Jacobi matrices on trees generated by Angelesco systems: asymptotics of coefficients and essential spectrum

We continue studying the connection between Jacobi matrices defined on a tree and multiple orthogonal polynomials (MOPs) that was discovered previously by the authors. In this paper, we consider Angelesco systems formed by two analytic weights and obtain asymptotics of the recurrence coefficients and strong asymptotics of MOPs along all directions (including the marginal ones). These results are then applied to show that the essential spectrum of the related Jacobi matrix is the union of intervals of orthogonality.

Asymptotics of a n,1 , a n,2 along Non-fully Marginal Sequences 43 8.3. Asymptotics of b n,1 , b n,2 along Non-fully Marginal Sequences 44

Introduction
It is well-known [2] that the spectral theory of one-sided self-adjoint Jacobi matrices can be naturally studied in the context of polynomials orthogonal on the real line and, conversely, many results in the latter topic find operator-theoretic interpretation. In [7], we discovered that a wide class of multiple orthogonal polynomials, e.g., celebrated Angelesco systems, is connected to self-adjoint Jacobi matrices defined on rooted Cayley trees. The present paper makes further step in this direction. We perform a case study of Angelesco systems with two measures of orthogonality given by analytic weights. Our analysis of the related matrix Riemann-Hilbert problem provides the asymptotics of the recurrence coefficients and strong asymptotics of MOPs for all large indices. One application of this precise asymptotic analysis is a characterization of the essential spectrum of the associated Jacobi matrix.
We start this introduction by recalling some definitions and main relations connecting Jacobi matrices on trees and MOPs and then state the main results of the paper. In what follows, we let N :" t1, 2, . . .u and Z ě0 :" t0, 1, 2, . . .u. We write | n| :" n 1`¨¨¨`nd for n " pn 1 , . . . , n d q P Z d ě0 , and let e 1 " p1, 0, . . . , 0q, . . . , e d " p0, . . . , 0, 1q, 1 " p1, . . . , 1q " e 1`¨¨¨` e d . Given an operator A in the Banach space, the symbols σpAq and σ ess pAq will denote its spectrum and essential spectrum, respectively [33]. In a metric space, B r pXq denotes the closed ball with center at X and radius r. For a complex number z, z and z are its real and imaginary parts, respectively. For a function f pzq, holomorphic in C`, the upper half-plane, its boundary values on R are denoted by f`pxq.
1.1. Jacobi matrices on trees. Denote by T an infinite pd`1q-homogeneous rooted tree (rooted Cayley tree) and by V the set of its vertices with O being the root. On the lattice N d , consider an infinite path t n p1q , n p2q , . . .u that starts at 1 (i.e., n p1q " 1) and satisfies n pj`1q " n pjq` e kj , k j P t1, . . . , du for every j " 0, 1, . . .. Clearly, these are paths for which, as we move from 1 to infinity, the multi-index of each next vertex is increasing by 1 at exactly one position. Each such path can be mapped to non-selfintersecting path in T that starts at O (see Figure 1 for d " 2) and this map is one-to-one. This construction defines a projection Π : V Ñ N d as follows: given X P V we consider a path from O to X, map it to a path on N d and let ΠpXq be the endpoint of the mapped path. Every vertex Y P V, Y ‰ O, has the unique parent, which we denote by Y ppq . This allows us to define the following index function: imath imath (1.1) ı : V Ñ t1, . . . , du, Y Þ Ñ ı Y such that ΠpY q " ΠpY ppq q` e ı Y , and therefore to distinguish the "children" of each vertex Y P V by denoting Z " Y pchq,ı Z when Y " Z ppq , see Figure 1 (for d " 2).
p3, 1q p2, 2q p2, 2q " Y pchq,1 p1, 3q " Y pchq,2 For a function f on V, we denote by f Y its value at a vertex Y P V. Given P satisfying (1.2) and κ P R d with | κ| " 1, we define the corresponding Jacobi operator J κ by Jacobi Jacobi Thus defined operator J κ is bounded and self-adjoint on 2 pVq. The spectral theory of Jacobi matrices on trees enjoyed considerable progress in the last decade, see, e.g., [1,10,19,21,26,27]. In this paper, we will study Jacobi matrices on trees that are generated by multiple orthogonality conditions. 1.2. Multiple orthogonal polynomials and recurrence relations. In [7], we investigated properties of the operator J κ in the case when the coefficients P are the recurrence coeffcients for MOPs. We now recall some basic facts about multiple orthogonal polynomials.
Let µ :" pµ 1 , . . . , µ d q, d P N, be a vector of positive finite Borel measures defined on R and n be a given a multi-index in Z d ě0 , | n| ě 1. Type I MOPs A x l Q n pxq " 0, l ă | n|´1, A piq 1´ ei " 0.
The polynomials of both types always exist, but their uniqueness is not guaranteed. If degpP n q " | n| for every non-identically zero polynomial P n pxq satisfying (1.5), then the multi-index n is called normal. In this case P n pxq is unique up to a multiplicative factor and we normalize it to be monic, i.e., P n pxq " x | n|`¨¨¨. It turns out that n is normal if and only if the linear form Q n pxq is defined uniquely up to multiplication by a constant. In this case, we will normalize it by n_2 n_2 (1.6) ż x | n|´1 Q n pxq " 1 .
We will say that vector µ is called perfect if all the multi-indices n P Z d ě0 are normal. When µ is perfect, it is known [35] that the polynomials P n pxq and the forms Q n pxq satisfy the following nearest-neighbor recurrence relations (NNRRs): recurrence recurrence (1.7) # zP n pzq " P n` ej pzq`b n,j P n pzq`ř d i"1 a n,i P n´ ei pzq, zQ n pzq " Q n´ ej pzq`b n´ ej ,j Q n pzq`ř d i"1 a n,i Q n` ei pzq, for each j P t1, . . . , du.
For the coefficients ta n,i , b n,i u, we have representations [7]: por1 por1 (1.8) a n,j " ş P n pxq x nj dµ j pxq ş P n´ ej pxq x nj´1 dµ j pxq , b n´ ej ,j " ż x | n| Q n pxq´ż x | n|´1 Q n´ ej pxq .
If d ą 1, unlike in one-dimensional case, we can not prescribe ta n,j u and tb n,j u arbitrarily. In fact, these coefficients satisfy the so-called "consistency conditions" which is a system of nonlinear difference equations. This discrete integrable system and the associated Lax pair were studied in [8,35].
Angelesco systems form an important subclass of the perfect systems. They were studied by Angelesco already in 1919, [4]. It is not difficult to see [7] that the corresponding NNRR coefficients satisfy conditions (1.2) and thus define the Jacobi matrix J κ by (1.3).
The asymptotic behavior of these coefficients ta n,j , b n,j u for the ray sequences regime, namely when multi-indices multi-indices (1.10) N c " t nu : n i " c i | n|`o` n˘, i P t1, . . . , du, | c | :" was studied in [7] for c " pc 1 , . . . , c d q P p0, 1q d (hereafter, lim N c stands for the limit as | n| Ñ 8, n P N c ). The following theorem was proved.
m:recurrenceOld Theorem 1.1 ([7]). Let µ be Angelesco system (1.9) such that for each i P t1, . . . , du the measure µ i is absolutely continuous with respect to the Lebesgue measure on ∆ i and the density µ 1 i pxq :" dµ i pxq{dx extends to a holomorphic and non-vanishing function in some neighborhood of ∆ i . Then the ray limits (1.10) of coefficients a n,i , b n,i ( from (1.7) exist for any c P p0, 1q d : limit limit (1.11) lim N c a n,i " A c,i and lim N c b n,i " B c,i , i P t1, . . . , du.
This result and expressions for A c,i and B c,i were obtained from the strong asymptotics of the MOPs also established in [7]. In Section 2, we will recall the formulas for A c,i , B c,i which show that these numbers depend on t∆ i u d i"1 only.
1.4. Results and structure of the paper. In this paper, we restrict ourselves to the case d " 2.
Our main technical achievement is an extension of the results in [7] on the strong asymptotics of the Angelesco MOPs to the full range of c : c P r0, 1s 2 . As a corollary of this extension, we get the following result. exist for any c P r0, 1s and i P t1, 2u, where N c :" N pc,1´cq is any sequence satisfying (1.10).
The dependence of the intervals ∆ c,i on the parameter c is described in greater detail in Section 4. Let R c , c P p0, 1q, be a 3-sheeted Riemann surface realized as follows: cut a copy of C along ∆ c,1 Y ∆ c,2 , which henceforth is denoted by R p0q c , the second copy of C is cut along ∆ c,1 and is denoted by R p1q c , while the third copy is cut along ∆ c,2 and is denoted by R p2q c . These copies are then glued to each other crosswise along the corresponding cuts, see Figure 2. It can be easily verified that thus constructed Riemann surface has genus 0. We denote by π the natural projection from R c to C and employ the notation z for a generic point on R c with πpzq " z as well as z piq for a point on R piq c with πpz piq q " z. Since R c has genus zero, one can arbitrarily prescribe zero/pole divisors of rational functions on R c as long as the degree of the divisor is zero. Clearly, a rational function with a given divisor is unique up to multiplication by a constant.
prop:angelesco Proposition 2.1. Let R c , c P p0, 1q, be as above and χ c pzq be the conformal map of R c onto C such that chi chi (2.4) χ c`z p0q˘" z`O`z´1˘as z Ñ 8.
Even though the expression for B 0,1 might seem strange, it has a meaning from the point of view of spectral theory of Jacobi matrices, see (A.8).
We prove Proposition 2.1 in Section 5. It is worth noting that the constants A c,1 and A c,2 are always positive. Indeed, denote by α 1 , β c,1 , α c,2 , β 2 the ramification points of R c with natural projections α 1 , β c,1 , α 2 , β c,2 , respectively. Then the symmetries of R c and χ c pzq yield that χ c pzq is real and changes from´8 to 8 when z moves along the cycle whose natural projection is the extended real line. Thus, χ c pzq is increasing when it moves past 8 p1q and 8 p2q , which yields the claim (this argument also shows that B c,1 ă B c,2 ). ss:2.2 2.2. Proof of Theorem 1.3. Our proof will be based on a characterization of the essential support of a Jacobi matrix on a tree obtained in [11,Theorem 4]. We need some preliminaries to formulate this result. Suppose T is a 3-homogeneous rooted tree with root at O (a binary tree), which means that O has two neighbors and any other vertex has three neighbors. Later in the text, we will use notation Z " Y to indicate that vertices Z and Y are neighbors and the symbol V will denote the set of all vertices of T . Given a real function V defined on V and a real positive function W defined on all edges, we make an assumption Isad1 Isad1 defined on 2 pVq. One example one can think of is J κ introduced in (1.3). Consider a set of distinct vertices (a path) tY n u, n P N, in V such that Y n " Y n`1 for every n. Clearly, every such path on the tree escapes to infinity, i.e., distpO, Y n q Ñ 8, n Ñ 8. We want to define R-limit (or "right limit") of J along this path. To do that, suppose G is a 3-homogeneous tree (without a root), O 1 is a fixed vertex on G, and J 1 is a bounded self-adjoint operator on G. Recall that B r pY q stands for the ball of radius r centered at Y and denote the restriction operator to this ball by P BrpY q . Consider two finite matrices: P BrpYn j q J P BrpYn j q and P BrpO 1 q J 1 P BrpO 1 q . If we identify 2 pB r pO 1 qq and 2 pB r pY nj qq by following the structure of the tree (and there are many ways to do that), then these matrices are defined on the same finite dimensional Euclidean space. If this identification can be done so that all sections of J 1 appear as the limits, we call J 1 an R-limit or right limit: Definition. We say that J 1 is an R-limit of J along tY n u if there is a subsequence tn j u such that P BrpYn j q J P BrpYn j q Ñ P BrpO 1 q J 1 P BrpO 1 q as j Ñ 8 for every fixed r P N. Matrix J 1 is called simply an R-limit of J if there exists a path along which J 1 is an R-limit of J .
Remark. For the rigorous definition of R-limit on more general graphs, see [11].
Theorem 2.1 (Theorem 4 in [11]). We have Remark. [11,Theorem 4] was stated for the regular trees only, but the proof is valid for rooted trees as well. Auxiliary operators L p1q c and L p2q c . Recall that T denotes the 3-homogeneous rooted tree with the root denoted by O and V stands for the set of all its vertices. There are two edges meeting at the root O. We label one of them type 1 and the other one -type 2. Now, consider two vertices that are at distance 1 from O. Each of them is coincident with exactly three edges. One of the edges for each vertex was labelled already, and we label the remaining two as an edge type 1 and an edge of type 2. We continue inductively by considering all edges that are at distance 2, 3, etc. from O and calling one of the unlabelled edges type 1 and the other one type 2. Now that all edges of T have types assigned to them, we continue by labeling the vertices. If a vertex Y meets two edges of type 1 and one edge of type 2, we call it a vertex of type 1; otherwise, if it is incident with two edges of type 2 and one edge of type 1, we call it type 2. We do not need to assign any type to the root O. At a vertex Y ‰ O of type ι Y , see (1.1), we define both operators L p1q c and L p2q c by the same formula: and at the root O we define the operators L p1q c and L p2q c differently from each other by Notice that these operators represent Jacobi matrices on T when c P p0, 1q. However, if c P t0, 1u either A c,1 or A c,2 becomes zero and L p1q c , L p2q c are no longer Jacobi matrices, strictly speaking.
Remark. The operators L p1q c and L p2q c already appeared in [7] as the strong limits of Jacobi matrices on finite trees that correspond to tP n u, the polynomials of the second type (see formula (3.3) and Subsection 4.5 in [7]).
Lemma 2.1. If J has coefficients in P Ang p∆ 1 , ∆ 2 q, then the R-limits of J and the R-limits of L plq c , l P t1, 2u, are related by the following identity sad1 sad1 Proof. This follows from the definition of the R-limit, construction of L p1q c and L p2q c , and from the assumption (1.13).
We further study auxiliary operators L p1q c and L p2q c in Appendix A.
Proof of Theorem 1.3. Assumptions (1.13) characterize the behavior of the coefficients at infinity. Thus, Weyl's theorem on the essential spectrum [33] implies that any two Jacobi matrices with parameters in P Ang p∆ 1 , ∆ 2 q have the same essential spectra. Moreover, by the same Weyl's theorem, this essential spectrum is independent of the choice of parameter κ in (1.3). Hence, it is enough to prove the theorem for the Jacobi matrix J κ generated by some Angelesco system with analytic weights and with κ " e 2 . In [7, Section 4] we established that ∆ 1 Y ∆ 2 Ď σpJ e2 q. Thus, ∆ 1 Y ∆ 2 Ď σ ess pJ e2 q as follows from the definition of the essential spectrum.
To prove the opposite inclusion, take any J for which the coefficients belong to P Ang p∆ 1 , ∆ 2 q. The application of Theorem 2.1 and Theorem A.
which yields an inclusion sad3 sad3 where the last equality follows from the properties of ∆ c,1 and ∆ c,2 (which we also discuss later in Proposition 4.1). Moreover, since In this section we state the results on asymptotic behavior of the forms Q n pxq and polynomials P n pxq defined in (1.4) and (1.5), respectively, along ray sequences N c " N pc,1´cq defined in (1.10) under the assumption that the measures of orthogonality are as in Theorem 1.1. Study of strong asymptotics of multiple orthogonal polynomials has long history, see for example [25,29,5,37]. Below, we follows the Riemann-Hilbert approach used in [37].
Throughout the paper, we use the following notation: given a system of Jordan arcs and curves Σ, we denote by Σ˝the subset of Σ consisting of points that possess a neighborhood that is separated by Σ into exactly two connected components. In particular, ∆i " pα i , β i q, i P t1, 2u.
3.1. Fully Marginal Ray Sequences. In this subsection we consider solely infinite ray sequences of the form fullymarginal fullymarginal (3.1) N i´1 " t n : there exists C ą 0 such that n i ď Cu , i P t1, 2u.
To describe the asymptotics we need to introduce the so-called Szegő functions of the measures µ 1 , µ 2 . To this end, let us set rhoi rhoi Observe that pρ i w i`q pxq ą 0 for x P ∆i :" pα i , β i q, where w i pzq was introduced in Proposition 2.1. Put szego szego Then each S ρi pzq is a holomorphic and non-vanishing function in Cz∆ i that is uniquely (up to a sign) characterized by the properties 1 szego-pts szego-pts Then the following theorem holds.
We prove Theorem 3.1 in Section 6.
1 Apzq " Bpzq as z Ñ z 0 means that the ratio Apzq{Bpzq is uniformly bounded away from zero and infinity as z Ñ z 0 .

3.2.
Szegő Functions on R c . Let us set ∆ c,i :" π´1p∆ c,i q, i P t1, 2u, and orient it so that R p0q c remains on the left when the cycle is traversed in the positive direction. Put wi wi to be the branch holomorphic outside of ∆ c,i . In what follows, it will be convenient to introduce the following notation F pkq pzq :" F`z pkq˘, k P t0, 1, 2u, for a function F pzq defined on R c zp∆ c,1 Y ∆ c,2 q. Then the following proposition holds.
We start by introducing an analog of the functions ϕ 1 pzq, ϕ 2 pzq in the non-fully marginal and non-marginal cases. Given a multi-index n, let cn cn (3.12) c n :" n 1 {| n|.
To alleviate the notation, in what follows we shall use the subindex n instead of c n for quantities depending on c n such that R n " R c n , S n pzq " S c n pzq, etc. We shall denote by Φ n pzq a rational function on R n which is non-zero and finite everywhere except at the points on top of infinity, has a pole of order | n| at 8 p0q , a zero of multiplicity n i at 8 piq for each i P t1, 2u, and satisfies normalization normalization n˘p zq " 1, z P C. Equality in (3.13) is a simple matter of a normalization since the logarithm of the absolute value of the left-hand side of (3.13) extends to a harmonic function on C which has a well defined limit at infinity and therefore is a constant.  Given c P r0, 1s, let N c " t nu be a sequence for which (3.11) holds. Then for all | n| large enough, n P N c , we have that $ & % P n pzq " p1`op1qqγ n`S n Φ n˘p 0q pzq, where the relations holds uniformly on closed subsets of Czp∆ c,1 Y∆ c,2 q and compact subsets ∆c ,1 Y∆c ,2 , respectively, and γ n is the constant such that lim zÑ8 γ n z | n|`S n Φ n˘p 0q pzq " 1.
When c ‰ c˚, c˚˚, see Proposition 4.1 further below, the the error rate op1q can be replaced by O c pε n q, where the dependence of O c pε n q on c is uniform for c on compact subsets r0, 1sztc˚, c˚˚u.
In the above theorem the functions S p0q n pzq could be replaced by their limits as discussed in Proposition 3.1. However, we can do this only at the expense of the error rate O c pε n q.
To describe asymptotic behavior of the forms Q n pxq, we need to introduce one additional function. Let Π n pzq be a rational function on R n with the zero/pole divisor and the normalization given by 1´β n,1´α n,2´β2 and Π p0q n p8q " 1, where α 1 , β n,1 , α n,2 , β 2 are the ramification points of R n . Then the following theorem holds. i P t1, 2u. Given c P r0, 1s, let N c " t nu be a sequence for which (3.11) holds. Then for all | n| large enough, n P N c , we have that A piq n pzq "´p1`op1qq pΠ piq n w n,i qpzq γ n pS n Φ n q piq pzq , uniformly on closed subsets of Cz∆ c,i for i P t1, 2u when c P p0, 1q, i " 2 when c " 0, and i " 1 when c " 1, while A piq n pzq " op1q´τ n`w n,i Φ piq n˘p zq¯´1 , uniformly on closed subsets of Cz∆ 0,1 for i " 1 when c " 0 and of Cz∆ 1,2 for i " 2 when c " 1, where τ n :" γ n S p0q n p8q, i.e., it is a constant such that lim zÑ8 τ n |z| n Φ p0q n pzq " 1. Moreover, A piq n pxq "´p1`op1qq pΠ piq n w n,i q`pxq γ n pS n Φ n q piq pxq´p 1`op1qq pΠ piq n w n,i q´pxq γ n pS n Φ n q piq pxq , uniformly on compact subsets of ∆c ,i , i P t1, 2u. As in the case of Theorem 3.2, the error rate can be improved to O c pε n q when c P r0, 1sztc˚, c˚˚u with dependence on c being locally uniform.
Let`p µ 1 , p µ 2˘b e a vector of Markov functions of the measures µ i , that is, Observe also that pp µ i`´p µ i´q pxq " ρ i pxq, x P ∆i , by Sokhotski-Plemelj formulae. Then one can deduce from orthogonality relations (1.5) that there exist polynomials P piq n pzq such that R piq n pzq :"`P n p µ i´P It also follows from (1.4) that there exists polynomial A n pxq such that Ln Ln where the asymptotic formula is valid for z Ñ 8. Then the following result holds.

thm:asymp4
Theorem 3.4. Under the conditions of Theorems 3.2-3.2, it holds for all n P N c with | n| large enough that R piq n pzq " p1`op1qqγ n`S n Φ n˘p iq pzqw´1 n,i pzq, uniformly on closed subsets of Cz∆ c,i , that is, including the traces on ∆ i z∆ c,i for i P t1, 2u when c P p0, 1q, for i " 2 when c " 0, and for i " 1 when c " 1, while R piq n pzq " op1qτ n Φ piq n pzqw´1 n,i pzq uniformly on closed subsets of Cz∆ 0,1 for i " 1 when c " 0 and of Cz∆ 1,2 for i " 2 when c " 1. Moreover, uniformly on closed subsets of Czp∆ c,1 Y ∆ c,2 q. As in Theorems 3.2 and 3.3 the error rate can be improved to O c pε n q when c P r0, 1sztc˚, c˚˚u with dependence on c being locally uniform. In this section we discuss further properties of the vector equilibrium problem (2.2)-(2.3) as well as prove some auxiliary lemmas needed later.
With the notation introduced after (2.3), the following proposition holds. prop:1 Moreover, it holds that 2 for i P t1, 2u, where the convergence of potentials is uniform on compact subsets of C. Furthermore, and V ωc,i Ñ V ωi uniformly on compact subsets of C as c Ñ 2´i, i P t1, 2u.
Further, recall the surface R c constructed just before Proposition 2.1. Given a rational function F pzq on R c , we denote its divisor of zeros and poles by pF q and write pF q " m 1 z 1`¨¨¨`ml z l´k1 p 1´¨¨¨´kt p t to mean that F pzq has a zero of order m i at z i for each i P t1, . . . , lu, a pole of order k i at p i for each i P t1, . . . , tu, and otherwise it is non-vanishing and finite, where necessarily ř l i"1 m i " ř t i"1 k i . It can be easily checked using Schwarz reflection principle, as it was done in [37, Proposition 2.1] for c rational, that the function  H c pzq :" . Therefore, the function h c pzq :" 2B z H c pzq, where 2B z :" B x´i B y , is rational on R c . In fact, it holds that  for z P R n , where the constant C n should be chosen so that (3.13) is satisfied.
prop:2 Proposition 4.2. Let D c :" α 1`βc,1`αc,2`β2 be the divisor 3 of the ramification points of R c . It holds that divisor-hc divisor-hc (4.4) ph c q " 8 p0q`8p1q`8p2q`z c´Dc ‹ pz p2q q " 0 as this sum must be an entire function that vanishes at infinity. Normalize h ‹ pzq so that h ‹ pz p0q q " 1{z`Op1{z 2 q as z Ñ 8. Set c ‹ :"´lim zÑ8 zhpz p1q q. Then R ‹ " R c‹ , z ‹ " z c‹ , and respectively h ‹ pzq " h c‹ pzq. , it also holds that the left hand sides of (2.3) are strictly less than zero on ∆ 1 z∆ c,1 and ∆ 2 z∆ c,2 , respectively, see [22]. In particular, we can write where K Ď rα 1 , β 1 s is compact, cappKq is the logarithmic capacity of K, and ω K is the logarithmic equilibrium distribution on K (when K is an interval, ω K is the arcsine distribution on K). As mentioned before (2.3), the maximizer of this functional is an interval containing α 1 (this was proven in [22]). Therefore, it is enough to consider compact sets K only of the form rα 1 , βs. Thus, the functional F pKq reduces to the function where we used explicit expressions for the logarithmic capacity and the equilibrium measure of an interval. To find the maximum of F c pβq on ∆ 1 , let us compute its derivative. To this end, it can be readily checked that A divisor is any formal linear combination of points of Rc with integer coefficients.
for every differentiable function f pxq on ∆ 1 . Observe also that V ωc,2 pxq is harmonic off ∆ 2 and therefore f c pxq :" V ωc,2 pxq "´ş log |x´y|dω c,2 pyq is a smooth function on ∆ 1 . Hence, by taking the limit as h Ñ 0 in the above equality, we get Fprime Fprime It is also obvious that f 1 c pxq " ş py´xq´1dω c,2 pyq, which is an increasing positive function on ∆ 1 . Thus, F 1 c pβq is a decreasing function of β and therefore has at most one zero. Moreover, it holds that fprime-bounds fprime-bounds Hence, F 1 c pβ 1 q ă 0 for all c small. As lim βÑα1 F 1 c pβq "`8, we get that β c,1 P pα 1 , β 1 q for all c small. Using F 1 c pβ c,1 q " 0 and the above estimates, we get from (4.5) that bc1 bc1 for all small c. This, in particular, implies that β c,1 Ñ α 1 as c Ñ 0. An analogous argument shows that α c,2 approaches β 2 when c Ñ 1. It further follows from (4.6) that f 1 c pxq uniformly converges to zero on ∆ 1 as c Ñ 1. Thus, F 1 c pβq ą 0 for all β P ∆ 1 and all c close to 1. That is, ∆ c,1 " ∆ 1 in this case. Similarly, we also get that ∆ c,2 " ∆ 2 for all c small.
Let us now describe what happens to the components of the vector equilibrium measure and their potentials as c Ñ 0. Clearly, V ωc,1 pzq Ñ 0 uniformly on compact subsets of Cz∆ 0,1 in this case. To show that ω c,2Ñ ω 2 as c Ñ 0, notice that for any Borel measure σ supported on ∆ 2 since ω 2 is a probability measure. It follows from (2.3) that V ωc,2 pxq is continuous on ∆ 2 " ∆ c,2 . Therefore, which implies that c,2 " 2 2`o p1q as c Ñ 0. Let ω be a weak˚limit point of ω c,2 as c Ñ 0. Then ω is a probability measure and where the first inequality follows from the Principle of Descent [34, Theorem I.6.8]. Therefore, Epω, ωq ď 2 " Epω 2 , ω 2 q, which implies that ω " ω 2 by the uniqueness of the equilibrium measure. To deduce the behavior of the constants c,1 as c Ñ 0, observe that # V 2ωc,1`ωc,2 pxq ď c,1 , x P p´8, α 1 s, where the first claim can be easily obtained from (2.3) and the second one was already mentioned at the beginning of the proof. Then V 2ωc,1`ωc,2 pα 1´ q ď c,1 ď V 2ωc,1`ωc,2 pα 1` q for any ą 0 since β c,1 ă α 1` for all c small enough. Hence, we get that Since V ω2 pxq is continuous on the real line and is arbitrary, we get that c,1 Ñ V ω2 pα 1 q as c Ñ 0. The respective claims for the limits as c Ñ 1 can be shown in a similar fashion. Let us point out one consequence of the fact that ω c,2Ñ ω 2 as c Ñ 0 that will be useful to us later. It holds that locally uniformly in Cz∆ 2 , where, as before, w 2 pzq :" a pz´α 2 qpz´β 2 q. Therefore, we can improve (4.7) to c-rate c-rate as c Ñ 0, where we again used (4.5).
The facts that ω c,iÑ ω c‹,i and c,i Ñ c‹,i as c Ñ c ‹ P p0, 1q, i P t1, 2u, were shown in the proof of [37, Proposition 2.1]. Let us now show that β c,1 Ñ β c‹,1 in this case (that is, that β c,1 is a continuous function of c). Weak˚convergence of measures necessitates that lim inf cÑc‹ β c,1 ě β c‹,1 . Assume to the contrary that there exists a subsequence c n Ñ c ‹ such that β c‹,1 ă β˚:" lim inf nÑ8 β cn,1 . Then for x P pβ c‹,1 , β˚q due to the Principle of Descent [34, Theorem I.6.8]. However, the above conclusion clearly contradicts the claim c,1 Ñ c‹,1 as c Ñ c ‹ . The convergence α c,2 Ñ α c‹,2 as c Ñ c ‹ can be shown analogously (unfortunately, this convergence of the endpoints was asserted without justification in the proof [37, Proposition 2.1]). Given the convergence of the endpoint, the uniform convergence of the potentials as c Ñ c ‹ P p0, 1q was established in the proof of [37, Proposition 2.1] using harmonicity of H c pzq. The same arguments can be applied to show that V ωc,i Ñ V ωi uniformly on compact subsets of C as c Ñ 2´i, i P t1, 2u.
Let us now establish the existence of the constants 0 ă c˚ă c˚˚ă 1 and the monotonicity properties of β c,1 and α c,2 . Claim (4.4) was obtained in [37, Proposition 2.3]. There it was further shown that special-zero special-zero Assume now that β c1,1 " β c2,1 ă β 1 . Then the functions h c1 pzq and h c2 pzq are defined on the same Riemann surface. Their difference has at least four zeros (double zero at 8 p0q and simple zeros at 8 p1q and 8 p2q ) and at most three poles α 1 , α 2 , β 2 . This is possible only if the function is identically zero and therefore c 1 " c 2 as h p1q c pzq " cz´1`Opz´2q by (4.2). Since β c,1 Ñ α 1 as c Ñ 0, this shows the existence of c˚and proves monotonicity of β c,1 as a function of c (it is a continuous and injective function of c). The existence of c˚˚and monotonicity of α c,2 are proven analogously. It also follows from (4.9) that c˚ď c˚˚. As it was shown in [37, Proposition 2.3] that z c˚" β c˚,1 p" β 1 q and z c˚˚" α c˚˚,2 p" α 2 q, we in fact get that c˚ă c˚˚.
It only remains to prove that z c is a continuous increasing function of c on rc˚, c˚˚s. To show monotonicity, take c˚ď c 1 ă c 2 ď c˚˚. It follows easily from (4.2) that each h c px p0q q is a decreasing function of x P pβ 1 , α 2 q. Thus, to prove that z c1 ă z c2 , it is enough to show that hpx p0q q ą 0 in pβ 1 , α 2 q, where hpzq :" ph c2´hc1 qpzq. Notice that hpx p0q q "´hpx p1q q´hpx p2q q by (4.2) and therefore it is sufficient to argue that hpx p1q q ă 0 on pβ 1 , 8q and hpx p2q q ă 0 on p´8, α 2 q. These claims are obvious for all |x| large enough since nd hpz p2q q " c 2´c1 z`O`z´2ȃ s z Ñ 8 according to (4.2). As explained after (4.9), hpzq vanishes only at 8 p0q , 8 p1q , and 8 p2q . Therefore, hpz p1q q and hpz p2q q cannot change sign on pβ 1 , 8q and p´8, α 2 q, respectively. Hence, these functions are negative everywhere on the considered rays by continuity.
To show continuity of z c as a function of c P rc˚, c˚˚s, we shall once again use the fact that h c px p0q q is a decreasing function on pβ 1 , α 2 q. When c P pc˚, c˚˚q, h c px p0q q is unbounded on both ends of pβ 1 , α 2 q and therefore changes sign from`to´when passing through z c (recall that h c pzq has poles at β 1 and α 2 in this case). When c " c˚, h c px p0q q is unbounded only at α 2 and, since it is non-vanishing, is negative on rβ 1 , α 2 q. Similarly, when c " c˚˚, it is unbounded at β 1 only and therefore is positive on pβ 1 , α 2 s. In any case, z c is the point where the potential V ωc,1`ωc,2 pxq achieves its minimum on rβ 1 , α 2 s. Thus, if z cn Ñ z ‹ as c n Ñ c ‹ when n Ñ 8, c n , c ‹ P pc˚, c˚˚q, then where the first inequality follows from the weak˚convergence of measures and the Principle of Descent [34, Theorem I. 6.8], the second one from the just discussed extremal property of z cn , and the last equality holds due to the weak˚convergence of measures and the fact that z c‹ does not belong to the supports of the measures in question. Since V ωc ‹,1`ωc‹,2 pxq is smallest at z c‹ , we get that z ‹ " z c‹ .
When c ‹ " c˚, essentially the same argument works. One just needs to replace z c‹ " β 1 with β 1` for any ą 0. Since V ωc ‹ ,1`ωc‹ ,2 pxq is increasing on rβ 1 , α 2 s, this shows that z ‹ ď z c‹` for any ą 0 and therefore z ‹ " z c‹ . Clearly, an analogous modification works when c ‹ " c˚˚.

Proof of Propositions 2.1 and 3.1 sec:5
On several occasions we shall refer to the following consequences of Koebe's 1{4-theorem, [30, where f pDq stands for image of a domain D under the function f pzq.
c . This is a meromorphic function in`R p0q c Y R p2q c˘z ∆ c,1 with a simple pole at 8 p0q , a simple zero at 8 p2q , and otherwise non-vanishing and finite. It is normalized so that ϕpz p0q q " z`Op1q as z Ñ 8.
Observe that ϕpzq continuously extends to the closed set R p0q Notice also that ϕ p0q pzq " ϕ 2 pzq for z P Cz∆ 2 , see (2.6). Define f c pzq :"`χ c`ϕ´1 pzq˘´B c,2˘{ z. Then f c pzq is a holomorphic function in CzI c,1 (there is no pole at the origin as ϕ´1p0q " 8 p2q and χ c pzq´B c,2 vanishes there) with bounded traces on I c,1 that assumes value 1 at infinity. Hence, it follows from Cauchy's integral formula that Since the traces f c˘p zq are bounded above in absolute value on I c,1 independently of c and |I c,1 | Ñ 0 as c Ñ 0, we see that f c pzq Ñ 1 as c Ñ 0 locally uniformly in Cztϕpα 1 qu. Hence, it holds that the desired limits (2.7) easily follow. Continuity of A c,1 , A c,2 , B c,1 , B c,2 as functions of c comes from the continuous dependence of α c,2 and β c,1 on c, see Proposition 4.2, and therefore the continuous dependence χ c pzq on c.

5.2.
Auxiliary Estimates, I. In the forthcoming analysis, the following functions will play an important role: It follows from the properties of χ c pzq, see (2.4) and (2.5), that Υ c,i pzq is a conformal map of R c onto C that maps 8 piq into 8 and 8 p0q into 0. Moreover, it holds that Upsilon2 Upsilon2 an open set such that π`R pkq c‹ zU˘" π`R pkq c zU c˘f or each k P t0, 1, 2u, then the bordered Riemann surfaces R c‹ zU and R c zU c are identical for all c sufficiently close to c ‹ and we can think of Υ c,i pzq as a function on R c‹ zU). On the other hand, when c Ñ 0, the following is true. lem:aux1 c , as c Ñ 0, where op1q holds uniformly on the entire surface R c and ψpzq :" s the conformal map of Cz∆ c,1 onto t|z| ą 1u that fixes the point at infinity and has positive derivative there. In addition, it holds that Υ p1q c,1 pzq " z´α 1`O pcq uniformly in C as c Ñ 0. Proof. Formula (5.6) follows immediately from (5.2), the very definition (5.3), and the first limit in (2.7). It also is immediate from (5.3) and (5.2) that lem51-4 lem51-4 in C (including the traces on ∆ 2 ) as c Ñ 0 since |ϕ p2q pzq| ď pβ 2´α2 q{4 ă |ϕpα 1 q|, see (2.6). It can be readily verified that the symmetric functions of the branches of a rational function on R c must be rational functions on C. Since Υ p1q c,1 pzq has a simple pole at infinity, Υ p0q c,1 pzq has a simple zero there, and Υ pkq c,1 pzq, k P t0, 1, 2u, are otherwise non-vanishing and finite, the product of three branches of Υ c,1 pzq must be a constant. Thus, similarly to (5.9), it holds that "´A Υ c,1 pzq. Therefore, Υ c,1 pzq " Υ c,1 pzq. In particular, Υ p2q c,1 pxq is real on ∆ c,1 and the traces of Υ pkq c,1 pzq on ∆ c,1 , k P t0, 1u, are conjugate-symmetric. Hence, we get from (5.9) and (5.10) that lem51-6 lem51-6 c,1˘p xqˇˇ2, x P ∆ c,1 , as c Ñ 0. Thus, (5.9), (5.11), and the maximum modulus principle applied to Υ p0q c,1 pzqφ c pzq and Υ p1q c,1 pzq{φ c pzq yield (5.7) with c 2 replaced by A c,1 . That is, we need to show that A c,1 " c 2 as c Ñ 0. As is mentioned above, the sum Υ p0q c,1 pzq`Υ p1q c,1 pzq`Υ p2q c,1 pzq is a rational function on C. Since it has only one pole, which is simple and located at infinity, it is a monic (see (5.4)) polynomial of degree 1.
To prove the last claim of the lemma, observe that Υ p1q as c Ñ 0 by (4.7) and (5.11). The desired claim now follows from the maximum modulus principle.

Thus, it holds that
Since A c,1 Ñ ppβ 1´α1 q{4q 2 by the limit analogous to the one for A c,2 in (2.7), this establishes the desired bounds in (5.15) and (5.16) for all c P r , 1q and any ą 0 fixed with the constants of proportionality dependent on and δ. On the other hand, the bounds for c P p0, s readily follow from (5.7) and (5.8) as and c|φ c pzq| " |z´α 1 | on K c,δ,1 and K c,δ,2 , respectively, as c Ñ 0 by elementary estimates and (4.7). The estimates of Υ pkq c,2 pzq can be verified similarly.
Let a function Π c pzq be defined on R c analogously to the way Π n pzq was defined on R n just before Theorem 3.3. Further, let Π c,i pzq, i P t1, 2u, be rational functions on R c with the divisors and normalization given by Pies Pies where D c is the divisor of the ramification points of R c , see Proposition 4.2. lem:aux2 Moreover, it holds that as c Ñ 0, where the first relation holds uniformly in C (that is, including the traces on ∆ c,1 Y ∆ 2 ) and the second one locally uniformly in Cz∆ 0,1 .
where C z psq is the third kind differential on R c with three simple poles at z, z 1 , z 2 that have the same natural projection z and respective residues´2, 1, 1. Limit (3.8) was in fact proven in [37,Section 7]. Thus, it only remains to show the validity of (3.9) and (3.10). In order to do that we shall use an alternative construction of S c pzq that is more amenable to asymptotic analysis.
Since we are interested in what happens when c Ñ 0, we shall assume that c ď mint1{2, c˚˚u (the choice of 1{2 is rather arbitrary, but convenient to use in (4.7)). Set where we take the branch of the square root such that D c,1 pzq is holomorphic and non-vanishing in the domain of the definition and has value 1 at infinity. The traces of D c,1 pzq on ∆ c,1 satisfy Let δ ą 0 be as in Lemma 5.2, that is, δ ď pα 2´β1 q{2. Then it follows from (4.7) that δc ď |∆ c,1 |{8. Using (4.7) once more together with our assumption that c ď 1{2, we get that (the constants in the above inequalities are in no way sharp, but sufficient for our purposes). Therefore, (5.23) and similar straightforward estimates of |2z´α 1´βc,1 | using (4.7) as well as (5.22) and the maximum modulus principle for holomorphic functions applied to both D c,1 pzq and D´1 c,1 pzq yield that uniformly on the respective sets, where the constants of proportionality do not depend on c, δ. Additionally, since β c,1 Ñ α 1 as c Ñ 0 and therefore w c,1 pzq " z´α 1`o p1q locally uniformly in Cz∆ 0,1 as c Ñ 0, it holds locally uniformly in Cz∆ 0,1 that Now, let D c,ρ1 pzq be the Szegő function of the restriction of ρ 1 pxq to ∆ c,1 normalized to have value 1 at infinity. That is, by Cauchy's theorem and integral formula. Hence, D c,ρ1 pzq " D c,µ 1 1 pzq is a holomorphic and nonvanishing function in Cz∆ c,1 with continuous and conjugate-symmetric traces on ∆ c,1 that satisfy according to Plemelj-Sokhotski formulae. Now, analyticity of ρ 1 pxq in a neighborhood of ∆ 1 implies that max xP∆c,1 |ρ 1 pxq{ρ 1 pα 1 q´1| Ñ 0 as c Ñ 0. Combining this estimate with (5.27) yields that when c Ñ 0 as well as that w c,1 pzq πi uniformly on compact subsets of Cz∆ 0,1 when c Ñ 0. Thus, it follows from the maximum modulus principle that prop-szego3 prop-szego3 (5.29) D c,ρ1 pzq " 1`op1q and G c,ρ1 "`1`op1q˘ρ 1 pα 1 q locally uniformly in Cz∆ 0,1 as c Ñ 0. One can also see from its very definition in (5.28) combined with the second formula of (5.29) that G c,ρ1 extends to a non-vanishing continuous function of c P r0, 1s (it is constant for all c ě c˚). This observation as well as (5.28) combined with positivity of ρ 1 pxq on ∆ 1 show that |D c,ρ1˘p xq| " 1 uniformly on ∆ c,1 for all c P p0, 1q. Then the maximum modulus principle for holomorphic functions applied to D c,ρ1 pzq and D´1 c,ρ1 pzq yields that uniformly in C for all c P p0, 1q (notice that |D c,ρ1 pzq| is a continuous function on the entire sphere C independent of c when c ě c˚). Let Γ c,2 :" χ c p∆ c,2 q, which are clockwise oriented analytic Jordan curves (recall that ∆ c,2 is oriented so that R p0q c remains on the left when ∆ c,2 is traversed in the positive direction and that χ c pzq is conformal on R c and maps 8 p0q into 8). The function is holomorphic and bounded in R c z∆ c,2 and has value 1 at 8 p0q . It follows from Plemelj-Sokhotski formulae that Observe also that pD c,1 D c,ρ1 qpπpzqq is holomorphic in a neighborhood of ∆ c,2 . Therefore, S c,2 pzq can be continued analytically across each side of ∆ c,2 . In fact, this continuation has an integral representation similar to (5.31), where one simply needs to homologously deform Γ c,2 within the domain of holomorphy of pD c,1 D c,ρ1 q`π`χ´1 c psq˘˘. Moreover, it holds that prop-szego5a prop-szego5a (5.33) S c,2 pzq " 1`op1q as c Ñ 0 and |S c,2 pzq| " 1, c P p0, c˚˚s, uniformly on R c (again, this means including the traces on ∆ c,2 ). Indeed, observe that the analytic curves Γ c,2 approach the circle |z´B 0,2 | " pβ 2´α2 q{4 ( by (2.7) and (5.2). Let δ ą 0 be small enough so that the integrand in (5.31) is analytic in a neighborhood of the closure of the annular domain bounded by Γ c,2 and C δ :" t|z´B 0,2 | " 2δ`pβ 2´α2 q{4u. Assuming that C δ is clockwise oriented, it follows from Cauchy's theorem that Γ c,2 can be replaced by C δ whenever z P R p2q c , i.e., whenever χ c pzq is interior or on Γ c,2 . Then it trivially holds that c now follows from (5.25) and (5.29) while the uniform boundedness follows from (5.24) and (5.30). Clearly, the estimates in the remaining part of R c can be obtained analogously by deforming Γ c,2 into the circles t|z´B 0,2 | " 2δ`pβ 2´α2 q{4u. As a part of the final piece of our construction, let Γ c,1 :" χ c p∆ c,1 q. Similarly to Γ c,2 , these are clockwise oriented analytic Jordan curves that collapse into a point B 0,1 by (2.7) and (5.2). Let which is a holomorphic and bounded function on R c that has value 1 at 8 p1q and whose traces on ∆ c,1 are continuous and satisfy by Plemelj-Sokhotski formulae. Notice that all the observation about analytic continuations (contour deformation) made for S c,2 pzq apply to S c,1 pzq as well. Since the Cauchy kernel is integrated against the pullback of a fixed function S ρ2 pzq{S ρ2 p8q from ∆ c,1 while the curves Γ c,1 collapse into a point, straightforward estimates of Cauchy integrals as well as analytic continuation (deformation of a contour) technique yield that prop-szego5b prop-szego5b (5.36) S c,1 pzq " 1`op1q as c Ñ 0 and |S c,1 pzq| " 1, c P p0, c˚˚s, locally uniformly on pR p0q c YR p2q c˘z ∆ c,1 and uniformly on R c , respectively. To examine what happens to S c,1 pzq on R p1q c , given ą 0, let C :" t|z´B c,1 | " u be clockwise oriented circle. It follows from (5.2) that the Jordan curve χ´1 c pC q belongs to R p0q c and is homologous to ∆ c,1 for all c sufficiently small. A straightforward computation shows that It further follows from (2.7) and (5.2) that Jordan curves π`χ´1 c pC q˘converge to the analytic Jordan curve pϕ 2`B0,2 q´1pC q (recall that ϕ 2 pzq " ϕ p0q pzq) and the latter curves collapse into a point α 1 as Ñ 0. Hence, by taking the limit as c Ñ 0 and then the limit as Ñ 0 of the Op¨q in (5.37) gives 0. Therefore, analytic continuation (deformation of a contour) technique and (5.34) imply that Finally, we are ready to state an alternative formula for the functions S c pzq when c ď c˚˚.  S p0q c pzq{S p0q c p8q " p1`op1qqS ρ2 pzq{S ρ2 p8q locally uniformly in Cz∆ 0,1 as c Ñ 0. Further, it follows from the middle relation in (3.7) and the last two asymptotic formulae that prop-szego6c prop-szego6c " 2πµ 1 1 pα 1 q|w 2 pα 1 q|S ρ2 pα 1 q S ρ2 p8q .
Plugging in the second asymptotic formula of (5.43) into (5.44) yields the first limit in (3.10). The other two now follow from (5.43).
Observe that the polynomials tP n,1 pxqu nPN0 form a normal family in a neighborhood of ∆ 2 . As degpP n,2 q " n 2 and it holds that ż x l P n,2 pxqP n,1 pxqdµ 2 pxq " 0, l P t0, . . . , n 2´1 u, by (1.5), the asymptotics of P n,2 pzq follows from [9, Theorem 2.7]. Namely, it holds that 6.1 6.1 (6.1) P n,2 pzq " p1`op1qq`S ρ2 pzq{S ρ2 p8q˘˜n Spz; x n,i q¸ϕ n2 2 pzq uniformly on compact subsets of Cz∆ 2 . Thus, to obtain the asymptotic formula for P n pzq, we only need to show that all the zeros tx n,i u n1 i"1 approach α 1 . We shall do it in a slightly more general setting.
Since t n` e 2 : n P N 0 u is a marginal sequence as well and the zeros of P n pzq and P n` e2 pzq also interlace, the limit in (6.2) for i " 2 follows similarly to the case i " 1. To prove Theorems 3.2-3.4 we use the extension to multiple orthogonal polynomials [20] of by now classical approach of Fokas, Its, and Kitaev [16,17] connecting orthogonal polynomials to matrix Riemann-Hilbert problems. The RH problem is then analyzed via the non-linear steepest descent method of Deift and Zhou [14].
As was agreed in Section 3.3, we label quantities dependent on c n only by the subindex n as in β n,1 :" β c n ,1 , ∆ n,i :" ∆ c n ,i , etc. If ∆ is a closed interval, we denote by ∆˝the open interval with the same endpoints. Moreover, when convenient, we write α n,1 p" α 1 q and β n,2 p" β 2 q even though they do not depend on the index n.
Throughout this section, the reader must keep in mind the definition of constants c˚and c˚˚in Proposition 4.1. Moreover, we would like to use the symbol c as a free parameter from the interval r0, 1s, as was done in the previous sections. Thus, we slightly modify the notation from the statement of Theorems 3.2-3.4 and assume that we deal with a sequence of multi-indices N c‹ such that c n " n 1 {| n| Ñ c ‹ P r0, 1s and n 1 , n 2 Ñ 8 as | n| Ñ 8, n P N c‹ .
We let rAs i,j to stand for pi, jq-th entry of a matrix A and E i,j to be the matrix whose entries are all zero except for rE i,j s i,j " 1. We set I to be the identity matrix, σ 3 :" diagp1,´1q to be the third Pauli matrix, and σp nq :" diag p| n|,´n 1 ,´n 2 q. Finally, for compactness of notation, we introduce transformations T i , i P t1, 2u, that act on 2ˆ2 matrices in the following way: where P n pzq is the polynomial satisfying (1.5), R piq n pzq, i P t1, 2u, are its functions of the second kind, see (3.14), m n,i are constants such that lim zÑ8 m n,i R piq n´ ei pzqz ni " 1 and e 1 :" p1, 0q, e 2 :" p0, 1q. ss:OL 7.2. Opening of the Lenses. Given c P p0, 1q and δ ą 0, denote by U c,δ,e an open square with vertices e˘cδ, e˘icδ when e P tα 1 , β c,1 u and e˘p1´cqδ, e˘ip1´cqδ when e P tα c,2 , β 2 u. Define mintβ 2´α2 , α 2´β1 u, c ď c˚˚, mintβ 2´αc,2 , α c,2´α2 u, c˚˚ă c.
Of course, it holds that cδ 1 pcq (resp. p1´cqδ 2 pcq) is constant for c ě c˚(resp. c ď c˚˚). Moreover, δ 1 pcq (resp. δ 2 pcq) approaches a non-zero constant as c Ñ 0`(resp. c Ñ 1´) by (4.8) and it approaches 0 as c Ñ c˚´(resp. c Ñ c˚˚`). Set δpcq :" mintδ 1 pcq, δ 2 pcqu. For brevity, we write U e :" U c n ,δ,e , n P N c‹ , e P E n :" E c n , E c :" tα 1 , β c,1 , α c,2 , β 2 u, assuming that δ P p0, δpc ‹ qq. In particular, all the domains U e are disjoint and β 1 R U βc,1 when c ‹ ă cẘ hile α 2 R U αc,2 when c ‹ ą c˚˚, again, for all | n| large enough, n P N c‹ . Section 7.4 contains a construction of maps ζ e pzq, conformal in U e , e P E c , such that ζ e pzq is real on the real line, vanishes at e, and maps p∆ c,1 Y ∆ c,2 q X U e into the negative reals (these subsets of ∆ c,1 Y∆ c,2 are covered by the darker shading on Figure 3). Using these conformal maps corresponding to c n for n P N c‹ , we can select piecewise smooth open Jordan arcs Γ˘ n,i , connecting α n,i to β n,i , defined by the following properties: Ipm1 Ipm1 (7.2) ζ β n,i`Γ˘ n,i X U β n,i˘Ă I˘:" z : argpzq "˘2π{3 ( , ζ α n,i`Γ˘ n,i X U α n,i˘Ă I¯, and Γ˘ n,i consist of straight line segments outside of U α n,i and U β n,i , see Figure 3. When c ‹ " c˚, we Ω´ n,1 Ω` n,2 " Ω2 Ω´ n,2 " Ω2 slightly modify (7.2) and require that Ipm2 Ipm2 (7.3) r ζ β n,1`Γ˘ n,1 X U β n,1˘Ă I˘, r ζ β n,1 pzq :" ζ β n,1 pzq´ζ β n,1 pβ 1 q, with an analogous modification holding for c ‹ " c˚˚at α n,2 . We denote by Ω˘ n,i the domains delimited by Γ˘ n,i and ∆ n,i , see Figure 3. Given Y pzq, the solution of RHP-Y , set eq:x eq:x It can be readily verified that Xpzq solves the following Riemann-Hilbert problem (RHP-X): rhx (a) Xpzq is analytic in Cz Xpzqz´σ p nq " I; (b) Xpzq has continuous traces on for each i P t1, 2u; (c) the entries of the first and pi`1q-st columns of Xpzq behave like O plog |z´ξ|q as z Ñ ξ P tα i , β i u, while the remaining entries stay bounded, i P t1, 2u. (b) N pzq has continuous traces on ∆˝ n,i that satisfy N`psq " N´psqT iˆ0 ρ i psq 1{ρ i psq 0˙; (c) it holds that N pzq " O`|z´e|´1 {4˘a s z Ñ e P E n . Let S n pzq :" S c n pzq be the one granted by Proposition 3.1. Put Spzq :" diag`S p0q n pzq, S p1q n pzq, S p2q n pzqf or z P Czp∆ n,1 Y ∆ n,2 q. Further, let Φ n pzq, w n,i pzq :" w c n ,i pzq, and Υ n,i pzq :" Υ c n ,i pzq be the functions given by (3.13) Since the jump matrix in RHP-N (b) has determinant 1, it follows from RHP-N (a,b) that detpN qpzq is holomorphic in CzE n with at most square root singularities at the points of E n . Thus, detpN qpzq is a constant and detpN qpzq " 1 by RHP-N (a). Therefore, it holds that detpM qpzq " detpDqpzq " detpCq " 1 due to the second relation in (3.7)
Finally, as detpM qpzq " 1, the estimates of M´1pzq follow in a straightforward fashion from the ones for M pzq.
Besides N pzq, we shall also need matrix functions that solve RHP-X within the domains U e , introduced at the beginning of Section 7.2, with an additional matching condition on the boundary. More precisely, let ε n be given by (3.11). For each e P tα 1 , β n,1 , α n,2 , β 2 u we are seeking a solution of the following RHP-P e : rhp (a,b,c) P e pzq satisfies RHP-X(a,b,c) within U e ; (d) P e psq " M psqpI`op1qqDpsq uniformly on BU e z e " β n,1 when c ‹ ă c˚, δpz c‹´β1 q˘´1 {2 , e " β 1 when c ‹ ą c˚, for some constant C ą 0 independent of n and δ, and analogous estimates hold around α n,2 , β 2 (in the cases c ‹ " c˚and c ‹ " c˚˚we cannot specify the exact rate of the error term), where the point z c , or more precisely z c was defined in Proposition 4.2. We will solve RHP-P e only for e P tα 1 , β n,1 u understanding that the solutions for e P tα n,2 , β 2 u can be constructed similarly. Solution of each RHP-P e will require a construction, carried out in the next subsection, of a local conformal map around α 1 and β n,1 . Recall that these maps were already used in (7.2). Since h p0q c˘p xq " h p1q c¯p xq on ∆c ,1 , the function ζ c,α1 pzq is holomorphic in the region of definition. When ω is a real measure on the real line, it trivially holds that Therefore, if the traces of ş px´zq´1dωpxq exist at x 0 , they are necessarily conjugate-symmetric. In particular, it follows from (4.2) that the integrand in (7.8) is purely imaginary on ∆c ,1 and therefore ζ c,α1 pxq ă 0 for x P ∆c ,1 . It also clearly follows from (4.2) that ζ c,α1 pxq ą 0 for x ă α 1 . Moreover, since h c pzq has a pole at α 1 , a ramification point of R c of order 2, ζ c,α1 pzq has a simple zero at α 1 . lem:4.1 Lemma 7.4. There exist δ α1 ą 0, A α1 ą 0, and D α1 ą 0, independent of c, such that each ζ c,α1 pzq is conformal in t|z´α 1 | ă δ α1 cu, 4A α1 c ď |ζ 1 c,α1 pα 1 q|, and |ζ 1 c,α1 pzq| ď D α1 c when t|z´α 1 | ă δ α1 cu for all c P p0, 1q.
When c P " c˚, c˚˚‰ the surface R c is always the same. Hence, one can argue using local coordinates that the pull-backs of h c pzq from a fixed circular neighborhood of α 1 to a fixed neighborhood in C continuously depend on c. Since each |ζ 1 c,α pα 1 q| ą 0 for c P p0, 1q, the desired estimate follows from compactness of rc˚, c˚˚s. When c˚˚ď c, h c pzq satisfies an equation similar to (7.9). Using this equation, we again can argue that the estimate holds as c Ñ 1, thus, proving that it holds uniformly for all c P p0, 1q.
Since the branchesĥ p0q c psq andĥ p1q c psq have 1 as a branch point, their limits come from the quadratic factor in (7.15). This observation together with with (4.8) readily yield that that 4.1.8 4.1.8 locally uniformly in t|1´s| ă 2u, where the branches of the square roots and the logarithm are principal and thereforeζ α1 psq is holomorphic in Czp´8,´1s and is positive for s P p1, 8q. Using the explicit expression forζ α1 psq, we can conclude that it is conformal in t|1´s| ă 2u and therefore lim inf cÑ0 δ α1 pcq ě 4|w 2 pα 1 q| by (4.8). When c Ñ 1, we can similarly get from the algebraic equation for h c pzq that ζ c,α1 pzq converges to 4.1.9 4.1.9 (7.17) which allows us to conclude that lim inf cÑ1 δ α1 pcq ą 0 as desired. Finally, let D α1 pcq :" c´1 max |z´α1|ďδα 1 c |ζ 1 c,α1 pzq|. These constants are finite for each c P p0, 1q since each ζ c,α1 pzq is, in fact, analytic in t|z´α 1 | ă 2δ α1 cu. Moreover, since ζ c,α1 pzq continuously depends on c, so do the constants D α1 pcq. Thus, we only need to check their limits as c Ñ 0 and c Ñ 1. The finiteness of D α1 :" sup cPp0,1q D α1 pcq now easily follows from (4.8), (7.16), and (7.17).

7.4.2.
Local maps around β c,1 when c P p0, c˚s. Given c P p0, c˚s, define where the choice of the root function can be made such that ζ βc,1 pzq is holomorphic with a simple zero at β c,1 and is positive for x ą β c,1 . Indeed, since h c pzq is bounded at β c,1 , which is a ramification point of order 2, we can write c pxq assume any non-zero real number somewhere on p´8, α 1 q Y pβ 2 , 8q and p´8, α 2 q Y pβ 2 , 8q, respectively. Thus, if v c " 0, then the function h c pzq´hpβ c,1 q would have at least four zeros (the zero at β c,1 would be at least a double one), but only three poles, which is impossible. Hence, v c ‰ 0, or more precisely, v c ą 0 since h p0q c pxq is a decreasing function on pβ c,1 , α 2 q as can be seen (4.2). Therefore, the integrand in (7.18) vanishes as a square root at β c,1 . Thus, ζ βc,1 pzq has a simple zero there. Again, as in (7.8), we select such a branch of the root function so that ζ βc,1 pzq is negative on ∆c ,1 . Since the difference h p0q c pxq´h p1q c pxq is real in the gap pβ c,1 , α 2 q, the map ζ βc,1 pzq is positive there. lem:4.2 Lemma 7.5. There exist δ β1 ą 0 and A β1 ą 0, independent of c P p0, c˚s, such that each ζ βc,1 pzq is conformal in t|z´β c,1 | ă δ β1 cu and 4A β1 c´1 {3 ď ζ 1 βc,1 pβ c,1 q for all c P p0, c˚s.
Since eachĥ p0q c psq has branchpoints at˘1 and is negative for s ă´1, see (4.2), the same must be true for their limitĥ p0q psq. Thus, solving the above quadratic equation gives uŝ Plugging the above limit and the substitution x " β c,1`| ∆ c,1 |ps´1q{2 into ( where v c was introduced in (7.19). This finishes the proof of the second claim of the lemma. To prove the first one, it is enough to observe that 4.2.5 4.2.5 as c Ñ 0, where the limit is conformal around 1.

7.4.3.
Local maps around β 1 for c close to c˚from the right. This construction will be used only for the ray sequences N c˚w ith infinitely many indices n such that c n ą c˚. By Proposition 4.2, h c˚p zq is bounded at β 1 while h c pzq has a simple pole at β 1 for all c ą c˚and a simple zero z c that approaches β 1 as c Ñ c˚`. Since the functions h c pzq converge around β 1 to h c˚p zq as c Ñ c˚`by (4.2) and Proposition 4.1, we can write for some c ą 0 such that c Ñ 0`as c Ñ c˚`, where f c pzq is a holomorphic function that is real on pα 1 , α 2 q (observe that the Puisuex expansion of ph p0q c´h p1q c qpxq around β c,1 does not have the integral powers of x´β 1 ). Similarly, it holds that ζ 3{2 β c˚,1 pzq " pz´β 1 q 3{2 f c˚p zq for some holomorphic function f c˚p zq that is real on pα 1 , α 2 q and is positive at β 1 . Since the right-hand side of (7.22) converges to ζ (we can adjust the constant δ β1 ą 0 from Lemma 7.5 so that the neighborhood of conformality is given by t|z´β 1 | ă δ β1 c 1 u). Moreover,ζ c,β1 pzq is positive for x ą β 1 and converges to ζ β c˚,1 pzq as c Ñ c˚`.
Proof. Let F pz; q be a family of holomorphic and non-vanishing functions in t|z| ă r 0 u that are positive at the origin and continuously depend on the parameter P r0, 0 s. Consider the equation where p ą 0 is a parameter that we shall fix in a moment. The solution of this cubic equation is formally given by # upz; q " 2p `v 1{3 pz; q`p 2 v´1 {3 pz; q, vpz; q " gpz; q´p 3 `agpz; qpgpz; q´2p 3 q.
Observe that g 1 px; q " px´ q " p3x´ qF px; q`xpx´ qF 1 px; q ‰ . The expression in the square brackets is negative at 0 and positive at . Since F p0; q ą δ ą 0, independently of P r0, 0 s for some δ, 0 ą 0 sufficiently small, the derivative of the expression in the square brackets, that is, 3F px; q`p5x´ qF 1 px; q`xpx´ qF 2 px; q, is positive on r0, s for all P r0, 0 s, where we might need to decrease 0 if necessary. Hence, there exists a unique point x P p0, q such that g 1 px q " 0. Since g 1 px; q " px´ q " 2xF px; q`px´ qpF px; q`xF 1 px; qq ‰ and F p0; q ą δ ą 0, independently of P r0, 0 s, we can decrease r 0 if necessary so that g 1 px; q ą 0 for x P p ; r 0 q and P r0, 0 s. Thus, there exists a unique y P p , r 0 q such that 2p 3 " gpy ; q for all P r0, 0 s, where, again, we might need to decrease 0 . Hence, we can choose vpz; q to be holomorphic in t|z| ă r 0 uzr0, y s and v 1{3 pz; q such that v 1{3 px; q Ñ´p as x Ñ 0´. Now, since gpx; q´p 3 is real on r0, y s and changes sign exactly once on each interval r0, x s, rx , s, and r , y s while the square root vanishes at the endpoint of these intervals, the change of the argument of v˘px; q is equal to 3π. Thus, we can define v 1{3 pz; q holomorphically in t|z| ă r 0 uzr0, y s as well, where it also holds that v 1{3 px; qv 1{3 px; q " p 2 and v˘p ; q "´e¯2 πi{3 p . In this case upz; q is in fact holomorphic in t|z| ă r 0 u, has a simple zero at the origin, is positive for x ą 0, and satisfies up ; q " 2p . Since upz; 0q " zp2F pz; 0qq 1{3 and upz; q continuously depends on , we can decrease r 0 if necessary so that all the function upz; q are conformal in t|z| ă r 0 u. Let upz; c q be the discussed solution of (7.24) and (7.25) with F pz; c q " f 2 c pz`β 1 q{2. Then the desired functionζ c,β1 pzq is given by upz´β 1 ; c q.

7.4.4.
Local maps around β 1 when c ą c˚. This construction will be used only for the ray sequences N c‹ with c ‹ ą c˚. Similarly to (7.8), given c P pc˚, 1q, define Then ζ c,β1 pzq is holomorphic in the domain of the definition, has a simple zero at β 1 , is real positive for x ą β 1 , and is real negative for x ă β 1 .

lem:4.4
Lemma 7.7. There exists a continuous and non-vanishing function δ β1 pcq on pc˚, 1q with non-zero one-sided limit at 1 such that ζ c,β1 pzq is conformal in t|z´β 1 | ă δ β1 pcqu. Moreover, the constant A β1 in Lemma 7.5 can be adjusted so that 4A β1 pz c´β1 q ď |ζ 1 c,β1 pβ 1 q|, where z c is the zero of h c pzq described in Proposition 4.2.
Proof. Since ζ c,β1 pzq has a simple zero at β 1 , δ β1 pcq is simply the largest radius of conformality, which is clearly positive. Moreover, when c Ñ 1, the limiting behavior of ζ c,β1 pzq is similar to the one described in (7.17) and therefore lim cÑ1´δβ1 pcq ą 0. To prove the second claim of the lemma observe that ζ 1 c,β1 pβ 1 q " u 2 c , where h p0q c pxq " u c px´β 1 q´1 {2`hp0q c pxq,h p0q c pxq " Op1q as x Ñ β 1 , exactly as in Lemma 7.4. Thus, we only need to investigate what happens when c Ñ c˚`(existence of a limit of ζ c,β1 pzq as c Ñ 1, which is conformal around β 1 , shows that |ζ 1 c,β1 pβ 1 q| is bounded from below as c Ñ 1). It follows from the second part of Proposition 4.1 and (4.2) that the Puiseux expansion of h p0q c pxq must converge to the Puiseux expansion of h p0q c˚p xq in some punctured neighborhood of β 1 . In particular, we have that u c Ñ 0 andh p0q c px c q Ñ h p0q c˚p β 1 q " h c˚p β 1 q as c Ñ c˚`for any sequence of points x c Ñ β1 as c Ñ c˚`. Since h p0q c pz c q " 0, it holds that u c pz c´β1 q´1 {2 "´h p0q c pz c q Ñ´h c˚p β 1 q as c Ñ c˚`, from which the estimate follows.
To prove (7.29), observe that for each x P ∆ c,1 fixed, the functions`H p0q c´H p1q c˘p x˘iyq are increasing for y P r0, 8q and vanish at y " 0 by (4.1) and (2.3). Moreover, since these functions have the same value at conjugate-symmetric points, it is enough to consider only the upper half-plane. As the right-hand side of (7.29) is positive whenever c, δ ą 0, we can assume without loss of generality that δ ă mintδ α1 , δ β1 , min cPrc 1 ,1q δ β1 pcqu, where δ α1 , δ β1 , c 1 , and δ β1 pcq were introduced in Lemmas 7.4, 7.5, 7.6, and 7.7, respectively.
In the considered case Argpx`iδcq P " π{2, π´arctanpδ{δ β1 q ‰ . Since the conformal maps ζ 3{2 βc,1 pzq continuously depend on c, have a rescaled limit when c Ñ 0, see (7.21), are positive for z ą β c,1 and negative for x ă β c,1 , (7.33) gets now replaced by 4.5.8 4.5.8 ‰ for all δ P p0, δ˚q and a possibly adjusted constant δ˚ą 0. Thus, combining the above observations with (7.31) gives us that 4.5.9 4.5.9 for some B 2 ą 0, independent of δ and c. Let now c 1 be the same as in Lemma 7.6 and |x`iδ´β 1 | ă δ β1 c for any c P pc˚, c 1 s, again, see Lemma 7.6. Then it follows from (7.23) that Sinceζ c,β1 pxq is positive for x ą β 1 and negative for x ă β 1 , it holds that or z with Argpzq P p0, πq. Since the mapsζ c,β1 pzq continuously depend on c P rc˚, c 1 s, where we set ζ c˚,β1 pzq :" ζ c˚,β1 pzq, see Lemma 7.6, the constant δ˚can be adjusted so that (7.35) remains valid with ζ c,β1 pzq replaced byζ c,β1 pzq for |δ| ă δ˚and c P rc˚, c 1 s. Hence, we can proceed exactly as in the case c P p0, c˚s, perhaps, at the expense of possibly adjusting the constant B 2 in (7.36). Further, when c P rc 1 , 1q, it follows from (7.26) that It also follows from Proposition 4.2 and Lemma 7.7 that |ζ 1 c,β1 pβ 1 q| is bounded away from 0 independently of c P rc 1 , 1q (the bound does depend on c 1 ). Notice also that in this case (7.33) remains valid with δ α1 replaced by min cPrc 1 ,1q δ β1 pcq. Therefore, (7.34) remains valid as well, where we need to replace ζ c,α1 pzq by ζ c,β1 pzq and, perhaps, adjust B 1 .
It only remains to examine what happens when α 1`δ 1 c ď x ď β c,1´δ 1 c for some δ 1 ą 0. To this end, let us denote byh c pxq the following function: Let us show thath c pxq ‰ 0 for x P ∆c ,1 . Indeed, ifh c px 1 q " 0 for some x 1 P ∆c ,1 , then h p0q c`p x 1 q " h p0q c´p x 1 q " h p1q c`p x 1 q " h p1q c´p x 1 q and this value is real. That is, there exist x 1 , x 2 P ∆ c,1 (πpx 1 q " πpx 2 q " x 1 ) at which h c pzq assumes the same non-zero real value. On the other hand, when c P pc˚, c˚˚q, h c pzq has simple poles at α 1 , β 1 , α 2 , β 2 . Therefore, it can be clearly seen from (4.2) that h p0q c pxq assumes every non-zero real value twice, once on p´8, α 1 q Y pβ 2 , 8q and once on pβ 1 , α 2 q. Furthermore, (4.2) also shows that h p1q c pxq and h p2q c pxq assume every non-zero real value once on p´8, α 1 q Y pβ 1 , 8q and p´8, α 2 q Y pβ 2 , 8q, respectively. As h c pzq has four zeros/poles, it assumes every value exactly four times. Thus, ifh c px 1 q were zero, h c pzq would assume a given real value six times, which is impossible. Since the proof for the case c P p0, c˚s Y rc˚˚, 1q is quite similar, the claim follows.
For that, it will be convenient to consider the rescaled functionĥ c psq :"h c pβ c,1`| ∆ c,1 |ps´1q{2q. These functions are purely-imaginary and non-vanishing on p´1, 1q. It follows from (4.8) that there exists δ 2 ą 0 such thath For each c fixed, the minimum over s is clearly non-zero and continuously depends on c. On the other hand, exactly as in Lemma 7.5, it holds that 4.5.10 4.5.10 (7.37)ĥ c psq Ñ´i |w 2 pα 1 q| c 1´s 1`s as c Ñ 0 uniformly on r´1`δ 2 , 1´δ 2 s, which again, has a non-zero minimum of the absolute value. Moreover, a computation similar to the one leading to (7.17) gives us that 4.5.11 4.5.11 1´s 2 as c Ñ 1 uniformly on r´1`δ 2 , 1´δ 2 s, which also has a non-zero minimum of the absolute value. Hence, it indeed holds thath min ą 0. Now, observe thath c pxq is a trace of a function analytic across ∆c ,1 , namely, of Therefore, for each x 1 P rα 1`δ 1 c, β c,1´δ 1 cs fixed, there exists δpc; x 1 q ą 0 such that 4.5.12 4.5.12 for all |z´x 1 | ă δpc; x 1 qc by (5.1). Notice that δpx 1 qc can be taken to be the radius of the largest disk of conformality of r H c pz; x 1 q. Observe also that δpc; x 1 q continuously depends on x 1 and therefore there exists δpcq ą 0 such that δpc; x 1 q ě δpcq for all x 1 P rα 1`δ 1 c, β c,1´δ 1 cs. Since δpcq can be made to continuously depend on c and the limits (7.37) and (7.38) hold not only on p´1, 1q, but in some neighborhood of p´1, 1q as well, the constant δ˚can be adjusted so that δpcq ą δ˚for all c P p0, 1q.
Since the functions r H c pz; x 1 q are conformal in |z´x 1 | ă δ˚c for each x 1 P rα 1`δ 1 c, β c,1´δ 1 cs and are purely imaginary on the real axis, the same continuity and compactness arguments we have been employing throughout the lemma imply that 4.5.13 4.5.13 (7.40) ´r H c px 1`i y; x 1 q¯ě Cˇˇr H c px 1`i y; x 1 qˇf or all y P p0, δ˚cq and x 1 P rα 1`δ 1 c, β c,1´δ 1 cs, where C ą 0 is constant independent of c. Since h c pzq " 2B z H c pzq, it follows from (7.39) and (7.40) that 4.5.14 4.5.14 The estimate in (7.29) now follows from (7.34), (7.36), and (7.41).
ss:LP 7.5. Local Parametrices. Below, we construct solutions of RHP-P e for e P tα 1 , β n,1 u, n P N c‹ . Recall that the squares U e have diagonals of length 2δc, where δ ď δpc ‹ q see Section 7.2. Additionally, we assume that δ ď mintδ α1 , δ β1 u or δ ď mintδ α1 , δ β1 pc ‹ qu, depending on c ‹ , see Lemmas 7.4-7.7. Then the maps constructed in Section 7.4 are conformal in the corresponding squares U e . sss:7.5.1 7.5.1. Matrix P α1 pzq. Let Ψpζq be a matrix-valued function such that rhpsi (a) Ψpζq is holomorphic in Cz`I`Y I´Y p´8, 0s˘, see (7.2); (b) Ψpζq has continuous traces on I`Y I´Y p´8, 0q that satisfy where I˘are oriented towards the origin; (c) Ψpζq " Oplog |ζ|q as ζ Ñ 0; (d) Ψpζq has the following behavior near 8: Solution of RHP-Ψ was constructed explicitly in [28] with the help of modified Bessel and Hankel functions. Observe that the jump matrices in RHP-Ψ(b) have determinant one. Therefore, it follows from RHP-Ψ(d) that detpΨpζqq " ? 2. Let ζ n,α1 pzq :" ζ c n ,α1 pzq, see (7.8), which is conformal in U α1 . It holds due to Lemma 7.4 and (5.1) that rhpsi-2 rhpsi-2 where A α1 is independent of δ and c n " n 1 {| n|. It also follows from (4.3) and (7.8) that Let Dpzq be given by (7.6). Note also that the matrix σ 3 Ψpζqσ 3 also satisfies RHP-Ψ, but with the orientation of all the rays in RHP-Ψ(b) reversed and i replaced by´i in the asymptotic formula of RHP-Ψ(d). Relation (7.43) and RHP-Ψ(a,b,c) imply that the matrix rhpsi-3 rhpsi-3 satisfies RHP-P α1 (a,b,c) for any holomorphic prefactor E α1 pzq. As ζ 1{4 " iζ 1{4 on p´8, 0q, where we take the principal branch, it can be easily checked that here. Then RHP-N (b) implies that rhpsi-4 rhpsi-4 is holomorphic in U α1 ztα 1 u. Since the first and second columns of M pzq has at most quarter root singularities at α 1 and the third one is bounded, see Lemma 7.3, E α1 pzq is in fact holomorphic in U α1 as desired. Finally, RHP-P α1 (d) follows from RHP-Ψ(d) and (7.42).

7.5.4.
Matrix P β1 pzq when c ‹ ą c˚. The construction of P β1 pzq in the considered case is absolutely identical to the one of P α1 pzq in Section 7.5.1. Clearly, we can assume that n P N c‹ is such that c n ą c˚. Let ζ n,β1 pzq :" ζ c n ,β1 pzq be the conformal map defined in (7.26), whose properties were described in Lemma 7.7. It follows from (4.3) and (7.26) that According to Lemma 7.7 and (5.1) theorem and since n 2 1 ď | n| 2 , it holds that |z| ă A β1 δpz c‹´β1 qn 2 where δ β1 pcq is continuous and non-vanishing on pc˚, 1s. Similarly to (7.44), a solution of RHP-P β1 is given by It again holds that detpP β1 pzqq " 1. 7.6. Solution of RHP-X. Set U n :" U α1 Y U β n,1 Y U α n,2 Y U β2 and Γ n :" Γ` n,1 Y Γ´ n,1 Y Γ` n,2 Y Γ´ n,2 . Put Σ n,δ :" BU n Y`pΓ n Y rβ n,1 , β 1 s Y rα 2 , α n,2 sq zU n˘, see Figure 4. For definiteness, we agree that all the segments in Σ n,δ are oriented from left to right and all the polygons are oriented counter-clockwise. We shall further denote by Σ n,δ,1 and Σ n,δ,2 the left and right, respectively, connected components of Σ n,δ .
For what is to come, we shall need uniform boundedness of the Cauchy operators on Σ n,δ . For convenience, we formulate this claim as a lemma.
BU β n,1 Γ` n,1 zU n Γ´ n,1 zU n Γ2 zU n Γ2 zU n Proof. Recall the following known fact, see [12,Equation (7.11)], if R 1 , R 2 are two semi-infinite rays with a common endpoint, then harmonic harmonic for some constant C r ą 0 (we can take C 2 " 1), where C R1 is the Cauchy operator defined on R 1 . Moreover, the same estimate holds when R 2 " R 1 and C R1 is replaced by the trace operators C R1˘, see [12, Equations (7.5)-(7.7)]. Trivially, the same estimate holds when R 2 is replaced by an interval disjoint from R 1 (may be for an adjusted constant C r ). Since we can embed any two segments with a common endpoint into semi-infinite rays with a common endpoint and embed a function from L r space of a segment into L r space of the corresponding ray by extending it by zero, the desired estimate then follows from (7.55) (again, with an adjusted constant C r ).
To show existence and prove size estimates of the matrix function Zpzq, let us first estimate the size of its jump: V psq :" Z´1 psqZ`psq´I, s P Σ n,δ .
More precisely, the following lemma holds.
Proof. We shall prove (7.57) separately for different parts of Σ n,δ . In fact, we shall do it only on Σ n,δ,1 understanding that the estimates on Σ n,δ,2 can be carried out in the same fashion. For s P BU e , e P tα 1 , β n,1 u, it holds that V psq " P e psqpM Dq´1psq´I. Therefore, the desired estimate (7.57) follows from Lemma 7.3 and RHP-P e (d). Let now s " x P ∆ 1 zp∆ n,1 Y U n q, which is non-empty when c ‹ ă c˚. In this case, it holds that Estimate (7.57) now follows from Lemma 7.3 and the estimate rhz-0 rhz-0 see (4.3) and (7.27). Lastly, let s P Γ˘ n,1 zU n . Then it holds that The desired estimate (7.57) can be deduced exactly as in the second step of the proof with (7.29) used instead of (7.27).
Proof. Let C and C´be the operators defined in Lemma 7.9 and C V : L r pΣ n,δ q Ñ L r pΣ n,δ q, r ą 1, be an operator defined by C V F :" C´pF V q for any 2ˆ2 matrix function F psq in L r pΣ n,δ q. Then it follows from Lemmas 7.9 and 7.10 that rhz-4 rhz-4 (7.60) }C V } r ď C r }V } L 8 pΣ n,δ q " op1q.
Let M pN c‹ q be such that the above norm is less than 1{2 for all n P N c‹ , | n| ě M pN c‹ q. Then the operator I´C V is invertible in L r pΣ n,δ q for all such n. Hence, one can readily verify that Zpzq " I`CpU V qpzq, U psq :" pI´C V q´1pIqpsq.
The above formula and Hölder inequality immediately yield that rhz-5 rhz-5 }U V } L r pΣ n,δ q distpz, Σ n,δ q δ´1}V } L 8 pΣ n,δ q for distpz, Σ n,δ q ě δ{5, where the constant in is independent of n and δ (it involves the arclengths of Σ n,δ , but the latter are uniformly bounded above and below). It can be readily seen from RHP-Z(b) that V psq can be analytically continued off each connected component of Σ˝ n,δ . Hence, solutions of RHP-Z for the same value of n and different values of δ are, in fact, analytic continuations of each other. Thus, using (7.61) together with (7.61) where δ is replaced by δ{2, we get that (7.61) in fact holds for distpz, prβ c‹,1 , β 1 s Y rα 2 , α c‹,2 sqzU n q ě δ{5. The set prβ c‹,1 , β 1 s Y rα 2 , α c‹,2 sqzU n is not empty only when c ‹ P r0, c˚q Y pc˚˚, 1s. In particular, we have finished the proof of the lemma for c ‹ P rc˚, c˚˚s. When c ‹ P p0, c˚q, set I n,δ :" rβ 1`i δc ‹ {3, β 1 s Y pβ c‹,1`i δc ‹ {3, β 1`i δc ‹ {3qzU n and let O n,δ be the bounded domain delimited by BU n , I n,δ , and rβ c‹,1 , β 1 qzU n . Observer that V psq extends as an analytic matrix function into O n,δ and still satisfies (7.58) there by (7.27). Thus, we can analytically continue Zpsq into O n,δ by multiplying it by I`V pzq there. This continuation will still have a jump matrix satisfying (7.57) and therefore itself will satisfy (7.61) away from its jump contour. This finishes the proof of the lemma when c ‹ P p0, c˚q Y pc˚˚, 1q (the proof for the case c ‹ P pc˚˚, 1q is identical). The proof in the case c ‹ " 0 (and therefore in the case c ‹ " 1) is similar and uses (7.28) instead of (7.27).
The fact that the above constructed matrix Zpzq has behavior as described in RHP-Z(c) follows from the fact that it admits an explicit local parametrix around β 1 (resp. α 2 ) when c ‹ ă c˚(resp. c ‹ ą c˚˚), see [37,Sections 8.3 and 9.1].

Proof of Theorems 3.2-3.4.
We are now ready to prove the main results of Section 3. We stop using the notation c ‹ and resume writing c as in the statements of Theorems 3.
uniformly in Cztα 1 , β 1 u when c " 0, in Cztβ 1 u when c P p0, c˚q, in C when c P rc˚, c˚˚s, in Cztα 2 u when c P pc˚˚, 1q, and in Cztα 2 , β 2 u when c " 1 by (7.57) and (7.59), where the dependence on c of O δ,c pε n q is uniform on compact subsets of r0, c˚q Y pc˚˚, 1s. Then it follows from (7.1), (7.63), the definition of M pzq in (7.5), and of C, Dpzq in (7.6) that P n pzq " rY pzqs 1,1 " rCs 1,1 rpZM qpzqs 1,1 rDpzqs 1,1 " γ n S Let now K be a closed subset of ∆c ,1 Y ∆c ,2 . Again, we can adjust δ so that K does not intersect U n for all | n| large enough. Hence, Y2 Y2 for i P t1, 2u, again by (7.4) and Lemma 7.12. Thus, we get for x P K X ∆ c,i that Since F p0q pxq " F piq pxq on ∆ n,i for any rational function F pzq on R n , the second asymptotic formula of the theorem now follows from (3.7), (7.65), and (5.5)-(5.8).
Hence, it follows from (7.63) that on closed subsets of Cz`∆ c,1 Y ∆ c,2˘i t holds that (as before, the contour Σ n,δ can be adjusted to accommodate any such closed set, moreover, one needs to write p Y˘pzq for z P ∆ i z∆ c,i Similarly to (7.64), set p B k pzq :" 0k " op1q, k P t0, 1, 2u.
Observe that all the jump matrices in RHP-Z(b) have determinant one. Since Zp8q " I, we therefore get that detpZpzqq " 1. Hence, the functions p B k pzq do obey the estimate of (7.65) as well. Again, it holds that p B k p8q " 0. Thus, where, as before, s n,l " S p0q n p8q{S plq n p8q. Now, observe that Π n,l pzq{Π n pzq "´A´1 n,l Υ n,l pzq, l P t1, 2u, which follows from comparing zero/pole divisors and the normalizations at 8 p0q of the left-and righthand sides of the above equality (recall that Π p0q n p8q " 1 and Π p0q n,l pzq "´z´1`Opz´2q, which can be seen from (5.19)). Therefore, it follows from (7.70) that Hence, the first asymptotic formula of the theorem follows from (7.65), (5.5)-(5.8) (here, one needs to recall that p B l p8q " 0 and therefore the estimate for pΥ plq n,l p B l qpzq around infinity follows from the maximum principle), (3.10), and the fact that A n,1 " c 2 n shown in the proof of Lemma 5.1. When c " 0 and i " 1, we also deduce from (7.71) and the maximum modulus principle that where we also used (3.9) and op1q behaves like the right-hand side of (7.65). Recall that Π p1q n pzq has a double zero at infinity. Therefore,ˇ`Π p1q n w 2 n,1˘p zqˇˇ"ˇˇˇˇ´Υ Finally, (7.66) and (7.68) give us p Y˘pxq " C´1`Z´1˘Tpxq`M´1˘TpxqD´1 pxq`I¯ρ´1 i pxqE 1,i`1ȏ n any compact subset of ∆c ,i , i P t1, 2u. Analogously to (7.71), the above formula yields that As in the previous two subsections, given a closed set K in Czp∆ 1 Y ∆ 2 q, we can adjust the contour Σ n,δ so that K lies in the unbounded component of its complement. Hence, using the notation of the previous two subsections, we get from (7.1), (7.5), (7.6), (7.63), and (7.65) that R piq n pzq " γ n S piq n pzqw´1 n,i pzq´1`B 0 pzq`s n,1 B 1 pzqΥ piq n,1 pzq`s n,2 B 2 pzqΥ piq n,2 pzq¯Φ piq n pzq for z P K, i P t1, 2u. The first asymptotic formula of the theorem now follows from (7.65), (5.5)-(5.8), (3.10), and the maximum modulus principle applied to pΥ piq n,i B i qpzq to extend the desired estimates to the neighborhood of infinity. As in the proof of Theorem 3.3, it holds when c " 0 and i " 1 that R p1q n pzq " op1qτ n Φ p1q n pzqw´1 n,1 pzq uniformly on closed subsets of Cz∆ 0,1 by (5.6)-(5.8) and (3.9)-(3.10). Since an analogous formula holds for c " 1 and i " 2, the second asymptotic formula of the theorem follows.
Finally, it follows from (7.67) and (7.68) that L n pzq "˜1`p B 0 pzq´Υ p0q n,1 pzq s n,1 A n,1 p B 1 pzq´Υ p0q n,2 pzq s n,2 A n,2 p B 2 pzq¸Π p0q n pzq γ n`S n Φ n˘p 0q pzq on closed subsets of Czp∆ c,1 Y ∆ c,2 q, from which the last asymptotic formula of the theorem follows, as usual, by (7.65) (holding for p B k pzq as well), (5.5)-(5.8), (3.10), and since A n,1 " c 2 n as shown in Lemma 5.1. While proving Theorem 1.2 we first consider the case of fully marginal sequences and then consider separately the asymptotic behavior of a n,1 , a n,2 and b n,1 , b n,2 .
8.1. Fully Marginal Ray Sequences. In this section we only consider sequences N 0 and N 1 satisfying (3.1). Again, we present the proof only in the case of c " 0. Recurrence formula (1.7) for P n pxq can be rewritten as z´b n,i " P n` ei pzq P n pzq`a n,1 P n´ e1 pzq P n pzq`a n,2 P n´ e2 pzq P n pxq , i P t1, 2u.
One can easily see from (8.1) that n` ei pzq P n pzq´z˙.

8.2.
Asymptotics of a n,1 , a n,2 along Non-fully Marginal Sequences. From now on we are assuming that ray sequences N c satisfy (3.11). It can be deduced from orthogonality relations (1.5) and definition (3.14) that R piq n pzq "´h i P t1, 2u. In particular, we have that m n,i "´2πi{h n´ ei,i in (7.1). Then it follows from the first and second asymptotic formulae of Theorem 3.4, the definition of constants γ n and τ n in Theorems 3.2 and 3.3, respectively, and the definition of the matrix C in (7.6) that where, as before, s n,i " S p0q n p8q{S piq n p8q, i P t1, 2u, the first formula holds for i P t1, 2u when c P p0, 1q, i " 2 when c " 0, and i " 1 when c " 1, and the second formula holds for the remaining cases. Furthermore, we get from (7.1) that 8.6 8.6 (8.6)´2 πi h n´ ei,i " m n,i " lim zÑ8 z 1´| n| rY pzqs i`1,1 .
Analogously to the computation after (7.63)-(7.65) we get that rY pzqs i`1,1 is equal to 8 , that a n,i " h n,i {h n´ ei,i . Therefore, it follows from (8.5) and (8.8) that a n,i " p1`op1qq`A n,i`s´1 n,i op1q˘or a n,i " op1q`s n,i A n,i`o p1qȋ P t1, 2u, where the first formula holds for i P t1, 2u when c P p0, 1q, i " 2 when c " 0, and i " 1 when c " 1, and the second formula holds for the remaining cases. The desired limits of a n,i therefore follow from continuity of the constants A c,i with respect to the parameter c, see Proposition 2.1, asymptotic formulae (3.10), and the estimates A c,1 " c 2 as c Ñ 0 (A c,2 " p1´cq 2 as c Ñ 1), see (5.11) and after. 8.3. Asymptotics of b n,1 , b n,2 along Non-fully Marginal Sequences. Excluding the cases i " 1 when c " 0 and i " 2 when c " 1, we get from (8.6)-(8.8) and (5.6)-(5.8) that 8.9 8.9 (8.9) P n´ ei pzq " p1`op1qqA´1 n,i Υ p0q n,i pzqγ n`S n Φ n q p0q pzq in some neighborhood of the point at infinity. Replacing the sequence N c with t n` e i : n P N c u, we get from (8.2), Theorem 3.2, and (8.9) that b n,i "´p1`op1qq lim zÑ8˜A n` ei Υ p0q n` ei,i pzq´z¸" p1`op1qqB n` ei , where we also used (2.5) and (5.3). The desired claim now follows from Proposition 2.1.
Out of the two exceptional cases, we shall only consider the case i " 1 when c " 0 understanding that the other one can be treated similarly. Assume for the moment that the measure µ 2 is, in fact, the arcsine distribution on ∆ 2 , that is, a px´α 2 qpβ 2´x q "´d x 2πiw 2`p xq .
Recall the notation of Section 6 where we wrote P n pzq " P n,1 pzqP n,2 pzq with polynomial P n,i pzq having all its zeros on ∆ i . We would like to show that when µ 2 is of the form (8.10), formula (6.1) still holds along any marginal ray sequence N 0 . To this end, we shall use 2ˆ2 Riemann-Hilbert analysis of orthogonal polynomials. Since this method has been described in detail in Section 7, we shall only outline the main steps. It follows from (1.5) and (8.10) that the Riemann-Hilbert problem (a) Y pzq is analytic in Cz∆ 2 and lim zÑ8 Y pzqz´n 2 " I; (b) Y pzq has continuous traces on each ∆2 that satisfy Y`pxq " Y´pxqˆ1 pP n,1 {w 2`q pxq 0 1˙; (c) the entries of the first column of Y pzq are bounded and the entries of the second column behave like Op|z´ξ|´1 {2 q as z Ñ ξ P tα 2 , β 2 u; is solved by Y pzq :"˜P n,2 pzq R p2q n pzq m ‹ n,2 P ‹ n,2 pzq m ‹ n,2 R ‹ n,2 pzq¸, where P ‹ n,2 pzq is the monic polynomial of degree n 2´1 orthogonal to lower degree polynomials with respect to the weight P n,1 pxqdµ 2 pxq and R ‹ n,2 pzq " 1 2πi ż P ‹ n,2 pxqP n,1 pxqdµ 2 pxq x´z " 1 m ‹ n,2 z n2`O`z´n 2´1˘.
Appendix A.
appendix In this Appendix, we will study the operators L p1q c and L p2q c defined in (2.10). As we have already mentioned in Section 2.2, these operators appear in [7,Formula (4.20)] used with κ " e 1 and κ " e 2 , respectively. In what follows, we denote by δ pY q the delta function (Kronecker symbol) of the vertex Y . Consider two functions sad12 sad12 (A.1) m I pzq :" xpL p1q c´z q´1δ pOq , δ pOq y, m II pzq :" xpL p2q c´z q´1δ pOq , δ pOq y . Given the function χ c pzq from Proposition 2.1 and c P p0, 1q, where, as usual, χ p0q c pzq are the values taken from the zero-th sheet R p0q c . By the Spectral Theorem [3], they can also be written in the form where σ plq O is the spectral measure of δ pOq with respect to L plq c , l P t1, 2u. The properties of the conformal map χ c pzq imply that the functions m I pzq and m II pzq satisfy: (A) m I pzq and m II pzq have no poles since χ p0q c pzq ‰ B c,j for z P R p0q c by conformality; (B) both m I pzq and m II pzq are Herglotz-Nevanlinna functions in C`, i.e., they are analytic, have positive imaginary part, and are continuous up to the boundary. Moreover, m I pxq " m II pxq " 0 for x P Rzp∆ c,1 Y ∆ c,2 q and mÌ pxq ą 0, mÌ I pxq ą 0 for x P ∆c ,1 Y ∆c ,2 .
We will use the following notation. If Y, Z P V and Y " Z, then deleting the edge pY, Zq that connects them leaves us with two subtrees. The one containing Y will be called T rZ,Y s , the other one will be called T rY,Zs . The restriction of any Jacobi matrix J to a subtree T 1 will be denoted by J T 1 .
We learned from (A) and (B) above that σ p1q O and σ p2q O are absolutely continuous measures with supports equal to ∆ c,1 Y ∆ c,2 . We need this for the following lemma.  Proof. Suppose that L plq c , l P t1, 2u, has an eigenvector Ψ. Since σ plq O is purely absolutely continuous as just explained, the restriction of L plq c to the cyclic subspace generated by δ pOq has no eigenvalues by the spectral theorem. Therefore, we must have Ψ O " 0. Now, consider the restrictions of Ψ to T rO,O pchq,1 s and to T rO,O pchq,2 s . One of these functions is not identically equal to zero and the one that is not must be an eigenvector of the corresponding operator: either J T rO,O pchq,1 s or J T rO,O pchq,2 s . By construction, these operators are identical to either L p1q c or L p2q c and, as we established earlier, this implies that Ψ O pchq,1 " Ψ O pchq,2 " 0. Repeating the argument, we can now show that Ψ " 0 identically on the whole tree which gives a contradiction.
The following observation holds for a general Jacobi matrix (2.8) and (2.9). Let σ Y denote the spectral measure of δ pY q with respect to J , i.e., mly2 mly2 If we delete all edges connecting Y to its neighbors, say l of them, we will be left with the vertex Y and l subtrees tT rY,Yj s u l j"1 . The restrictions of J to these subtrees are also Jacobi matrices and we previously denoted them by J T rY,Y j s . Let Proof. Let f :" pJ´zq´1δ pY q . Clearly, J f " zf`δ pY q , that is, Y,Yj f Y˘´1 f |V rY,Y j s , which is a renormalized restriction of f to the set of vertices V rY,Yj s of T rY,Yj s . Observe that mly1 mly1 where both relations follow from the first line of (A.5) (for the second relation we need to separate the summand corresponding to Z " Y , bring it to the other side of the equation, and then divide by it). It follows immediately from (A.6) that J T rY,Y j s f pjq " zf pjq`δYj ñ f pjq " pJ T rY,Y j s´z q´1δ pYj q .
The claim of the lemma follows from the second equality in (A.5) since f Y " xpJ´zq´1δ pY q , δ pY q y " m Y pzq and similarly f Yj "´`W 1{2 Y,Yj f Y˘f pjq Yj "´W 1{2 Yj ,Y m Y pzqm rY,Yj s pzq.
Let us now return to the operators J " L plq c , l P t1, 2u. Take any vertex Y ‰ O. Deleting the edge pY, Y ppq q leaves us with two subtrees. As before, we denote by T rY,Y ppq s the one containing Y ppq , and let m , m plq rO,Zs pzq is equal to either m I pzq when ι " 2 or m II pzq when ι " 1. In any case, m plq rY,Os pzq is meromorphic outside ∆ c,1 Y ∆ c,2 . Suppose now that the claims are true for all vertices up to the distance n. Consider any Y such that its distance from the root is n`1. Let ι be the type of Y . As in the first part of the proof, apply (A.4) at the vertex Y ppq of the subtree T rY,Y ppq s to get where ι ppq is the type of Y ppq and Z is the "sibling" of Y . The first function in the denominator is meromorphic outside ∆ c,1 Y ∆ c,2 by the inductive assumption and the other one is either m I pzq or m II pzq. Thus, m plq rY,Y ppq s is also meromorphic outside ∆ c,1 Y ∆ c,2 . This way we get the claim for n`1 and so we proved the first statement of the lemma. Now, apply (A.4) to m plq Y pzq. The functions involved are m rY,Y pchq,j s pzq, j P t1, 2u, and m rY,Y ppq s pzq. The first two are m I pzq, m II pzq and they are analytic in the considered domain. The third one is meromorphic there by the first statement of the lemma. Notice that m plq Y pzq can not have poles by Lemma A.1 thus it is analytic outside ∆ c,1 Y ∆ c,2 . sad14 Lemma A.4. Let Y P V and c P p0, 1q. If σ plq Y is the spectral measure of Y with respect to L plq c , l P t1, 2u, then it is absolutely continuous and its support is equal to ∆ c,1 Y ∆ c,2 .
Proof. The measure σ for every interval I Ă ∆ c,1 Y ∆ c,2 . This implies that σ plq Y is purely absolutely continuous on I. By Lemma A.3, the measure σ plq Y is supported inside ∆ c,1 Y ∆ c,2 and Lemma A.1 implies that it has no mass points. Therefore, we conclude that σ plq Y is purely absolutely continuous, as claimed. sad15 Theorem A.1. We have that σpL plq c q " σ ess pL plq c q " ∆ c,1 Y ∆ c,2 , l P t1, 2u, where, as before, we understand that ∆ 0,1 :" tα 1 u and ∆ 1,2 :" tβ 2 u.
Proof. If c P p0, 1q, Lemma A.4 shows that δ pY q belongs to the absolutely continuous subspace of L plq c for all Y . Since all linear combinations of δ pY q must belong to this subspace and are dense in 2 pVq, this subspace is in fact the whole space 2 pVq. Thus, σpL plq c q " σ ess pL plq c q and it is equal to ∆ c,1 Y ∆ c,2 by Lemma A.4 and the Spectral Theorem.
Furthermore, since the restriction of A 2 from 2 pZ ě0 q to 2 pNq is equal to A 1 and therefore m r0,1s pzq " p m 1 pzq in the notation of (A.3), we get from (A.4) that p m 2 pzq :" xpA 2´z q´1δ p0q , δ p0q y "´1 A 0,2 p m 1 pzq`z´B 0,1 , where w 2 pzq was introduced in the Proposition 2.1. One can readily check that p m 2 pxq ą 0 for x P pα 2 , β 2 q, p m 2 pxq " 0 for x R rα 2 , β 2 s, and that p m 2 pzq has the unique pole at a point q x P R given by mly4 mly4