observing that by (48), − − υ − n e nψ − ( x ) e nψ + ( x ) 41 − υ − n e nψ − ( x ) 00 0 1 × − − υ − n e nψ + ( x ) = − υ − n − − υ − n . p + ∆ + ∆ − Γ + Γ − p − , p + ], as shown in Figure 2, and define the newmatrix T ( z ) = U ( z ) ∓ − υ − n e nψ ( z ) , (61)for z in the domains bounded by Γ ± and [0 , p + ] (we take “ − ” in (61) for (cid:61) z > 0, and “+”otherwise), and T ( z ) = U ( z ) elsewhere. Hence, T is holomorphic in C \ ( R ∪ Γ ± ∪ ∆ ± ), and T + ( z ) = T − ( z ) J T ( z ) , with J T ( z ) = − υ − n − − υ − n , x ∈ (0 , p + ) , J T ( z ) = − υ − n ) υ n e n ( g + g − − g + ω )( x ) 01 1 , x > p + , J T ( z ) = x / e − n ( g + g − − g ) , x ∈ ( p − , , J T ( z ) = ± z / υ n e n (2 g − g ) , z ∈ ∆ ± , J T ( z ) = − / (2 x / )2 x / , x ∈ ( −∞ , p − ) , J T ( z ) = − υ − n e nψ ( z ) , z ∈ Γ ± . By (49), we can always choose Γ ± in such a way that the (2 , 1) entry of J T is exponentially19ecaying on Γ ± , away from their endpoints 0 and p + . Obviously, by definition, T ( z ) = (cid:0) I + O ( z − ) (cid:1) (cid:18) A L ( z ) (cid:19) as z → ∞ , z ∈ C \ R . Since 1 − υ − n e nψ ( z ) = O ( | z | / ) , z → , we get that T has the same asymptotic behavior at z = p − and at the origin as X , see(RH-X3), when z = 0 is approached both from inside and outside the contours Γ ± . Observing the jumps J T above, we can infer that an appropriate model for T is a matrix N ,holomorphic in C \ (( −∞ , p − ] ∪ [0 , p + ]), such that N + ( x ) = N − ( x ) − / , x ∈ (0 , p + ) , (62) N + ( x ) = N − ( x ) − / (2 x / )0 2 x / , x ∈ ( −∞ , p − ) , (63)and N ( z ) = (cid:0) I + O ( z − ) (cid:1) (cid:18) A L ( z ) (cid:19) as z → ∞ , z ∈ C \ R (64)(matching asymptotically the behavior of T ). Observe that this behavior at infinity is con-sistent with the jump on ( −∞ , p − ), see (57).This RHP is solved using the Riemann surface R constructed gluing the three copies of C , as shown in Figure 3. This is a surface of genus 0.It was observed in [40] that z = 1 + ζζ (1 − ζ ) (65)establishes a one-to-one bijection between R and the extended ζ -plane . There are threeinverse functions to (65), which we choose such that as z → ∞ , ζ ( z ) = 1 − z − z + O (cid:18) z (cid:19) , (66) ζ ( z ) = 1 z / + 1 z + 32 z / + 3 z + 558 z / + O (cid:18) z (cid:19) , (67) ζ ( z ) = − z / + 1 z − z / + 3 z − z / + O (cid:18) z (cid:19) . (68) Formally, we can say that R is parametrized by ( z, ζ ), where z is the canonical projection on C , and bothvariables are related by (65). Then the bijection is ( z, ζ ) ↔ ζ . R R p + p + p − p − (cid:101) R (cid:101) R (cid:101) R q − − q + z ζ Figure 3: The Riemann surface R and the mapping (65).Again, all fractional powers are taken as principal branches, that is, positive on R + , with thebranch cut along R − .Figure 3 also shows the domains (cid:101) R j = ζ j ( R j ) , j = 1 , , , where R j is the j th sheet of the Riemann surface, and the location of the points ζ (0) = ∞ , ζ ( ∞ ) = 0 , q ± := ζ ( p ± ) = − ± √ 52 (69)in the ζ -plane. We observe that ζ ( −∞ , p − ) and ζ (0 , p + ) are in the lower half plane, while ζ − ( −∞ , p − ) and ζ − (0 , p + ) are in the upper half plane.Let us define the following functions: r ( ζ ) = (2 ζ ) / (1 + ζ ) / , r ( ζ ) = 14 r ( ζ ) , for ζ ∈ (cid:101) R ∪ (cid:101) R , (70)as well as r ( ζ ) = − ζ ) / (1 − ζ ) / , ζ ∈ (cid:101) R ∪ ( (cid:61) ζ > , ζ ) / (1 − ζ ) / , ζ ∈ (cid:101) R ∪ ( (cid:61) ζ < . for ζ ∈ (cid:101) R . (71)In these formulas we use the main branch of all square roots: the branch of (1 + ζ ) / in C \ ( −∞ , − 1] takes the value 1 at ζ = 0; the branch of (1 − ζ ) / is fixed in C \ [1 , + ∞ ) byits value 1 at ζ = 0, and ζ / is holomorphic in C \ ( −∞ , 0] and positive on the positive semiaxis.With this convention, each function r j is holomorphic in its corresponding domain (cid:101) R j , j = 1 , , 3. 21ow we define f j ( z ) = r j ( ζ j ( z )) , j = 1 , , z on the Riemann surface R , and its projectionon the complex plane. In this fashion, f is holomorphic in C \ [0 , p + ], f is holomorphicin C \ (( −∞ , p − ] ∪ [0 , p + ]), and f is holomorphic in C \ ( −∞ , p − ]. Observe also that by(66)–(68), as z → ∞ , f ( z ) = 1+ O (cid:18) z (cid:19) , f ( z ) = z − / / (cid:18) O (cid:18) z / (cid:19)(cid:19) , f ( z ) = i √ z / (cid:18) O (cid:18) z / (cid:19)(cid:19) . (72)For z ∈ (0 , p + ) we have f ± ( z ) = r ( ζ ± ( z )) = r ( ζ ∓ ( z )) = 4 r ( ζ ∓ ( z )) = 4 f ∓ ( z ) . On the other hand, let z ∈ ( −∞ , p − ); since ζ ( −∞ , p − ) is in the lower half plane, we have f − ( z ) = r ( ζ − ( z )) = r ( ζ ( z )) = 1(2 ζ ( z )) / (1 − ζ ( z )) / = 2 r ( ζ ( z )) (1 + ζ ( z )) / ζ ( z ) (1 − ζ ( z )) / . By (65) and (67), for z ∈ C \ ( −∞ , p + ],(1 + ζ ) / ζ (1 − ζ ) / = z / , ζ = ζ ( z ) . Hence, f − ( z ) = 2( z / ) + f ( z ) , z ∈ ( −∞ , p − ) . Analogously, f ( z ) = r ( ζ ( z )) = r ( ζ − ( z )) = − ζ − ( z )) / (1 − ζ − ( z )) / = − r ( ζ − ( z )) (1 + ζ − ( z )) / ζ − ( z ) (1 − ζ − ( z )) / = 2( z / ) + f − ( z ) , z ∈ ( −∞ , p − ) . Gathering all these formulas we conclude that if we define (cid:99) N ( z ) = N ( z ) diag( f ( z ) , f ( z ) , f ( z )) , z ∈ C \ (( −∞ , p − ] ∪ [0 , p + ]) , then we will obtain the following RH problem for (cid:99) N : it is holomorphic in C \ (( −∞ , p − ] ∪ [0 , p + ]), (cid:99) N + ( z ) = (cid:99) N − ( z ) − , z ∈ (0 , p + ) , (73) (cid:99) N + ( z ) = (cid:99) N − ( z ) − 10 1 0 , z ∈ ( −∞ , p − ) , (74)22nd (cid:99) N ( z ) = 2 − / (cid:18) I + O (cid:18) z (cid:19)(cid:19) diag(1 , z − / , z / ) − i i × diag(1 + O ( z − ) , O ( z − / ) , O ( z − / ))= 2 − / (cid:16) I + O ( z − / ) (cid:17) diag(1 , z − / , z / ) − i i , z → ∞ , z ∈ C \ R . In order to solve this RH problem we use the polynomial D ( ζ ) D ( ζ ) = ζ ( ζ − q + )( ζ − q − ) = ζ ( ζ + ζ − 1) (75)(see (69)). The square root D ( ζ ) / , which branches at 0 and q ± , is defined with a cut on ζ − ( −∞ , p − ) ∪ ζ − (0 , p + ), which, as noted before, are the parts of the boundary of (cid:101) R that arein the upper half of the ζ -plane. We assume that D ( ζ ) / > ζ > q + , so that D ( ζ ) / < q − , 0) and D ( ζ ) / ∈ i R − for z ∈ (0 , q + ).We construct the matrix (cid:99) N ( z ) as follows (cid:99) N ( z ) = F ( ζ ( z )) F ( ζ ( z )) F ( ζ ( z )) F ( ζ ( z )) F ( ζ ( z )) F ( ζ ( z )) F ( ζ ( z )) F ( ζ ( z )) F ( ζ ( z )) , (76)where F ( ζ ) = K ζ D ( ζ ) / , F ( ζ ) = K ζ ( ζ − D ( ζ ) / , F ( ζ ) = K ( ζ − ζ − ζ ∗ ) D ( ζ ) / , with D ( ζ ) given by (75), and K , K , K , ζ ∗ are constants to be computed.Because of the branch cut for D ( ζ ) / the functions F j , j = 1 , , , defined above satisfy F j + ( ζ ) = − F j − ( ζ ) , ζ ∈ ∂ (cid:101) R ∩ ( (cid:61) ζ > , and F j + ( ζ ) = F j − ( ζ ) , ζ ∈ ∂ (cid:101) R ∩ ( (cid:61) ζ < . Consequently, on (0 , p + ), for j = 1 , , (cid:99) N j ( z ) = F j ( ζ ( z )) = − F j ( ζ − ( z )) = − (cid:99) N j − ( z ) , (cid:99) N j ( z ) = F j ( ζ ( z )) = F j ( ζ − ( z )) = (cid:99) N j − ( z ) , (cid:99) N j ( z ) = F j ( ζ ( z )) = F j ( ζ − ( z )) = (cid:99) N j − ( z ) , and on ( −∞ , p − ), (cid:99) N j ( z ) = F j ( ζ ( z )) = F j ( ζ − ( z )) = (cid:99) N j − ( z ) , (cid:99) N j ( z ) = F j ( ζ ( z )) = F j ( ζ − ( z )) = (cid:99) N j − ( z ) , (cid:99) N j ( z ) = F j ( ζ ( z )) = − F j ( ζ − ( z )) = − (cid:99) N j − ( z ) . 23n other words, (cid:99) N constructed by formula (76) satisfies the jump conditions (73)–(74).It remains to analyze the asymptotic behavior at infinity. Notice that, by (75), D (1) = 1, sothat F ( ζ ) = K + O ( ζ − , ζ → , and F ( ζ ) = O ( ζ / ) , ζ → . Taking K = 1 and using (66)–(68) it follows that (cid:99) N ( z ) = 1 + O ( z − ) , z → ∞ , and (cid:99) N k ( z ) = O ( z − / ) , k = 2 , , z → ∞ . We also have that F j ( ζ ) = O ( ζ − , ζ → , j = 2 , (cid:99) N j ( z ) = O ( z − ) , z → ∞ , j = 2 , . With our convention about the branch of D ( ζ ) / , we see that D ( ζ ) / = − iζ / + O ( ζ / ) , ζ → , where the (main) branch cut of ζ / goes along the arc ζ − ( −∞ , p − ), which joins q − and 0in the upper half plane, and the ray ( −∞ , q − ]. Thus, F ( ζ ) = − iK ζ / + O ( ζ / ) , ζ → . Using (67) and (68), it follows that (cid:99) N ( z ) = − iK z − / + O ( z − / ) , z → ∞ . (cid:99) N ( z ) = − K z − / + O ( z − / ) , z → ∞ . Choosing K = 2 − / i, we find that, as z → ∞ , (cid:99) N ( z ) = 2 − / z − / (cid:16) O ( z − / ) (cid:17) , (cid:99) N ( z ) = − − / iz − / (cid:16) O ( z − / ) (cid:17) , as needed.Analogously, F ( ζ ) = K (cid:18) iζ ∗ ζ − / − i (2 + ζ ∗ )2 ζ / + O ( ζ / ) (cid:19) , ζ → , and substituting ζ ( z ) and ζ ( z ) into F , we find that (cid:99) N ( z ) = K (cid:16) iζ ∗ z / − i (1 + ζ ∗ ) z − / + O ( z − / ) (cid:17) , z → ∞ . (cid:99) N ( z ) = K (cid:16) − ζ ∗ z / − (1 + ζ ∗ ) z − / + O ( z − / ) (cid:17) , z → ∞ . ζ ∗ = − K = K = 2 − / i , we obtain (cid:99) N ( z ) = 2 − / z / (cid:16) O ( z − / ) (cid:17) , (cid:99) N ( z ) = 2 − / iz / (cid:16) O ( z − / ) (cid:17) , z → ∞ . The matrix (cid:99) N ( z ) has the following behavior near the (finite) branch points (cid:99) N ( z ) = | z − p − | − / | z − p − | − / | z − p − | − / | z − p − | − / | z − p − | − / | z − p − | − / , z → p − , (77) (cid:99) N ( z ) = | z − p + | − / | z − p + | − / | z − p + | − / | z − p + | − / | z − p + | − / | z − p + | − / , z → p + , (78)and (cid:99) N ( z ) = | z | − / | z | − / | z | − / | z | − / | z | − / | z | − / , z → . (79)Indeed, for j = 1 , , , we have that F j ( ζ ) = O (( ζ − q − ) − / ), ζ → q − . Now, ζ − ( q − ) = ζ − ( q − ) = p − is a first order finite branch point. On the other hand, p − is a regular pointof ζ and the image by ζ of a sufficiently small neighborhood of p − remains bounded awayfrom all the singularities of F j , j = 1 , , 3. This gives (77). Analogously, for j = 1 , , , wehave that F j ( ζ ) = O (( ζ − q + ) − / ), ζ → q + . Now, ζ − ( q + ) = ζ − ( q + ) = p + is a first orderfinite branch point. We also have that p + is a regular point of ζ and the image by ζ of asufficiently small neighborhood of p + remains bounded away from all the singularities of F j , j = 1 , , 3, and (78) follows. Finally, F ( ζ ) = O ( ζ / ), F ( ζ ) = O ( ζ / ), F ( ζ ) = O ( ζ − / ), as ζ → ∞ . Since ζ − ( ∞ ) = ζ − ( ∞ ) = 0 is a first order finite branch point, while 0 is a regularpoint of ζ , whose image of a small neighborhood of 0 is away from the singularities of F j , j = 1 , , 3, we obtain (79).Notice that (73)–(74) imply that det (cid:99) N ( z ) is analytic in C \ { , p − , p + } . This togetherwith (77)–(79) gives us that det (cid:99) N ( z ) is a entire function. From the asymptotic behavior of (cid:99) N ( z ) at ∞ , it follows that lim z →∞ det (cid:99) N ( z ) = − i/ 2; thereforedet (cid:99) N ( z ) ≡ − i/ , z ∈ C . We can summarize our findings as follows: Proposition 5. A solution of the model Riemann-Hilbert problem (62) – (64) is given by N ( z ) = F ( ζ ( z )) F ( ζ ( z )) F ( ζ ( z )) F ( ζ ( z )) F ( ζ ( z )) F ( ζ ( z )) F ( ζ ( z )) F ( ζ ( z )) F ( ζ ( z )) diag(1 /f ( z ) , /f ( z ) , /f ( z )) , where f j ( z ) = r j ( ζ j ( z )) , j = 1 , , , functions r j are given in (70) – (71) , and F ( ζ ) = ζ / ( ζ + ζ − / , F ( ζ ) = 2 − / i ζ / ( ζ − ζ + ζ − / , F ( ζ ) = 2 − / i ( ζ − ζ / ( ζ + ζ − / , with the branches chosen as specified above. + p + + δp + − δ Γ + Γ − ∂B δ Figure 4: Local analysis at p + . p + In the terminology of [7], we deal here with a (soft) band/void edge.Consider a small fixed disk, B δ , of radius 0 < δ < p + / 2, and center at p + (see Figure 4).We look for P holomorphic in B δ \ ( R ∪ Γ ± ), such that P + ( z ) = P − ( z ) J T ( z ), where, as wehave seen, J T ( z ) = − υ − n − − υ − n , x ∈ ( p + − δ, p + ) , J T ( z ) = − υ − n ) υ n e n (2 g − g + ω )( x ) 01 1 , ( p + , p + + δ ) , J T ( z ) = − υ − n e nψ ( z ) , z ∈ B δ ∩ Γ ± , and P is bounded as z → p + , z ∈ R \ Γ ± .Additionally, as n → ∞ , we need P ( z ) = N ( z ) ( I + O (1 /n )) z ∈ ∂B δ \ ( R ∪ Γ ± ) , where N is the matrix-valued function described in Proposition 5.We follow the well-known scheme, and build P in the form P ( z ) = E ( z ) Ψ (cid:16) n / f ( z ) (cid:17) diag (cid:32) − υ − n ) / e − n ψ ( z ) , (cid:0) − υ − n (cid:1) / e n ψ ( z ) , (cid:33) , (80)where E ( z ) = N ( z ) √ π −√ π − i √ π − i √ π 00 0 1 n / f / ( z ) 0 00 n − / f − / ( z ) 00 0 1 , (81)26nd f ( z ) = (cid:20) ψ ( z ) (cid:21) / (82)is a biholomorphic (conformal) map of a neighborhood of p + onto a neighborhood of theorigin such that f ( z ) is real and positive for z > p + ; recall that ψ was defined in (47). Wemay deform the contours Γ ± near p + in such a way that f maps Γ ± ∩ B δ to the rays withangles π and − π , respectively.Matrix Ψ is build using the Airy functions, as described for instance in [30, page 253].We put y ( s ) = Ai( s ) , y ( s ) = ω Ai( ωs ) , y ( s ) = ω Ai( ω s ) , ω = e πi/ , where Ai is the usual Airy function. Define the 2 × K by K ( s ) = (cid:18) y ( s ) − y ( s ) y (cid:48) ( s ) − y (cid:48) ( s ) (cid:19) , arg s ∈ (0 , π/ , K ( s ) = (cid:18) − y ( s ) − y ( s ) − y (cid:48) ( s ) − y (cid:48) ( s ) (cid:19) , arg s ∈ (2 π/ , π ) , K ( s ) = (cid:18) − y ( s ) y ( s ) − y (cid:48) ( s ) y (cid:48) ( s ) (cid:19) , arg s ∈ ( − π, − π/ , K ( s ) = (cid:18) y ( s ) y ( s ) y (cid:48) ( s ) y (cid:48) ( s ) (cid:19) , arg s ∈ ( − π/ , . Then we take the 3 × Ψ as Ψ ( s ) = (cid:18) K ( s ) (cid:19) . This construction uses also identity (50). p − In the terminology of [7], this is a (soft) band/saturated region edge.Consider a small fixed disk, B δ , of radius 0 < δ < | p − | / 2, and center at p − (see Figure 5).We look for P holomorphic in B δ \ ( R ∪ ∆ ± ), such that P + ( z ) = P − ( z ) J T ( z ), where, as wehave seen, J T ( z ) = x / e − n ( g + g − − g ) , x ∈ ( p − , p − + δ ) , J T ( z ) = − / (2 x / )2 x / , ( p − − δ, p − ) , J T ( z ) = ± z / υ n e n (2 g − g ) , z ∈ B δ ∩ ∆ ± , − p − + δp − − δ ∆ + ∆ − ∂B δ Figure 5: Local analysis at p − .and P is bounded as z → p − , z ∈ R \ ∆ ± .Additionally, as n → ∞ , we need P ( z ) = N ( z ) ( I + O (1 /n )) z ∈ ∂B δ \ ( R ∪ ∆ ± ) , where N is the matrix-valued function described in Proposition 5.Let us define h ( z ) = e n (2 g − g )( z ) / / z / υ n , z ∈ C \ ( −∞ , p + ] , and H ( z ) = (cid:40) diag (1 , h ( z ) , /h ( z )) , z ∈ B δ ∩ {(cid:61) z < } , diag (1 , ih ( z ) , − i/h ( z )) , z ∈ B δ ∩ {(cid:61) z > } . (83)Let also (cid:101) P ( z ) = P ( z ) H ( z ) , z ∈ B δ \ R . Then (cid:101) P + ( z ) = (cid:101) P − ( z ) J (cid:101) P ( z ), with J (cid:101) P ( z ) = − , x ∈ ( p − , p − + δ ) , J (cid:101) P ( z ) = − , ( p − − δ, p − ) , J (cid:101) P ( z ) = − , z ∈ B δ ∩ ∆ ± , and (cid:101) P ( z ) = O (1 , z − / , z / ) as z → p − , z ∈ R \ ∆ ± .Comparing it with the RH problem for K above (see e.g. [15, p. 213]) we see that non-trivial jumps for (cid:101) P coincide with those of σ σ K ( z ) σ σ , where σ σ = (cid:18) − 11 0 (cid:19) δ − δ Γ + Γ − ∂B δ Figure 6: Local analysis at the origin.(see also [7, formula (5.22)]).Then, taking Ψ ( s ) = (cid:18) σ σ K ( s ) σ σ (cid:19) , as before, we conclude that P ( z ) = E ( z ) Ψ (cid:16) n / (cid:98) f ( z ) (cid:17) H − ( z ) , (84)where E ( z ) = N ( z ) √ π −√ π − i √ π − i √ π n / (cid:98) f / ( z ) 00 0 n − / (cid:98) f − / ( z ) , (85)and (cid:98) f ( z ) = (cid:20) (cid:98) ψ ( z ) (cid:21) / (86)with (cid:98) ψ defined in (53), such that (cid:98) f is a biholomorphic (conformal) map of a neighborhood of p − onto a neighborhood of the origin such that f ( z ) is real and positive for z > p − . Following the ideas of [30] (see Section 8.2.1 therein), we consider a small fixed disk, B δ , ofradius 0 < δ < | p − | / 2, centered at the origin (see Figure 6). We look for P holomorphic in B δ \ ( R + ∪ Γ ± ), such that P + ( z ) = P − ( z ) J T ( z ), where, as we have seen, J T ( z ) = − υ − n − − υ − n , x ∈ B δ ∩ (0 , p + ) , J T ( z ) = − υ − n e nψ ( z ) , z ∈ B δ ∩ Γ ± , P ( z ) = O (cid:0) | z | − / | z | − / (cid:1) z → , (87)and P ( z ) = N ( z ) ( I + O (1 /n )) z ∈ ∂B δ \ ( R + ∪ Γ ± ) . Observe that at this stage we have disregarded the jump of T on ( − δ, x / e − n ( g + g − − g ) , because, according to item ( ii ) of Proposition 4, the off-diagonal term converges to 0 uniformlyin n .Let (cid:101) P ( z ) = P ( z ) diag (cid:32) √ − υ − n e − nψ ( z ) / , √ − υ − n e nψ ( z ) / , (cid:33) , z ∈ B δ \ R , with the square root (well defined for n large enough) positive on R + . Using (48) and (52)we conclude that (cid:101) P is also holomorphic in B δ \ ( R + ∪ Γ ± ), with (cid:101) P + ( z ) = (cid:101) P − ( z ) J (cid:101) P ( z ), where J (cid:101) P ( z ) = − , x ∈ B δ ∩ (0 , p + ) , J (cid:101) P ( z ) = , z ∈ B δ ∩ Γ ± . Also, the local behavior of (cid:101) P at the origin matches that of P (see (87)).Parametrix P will be built in terms of the modified Bessel functions of order 0 see [30,Section 8.2.1]. Namely, with the modified Bessel functions I and K , and the Hankel func-tions H (1)0 and H (2)0 (see [1, Chapter 9]), we define a 2 × L ( ζ ) for | arg ζ | < π/ L ( ζ ) = (cid:32) I (2 ζ / ) iπ K (2 ζ / )2 πiζ / I (cid:48) (2 ζ / ) − ζ / K (cid:48) (2 ζ / ) (cid:33) . (88)For 2 π/ < arg ζ < π we define it as L ( ζ ) = H (1)0 (2( − ζ ) / ) H (2)0 (2( − ζ ) / ) πζ / (cid:16) H (1)0 (cid:17) (cid:48) (2( − ζ ) / ) πζ / (cid:16) H (2)0 (cid:17) (cid:48) (2( − ζ ) / ) . (89)And finally for − π < arg ζ < − π/ L ( ζ ) = H (2)0 (2( − ζ ) / ) − H (1)0 (2( − ζ ) / ) − πζ / (cid:16) H (2)0 (cid:17) (cid:48) (2( − ζ ) / ) πζ / (cid:16) H (1)0 (cid:17) (cid:48) (2( − ζ ) / ) . (90)30 + ∆ + ∆ − Γ + Γ − p − Figure 7: Contours for R .With this definition we take Ψ ( s ) = (cid:18) σ L ( − s ) σ (cid:19) , σ = (cid:18) − (cid:19) . As in [30], we conclude that P ( z ) = E ( z ) Ψ (cid:0) n f ( z ) (cid:1) diag (cid:32) √ − υ − n e nψ ( z ) / , √ − υ − n e − nψ ( z ) / , (cid:33) , (91)where E ( z ) = N ( z ) diag (cid:18) √ (cid:18) ii (cid:19) , (cid:19) diag (cid:16) (2 πn ) / f ( z ) / , (2 πn ) − / f ( z ) − / , (cid:17) , (92)and f ( z ) = (cid:20) 34 ( ψ ( z ) − ψ (0)) (cid:21) / . (93) Recall that we denote generically by B δ the small disks around the branch points 0 and p ± ,and by P the local parametrices built in B δ . We define the matrix valued function R as R ( z ) = (cid:40) T ( z ) P − ( z ) , in the neighborhoods B δ , T ( z ) N − ( z ) , elsewhere. (94)Then R is defined and analytic outside the real line, the lips ∆ ± and Γ ± of the lenses andthe circles around the three branch points. The jump matrices of T and N coincide on( −∞ , p − ) and (0 , p + ) and the jump matrices of T and P coincide inside the three disks withthe exception of the interval ( − δ, R has an analytic continuation to thecomplex plane minus the contours shown in Figure 7.We can follow the arguments in [30, Section 9] to conclude that R ( z ) = I + O (cid:18) n ( | z | + 1) (cid:19) , n → ∞ , (95)uniformly for z in the complex plane outside of these contours.31 Asymptotics Now we unravel all the transformations in order to get the asymptotic results from Theorems 1and 2.Assume first that z lies outside the small disks B δ around the branch points 0 and p ± , sothat T ( z ) = R ( z ) N ( z ) = (cid:18) I + O (cid:18) n ( | z | + 1) (cid:19)(cid:19) N ( z ) . Assume further that z lies in one of the unbounded component of the complement to thecurves depicted in Figure 7. By (58), (60) and (61), Y ( z ) = diag (cid:0) e − nω , , (cid:1) (cid:18) I + O (cid:18) n ( | z | + 1) (cid:19)(cid:19) × N ( z ) diag (cid:16) e n ( g ( z )+ ω ) , e − n ( g ( z ) − g ( z )) , e − ng ( z ) (cid:17) (cid:18) A − ∗ ( z ) (cid:19) , where A ∗ stands either for A L or A R .Thus, Y ( z ) = (1 , , Y ( z ) = (cid:0) e − nω , , (cid:1) (cid:18) I + O (cid:18) n ( | z | + 1) (cid:19)(cid:19) N ( z ) e n ( g ( z )+ ω ) = e ng ( z ) (1 , , (cid:18) I + O (cid:18) n ( | z | + 1) (cid:19)(cid:19) N ( z ) = e ng ( z ) (1 , , (cid:18) I + O (cid:18) n ( | z | + 1) (cid:19)(cid:19) N ∗ ( z )= e ng ( z ) (cid:18) N ( z ) + O (cid:18) n ( | z | + 1) (cid:19)(cid:19) . It remains to use Proposition 5 to establish (18).In the same fashion, if z lies on the +-side of (0 , p + ), that is, in a domain of the formΩ = { z ∈ C : (cid:60) z ∈ ( ε, p + − ε ) , (cid:61) z ∈ [0 , ε ) } , where ε > Y ( z ) = diag (cid:0) e − nω , , (cid:1) (cid:18) I + O (cid:18) n (cid:19)(cid:19) N ( z ) − υ − n e nψ ( z ) × diag (cid:16) e n ( g ( z )+ ω ) , e − n ( g ( z ) − g ( z )) , e − ng ( z ) (cid:17) (cid:18) A − R ( z ) (cid:19) , 32o that Y ( z ) = (1 , , Y ( z ) = e ng ( z ) (1 , , (cid:18) I + O (cid:18) n (cid:19)(cid:19) N ( z ) − υ − n e nψ ( z ) = e ng ( z ) (1 , , (cid:18) I + O (cid:18) n (cid:19)(cid:19) (cid:18) N ∗ ( z ) + 1 − υ − n e nψ ( z ) N ∗ ( z ) (cid:19) , which proves (19) with the aid of Proposition 5. Now formula (20) follows from (48).In the same vein, Y + ( x ) = e nω diag (cid:0) e − nω , , (cid:1) R + ( x ) N + ( x ) e ng ( x )1 − υ − n ( x )4 e ng − ( x )) , and (cid:0) , w ,n ( y ) , w ,n ( y ) (cid:1) Y + ( y ) − = 4 e − nω − υ − n ( y ) (cid:16) − − υ − n ( y )4 e − ng ( y ) , e − ng − ( y ) , (cid:17) × N − ( y ) R − ( y ) diag ( e nω , , , where we have used the explicit expression for A R , the boundary values (48), and the equi-librium conditions ( i ) from Proposition 4. In consequence, by formula (22), K n ( x, y ) = 12 πi ( x − y ) 41 − υ − n ( y ) (cid:16) − − υ − n ( y )4 e − ng ( y ) , e − ng − ( y ) , (cid:17) × N − ( y ) R − ( y ) R + ( x ) N + ( x ) e ng ( x )1 − υ − n ( x )4 e ng − ( x )) . We have N − ( y ) R − ( y ) R + ( x ) N + ( x ) = N − ( y ) (cid:18) I + O (cid:18) x − yn (cid:19)(cid:19) N + ( x )= I + O ( x − y ) as y → x. K n ( x, y ) = 12 πi ( x − y ) 41 − υ − n ( y ) (cid:16) − − υ − n ( y )4 e − ng ( y ) , e − ng − ( y ) , (cid:17) × ( I + O ( x − y )) e ng ( x )1 − υ − n ( x )4 e ng − ( x )) = 12 πi ( x − y ) (cid:18) − e − n ( g ( x ) − g ( y )) + 1 − υ − n ( x )1 − υ − n ( y ) e − n ( g − ( x ) − g − ( y )) + O ( x − y ) (cid:19) = 12 πi ( x − y ) (cid:16) − e − n ( g ( x ) − g ( y )) + e − n ( g − ( x ) − g − ( y )) (cid:17) + O (1) , y → x. Using (44) we conclude that K n ( x, x ) = nλ (cid:48) ( x ) + O (1) , n → ∞ . On the other hand, if we take x n = x ∗ + xnλ (cid:48) ( x ∗ ) , y n = x ∗ + ynλ (cid:48) ( x ∗ ) , we get K n ( x n , y n ) = nλ (cid:48) ( x ∗ ) π ( x − y ) (cid:18) − e − n ( g ( x n ) − g ( y n )) + e − n ( g − ( x n ) − g − ( y n )) + O (cid:18) n (cid:19)(cid:19) = nλ (cid:48) ( x ∗ ) π ( x − y ) (cid:18) e πi ( x − y ) − e − πi ( x − y ) + O (cid:18) n (cid:19)(cid:19) . This concludes the proof of Theorem 2. Acknowledgements The first author (AIA) received support from RFBR grant 13-01-12430 (OFIm) and theExcellence Chair Program sponsored by Universidad Carlos III de Madrid and the Bank ofSantander. The second (GLL) and the third (AMF) authors were supported by MICINN ofSpain under grants MTM2012-36732-C03-01 and MTM2011-28952-C02-01, respectively, andby the European Regional Development Fund (ERDF). Additionally, AMF was supported byJunta de Andaluca (the Excellence Grant P11-FQM-7276 and the research group FQM-229)and by Campus de Excelencia Internacional del Mar (CEIMAR) of the University of Almera.This work was completed during a visit of AMF to the Department of Mathematics of theVanderbilt University. He acknowledges the hospitality of the hosting department, as well asa partial support of the University of Almer´ıa through the travel grant EST2014/046.34 eferences [1] M. Abramowitz and I. A. Stegun, editors. 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