RH problem for $Y(\cdot ) = Y_n(\cdot ;\vec \alpha,\vec \beta,V,W)$

1. (a) $Y : {\mathbb {C}} \setminus \mathbb {T} \to \mathbb {C}^{2 \times 2}$ is analytic.

2. (b) For each $z \in \mathbb {T}\setminus \{t_{0},\ldots,t_{m}\}$, the boundary values $\lim _{z' \to z}Y(z')$ from the interior and exterior of $\mathbb {T}$ exist, and are denoted by $Y_+(z)$ and $Y_{-}(z)$ respectively. Furthermore, $Y_+$ and $Y_{-}$ are continuous on $\mathbb {T}\setminus \{t_{0},\ldots,t_{m}\}$, and are related by the jump condition

   (2.1)\begin{equation} Y_+(z) = Y_-(z)\begin{pmatrix}1 & z^{{-}n}f(z) \\ 0 & 1\end{pmatrix}, \qquad z \in \mathbb{T}\setminus \{t_{0},\ldots,t_{m}\}, \end{equation}
   where $f$ is given by (1.2).

3. (c) $Y$ has the following asymptotic behaviour at infinity:

   \[ Y(z) = (1+{\mathcal{O}}(z^{{-}1}))z^{n \sigma_{3}}, \qquad \mbox{as } z \to \infty, \]
   where $\sigma _{3} = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$.

4. (d) As $z \rightarrow t_k$, $k=0, \ldots, m$, $z \in {\mathbb {C}} \setminus \mathbb {T}$,

   \[ Y(z) = \begin{cases} \begin{pmatrix} {\mathcal{O}}(1) & {\mathcal{O}}(1) + {\mathcal{O}}({|z-t_k|}^{\alpha_{k}}) \\ {\mathcal{O}}(1) & {\mathcal{O}}(1) + {\mathcal{O}}({|z-t_k|}^{\alpha_{k}}) \end{pmatrix}, & \mbox{if } {\rm Re\,} \alpha_k \ne 0, \\ \begin{pmatrix} {\mathcal{O}}(1) & {\mathcal{O}}(\log{|z-t_k|}) \\ {\mathcal{O}}(1) & {\mathcal{O}}(\log{|z-t_k|}) \end{pmatrix}, & \mbox{if } {\rm Re\,} \alpha_k = 0. \end{cases} \]

Suppose $\{p_k(z) = \kappa _{k} z^{k}+\ldots \}_{k\geq 0}$ and $\{\hat {p}_k(z)=\kappa _{k} z^{k}+\ldots \}_{k\geq 0}$ are two families of polynomials satisfying the orthogonality conditions

(2.2)\begin{equation} \begin{cases} \displaystyle\frac{1}{2\pi}\int_0^{2\pi} p_k(z)z^{{-}j}f(z){\rm d}\theta= \kappa_k^{{-}1}\delta_{jk}, \\ \displaystyle\frac{1}{2\pi}\int_0^{2\pi} \hat{p}_k(z^{{-}1})z^{j}f(z){\rm d}\theta= \kappa_k^{{-}1}\delta_{jk}, \end{cases} \quad z=e^{i\theta}, \quad j= 0,\ldots,k. \end{equation}

Then the function $Y(z)$ defined by

(2.3)\begin{align} Y(z) & = \left( \kappa_n^{{-}1}p_n(z) \quad \kappa_n^{{-}1}\int_{\mathbb{T}} \frac{p_n(s)f(s)}{2\pi i s^n(s-z)}{\rm d}s - \kappa_{n-1}z^{n-1}\hat{p}_{n-1}(z^{{-}1})\right.\notag\\ & \quad -\left. \kappa_{n-1}\int_{\mathbb{T}} \frac{\hat{p}_{n-1}(s^{{-}1})f(s)}{2\pi i s(s-z)}{\rm d}s \right) \end{align}

solves the RH problem for $Y$. It was first noticed by Fokas, Its and Kitaev [Reference Fokas, Its and Kitaev33] that orthogonal polynomials can be characterized by RH problems (for a contour on the real line). The above RH problem for $Y$, whose jumps lie on the unit circle, was already considered in e.g. [Reference Baik, Deift and Johansson3, eq. (1.26)] and [Reference Deift, Its and Krasovsky24, eq. (3.1)] for more specific $f$.

The monic orthogonal polynomials $\kappa _{n}^{-1}p_{n}, \kappa _{n}^{-1}\hat {p}_{n}$, and also $Y$, are unique (if they exist). The orthogonal polynomials exist if $f$ is strictly positive almost everywhere on $\mathbb {T}$ (this is the case if $W$ is real-valued, $\alpha _{k}>-1$ and $i\beta _{k} \in (-\frac {1}{2},\frac {1}{2})$). More generally, a sufficient condition to ensure existence of $p_{n}, \hat {p}_{n}$ (and therefore of $Y$) is that $D_{n}^{(n)} \neq 0 \neq D_{n+1}^{(n)}$, where $D_{l}^{(n)} =: \det (f_{j-k})_{j,k=0,\ldots,l-1}$, $l\geq 1$ (note that $D_{n}^{(n)}=D_{n}(\vec {\alpha },\vec {\beta },V,W)$), see e.g. [Reference Claeys and Krasovsky21, Section 2.1]. In fact,

(2.4)\begin{equation} p_{k}(z) = \frac{\begin{vmatrix} f_{0} & f_{{-}1} & \ldots & f_{{-}k}\\ \vdots & \vdots & \ddots & \vdots\\ f_{k-1} & f_{k-2} & \ldots & f_{{-}1}\\ 1 & z & \ldots & z^k \end{vmatrix}}{\sqrt{D_k^{(n)}}\sqrt{D_{k+1}^{(n)}}}, \quad \hat{p}_k(z) = \frac{\begin{vmatrix} f_{0} & f_{{-}1} & \ldots & f_{{-}k+1} & 1 \\ f_{1} & f_{0} & \ldots & f_{{-}k+2} & z \\ \vdots & \vdots & & \vdots & \vdots \\ f_{k} & f_{k-1} & \ldots & f_{1} & z^{k} \\ \end{vmatrix}}{\sqrt{D_k^{(n)}}\sqrt{D_{k+1}^{(n)}}}, \end{equation}

and $\kappa _k= (D_{k}^{(n)})^{1/2}/(D_{k+1}^{(n)})^{1/2}$. (Note that $p_{k}$, $\hat {p}_{k}$ and $\kappa _{k}$ are unique only up to multiplicative factors of $-1$. This can be fixed with a choice of the branch for the above roots. However, since $Y$ only involves $\kappa _{n}^{-1}p_{n}$ and $\kappa _{n-1}\hat {p}_{n-1}$, which are unique, this choice for the branch is unimportant for us.) If $D_{k}^{(n)}\neq 0$ for $k=0,1,\ldots,n+1$, it follows that

(2.5)\begin{equation} D_n(\vec{\alpha},\vec{\beta},V,W) = \prod_{j=0}^{n-1} \kappa_j^{{-}2}. \end{equation}

Lemma 2.1 Let $n \in \mathbb {N}$ be fixed, and assume that $D_k^{(n)}(f)\neq 0$, $k = 0,1,\ldots,n+1$. For any $z\ne 0$, we have

(2.6)\begin{equation} [Y^{{-}1}(z)Y'(z)]_{21}z^{{-}n+1} = \sum_{k = 0}^{n-1}\hat{p}_k(z^{{-}1})p_k(z), \end{equation}

where $Y(\cdot ) = Y_n(\cdot ;\vec \alpha,\vec \beta,V,W)$.

Proof. The assumptions imply that $\kappa _k= (D_k^{(n)})^{1/2}/(D_{k+1}^{(n)})^{1/2}$ is finite and nonzero and that $p_k, \hat {p}_k$ exist for all $k \in \{0,\ldots,n\}$. Note that (a) $\det Y: {\mathbb {C}} \setminus {\mathbb {T}} \to {\mathbb {C}}$ is analytic, (b) $(\det Y)_+(z) = (\det Y)_{-}(z)$ for $z \in {\mathbb {T}}\setminus \{t_{0},\ldots,t_{m}\}$, (c) $\det Y(z) = o(|z-t_{k}|^{-1})$ as $z \to t_{k}$ and (d) $\det Y(z) = 1+o(1)$ as $z\to \infty$. Hence, using successively Morera's theorem, Riemann's removable singularities theorem and Liouville's theorem, we conclude that $\det Y \equiv 1$. Using (2.3) and the fact that $\det Y \equiv 1$, we obtain

\begin{align*} \left[Y^{{-}1}(z)Y'(z)\right]_{21}& =\frac{z^n}{\kappa_n}\cdot\frac{\kappa_{n-1}}{z}\hat{p}_{n-1}(z^{{-}1})\frac{d}{dz}p_n(z)\\& \quad -\kappa_n^{{-}1}p_n(z)\frac{d}{dz}\left[z^n\cdot\frac{\kappa_{n-1}}{z}\hat{p}_{n-1}(z^{{-}1})\right]. \end{align*}

Using the recurrence relation (see [Reference Deift, Its and Krasovsky24, Lemma 2.2])

\[ \frac{\kappa_{n-1}}{z}\hat{p}_{n-1}(z^{{-}1})= \kappa_{n}\hat{p}_{n}(z^{{-}1})-\hat{p}_{n}(0)z^{{-}n}p_{n}(z), \]

we then find

\begin{align*} & \left[Y^{{-}1}(z)Y'(z)\right]_{21}\\ & \quad = z^{n-1}\left({-}np_n(z)\hat{p}_n(z^{{-}1})+z\left(\hat{p}_n(z^{{-}1})\frac{d}{dz}p_n(z)-p_n(z)\frac{d}{dz}\hat{p}_n(z^{{-}1})\right)\right). \end{align*}

The claim now directly follows from the Christoffel–Darboux formula [Reference Deift, Its and Krasovsky24, Lemma 2.3].

Proposition 2.2 Let $n \in \mathbb {N}_{\geq 1}:=\{1,2,\ldots \}$ be fixed and suppose that $f$ depends smoothly on a parameter $\gamma$. If $D_k^{(n)}(f)\neq 0$ for $k = n-1,n,n+1$, then the following differential identity holds

(2.7)\begin{equation} \partial_{\gamma} \log D_n(\vec{\alpha},\vec{\beta},V,W) = \frac{1}{2\pi}\int_{0}^{2\pi}[Y^{{-}1}(z)Y'(z)]_{21}z^{{-}n+1}\partial_{\gamma}f(z){\rm d}\theta, \qquad z=e^{i\theta}. \end{equation}

Proof. We first prove the claim under the stronger assumption that $D_k^{(n)}(f)\neq 0$ for $k = 0,1,\ldots,n+1$. In this case, $\kappa _k= (D_k^{(n)})^{1/2}/(D_{k+1}^{(n)})^{1/2}$ is finite and nonzero and $p_k, \hat {p}_k$ exist for all $k = 0,1,\ldots,n$. Replacing $z^{-j}$ with $\hat {p}_{j}(z^{-1})\kappa _j^{-1}$ in the first orthogonality condition in (2.2) (with $k=j$), and differentiating with respect to $\gamma$, we obtain, for $j = 0, \ldots, n-1$,

(2.8)\begin{align} -\frac{\partial_\gamma[\kappa_j]}{\kappa_j}& =\frac{\kappa_j}{2\pi}\partial_\gamma\left[\int_0^{2\pi}p_j(z)\hat{p}_j(z^{{-}1})\kappa_j^{{-}1}f(z){\rm d}\theta\right] \nonumber\\ & =\frac{1}{2\pi}\int_0^{2\pi}p_j(z)\hat{p}_j(z^{{-}1})\partial_\gamma[f(z)]{\rm d}\theta+\frac{\kappa_j}{2\pi}\int_0^{2\pi}\partial_\gamma\left[p_j(z)\hat{p}_j(z^{{-}1})\kappa_j^{{-}1}\right]f(z){\rm d}\theta. \end{align}

The second term on the right-hand side can be simplified as follows:

(2.9)\begin{align} & \frac{\kappa_j}{2\pi}\int_0^{2\pi}\partial_\gamma\left[p_j(z)\hat{p}_j(z^{{-}1})\kappa_j^{{-}1}\right]f(z){\rm d}\theta\nonumber\\& \quad = \frac{\kappa_j}{2\pi}\int_0^{2\pi}\partial_\gamma[p_j(z)]\hat{p}_j(z^{{-}1})\kappa_j^{{-}1}f(z){\rm d}\theta = \frac{\partial_\gamma[\kappa_j]}{\kappa_j}, \end{align}

where the first and second equalities use the first and second relations in (2.2), respectively. Combining (2.8) and (2.9), we find

(2.10)\begin{equation} -2\frac{\partial_\gamma[\kappa_j]}{\kappa_j}=\frac{1}{2\pi}\int_0^{2\pi}p_j(z)\hat{p}_j(z^{{-}1})\partial_\gamma[f(z)]{\rm d}\theta.\end{equation}

Taking the log of both sides of (2.5) and differentiating with respect to $\gamma$, we get

(2.11)\begin{align} \partial_{\gamma}\log D_{n}(\vec{\alpha},\vec{\beta},V,W)={-}2\sum_{j=0}^{n-1}\frac{\partial_\gamma[\kappa_j]}{\kappa_j}=\frac{1}{2\pi}\int_0^{2\pi}\left(\sum_{j=0}^{n-1}p_j(z)\hat{p}_j(z^{{-}1})\right)\partial_\gamma[f(z)]{\rm d}\theta.\end{align}

An application of lemma 2.1 completes the proof under the assumption that $D_k^{(n)}(f)\neq 0$, $k = 0,1,\ldots,n+1$. Since the existence of $Y$ only relies on the weaker assumption $D_k^{(n)}(f)\neq 0$, $k = n-1,n,n+1$, the claim follows from a simple continuity argument.

RH problem for $T$

1. (a) $T: {\mathbb {C}} \setminus \mathbb {T} \rightarrow {\mathbb {C}}^{2\times 2}$ is analytic.

2. (b) The boundary values $T_+$ and $T_{-}$ are continuous on $\mathbb {T}\setminus \{t_{0},\ldots,t_{m}\}$ and are related by

   \[ T_+(z) = T_-(z)\begin{pmatrix} e^{{-}2n\xi_+(z)} & e^{W(z)} \omega(z) \\ 0 & e^{{-}2n\xi_{-}(z)} \end{pmatrix}, \qquad z \in \mathbb{T} \setminus \{t_{0},\ldots,t_{m}\}. \]

3. (c) As $z \to \infty$, $T(z) = I + {\mathcal {O}}(z^{-1})$.

4. (d) As $z \rightarrow t_k$, $k = 0, \ldots, m$, $z \in {\mathbb {C}} \setminus \mathbb {T}$,

   \[ T(z) = \begin{cases} \begin{pmatrix} {\mathcal{O}}(1) & {\mathcal{O}}(1) + {\mathcal{O}}({|z-t_k|}^{\alpha_{k}}) \\ {\mathcal{O}}(1) & {\mathcal{O}}(1) + {\mathcal{O}}({|z-t_k|}^{\alpha_{k}}) \end{pmatrix}, & \mbox{if } {\rm Re\,} \alpha_k \ne 0, \\ \begin{pmatrix} {\mathcal{O}}(1) & {\mathcal{O}}(\log{|z-t_k|}) \\ {\mathcal{O}}(1) & {\mathcal{O}}(\log{|z-t_k|}) \end{pmatrix}, & \mbox{if } {\rm Re\,} \alpha_k = 0. \end{cases} \]

The jumps of $T$ for $z \in \mathbb {T}\setminus \{t_{0},\ldots,t_{m}\}$ can be factorized as

\begin{align*} & \begin{pmatrix} e^{{-}2n\xi_+(z)} & e^{W(z)} \omega(z) \\ 0 & e^{{-}2n\xi_{-}(z)} \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ e^{{-}W(z)}\omega(z)^{{-}1}e^{{-}2n \xi_{-}(z)} & 1 \end{pmatrix} \\ & \quad\times \begin{pmatrix} 0 & e^{W(z)}\omega(z) - e^{{-}W(z)}\omega(z)^{{-}1} & 0 \end{pmatrix} \begin{pmatrix} 1 & 0 \\ e^{{-}W(z)}\omega(z)^{{-}1}e^{{-}2n \xi_+(z)} & 1 \end{pmatrix}. \end{align*}

Before proceeding to the second transformation, we first describe the analytic continuations of the functions appearing in the above factorization. The functions $\omega _{\beta _{k}}$, $k =0,\ldots,m$, have a straightforward analytic continuation from $\mathbb {T} \setminus \{t_{k}\}$ to $\mathbb {C}\setminus \{\lambda t_{k}: \lambda \geq 0\}$, which is given by

(3.14)\begin{align} \omega_{\beta_{k}}(z) & = z^{\beta_{k}}t_{k}^{-\beta_{k}}\nonumber\\& \quad \times \begin{cases} e^{i \pi \beta_{k}}, & 0 \leq \arg_{0} z < \theta_{k}, \\ e^{{-}i \pi \beta_{k}}, & \theta_{k} \leq \arg_{0} z < 2 \pi, \end{cases}\enspace z \in \mathbb{C}\setminus \{\lambda t_{k}: \lambda \geq 0\}, \quad k =0,\ldots,m, \end{align}

where $\arg _{0} z \in [0,2\pi )$, $t_{k}^{-\beta _{k}} := e^{-i\beta _{k} \theta _{k}}$ and $z^{\beta _{k}} := |z|^{\beta _{k}}e^{i\beta _{k} \arg _{0} z}$. For the root-type singularities, we follow [Reference Deift, Its and Krasovsky24] and analytically continue $\omega _{\alpha _{k}}$ from $\mathbb {T}\setminus \{t_{k}\}$ to $\mathbb {C}\setminus \{\lambda t_{k}: \lambda \geq 0\}$ as follows

\begin{align*} \omega_{\alpha_{k}}(z) = \frac{(z-t_{k})^{\alpha_{k}}}{(z t_{k} e^{i \ell_{k}(z)})^{\alpha_{k}/2}} := \frac{e^{\alpha_{k}(\log |z-t_{k}|+i \hat{\arg}_{k}(z-t_{k}))}}{e^{\frac{\alpha_{k}}{2}(\log |z| + i \arg_{0}(z) + i \theta_{k} + i \ell_{k}(z))}},\quad z \in \mathbb{C}\setminus \{\lambda t_{k}: \lambda \geq 0\},\\ k=0,\ldots,m, \end{align*}

where $\hat {\arg }_{k} z \in (\theta _{k},\theta _{k}+2\pi )$, and

\[ \ell_{k}(z) = \begin{cases} 3 \pi, & 0 \leq \arg_{0} z < \theta_{k}, \\ \pi, & \theta_{k} \leq \arg_{0} z < 2 \pi. \end{cases} \]

Now, we open lenses around $\mathbb {T}\setminus \{t_{0},\ldots,t_{m}\}$ as shown in figure 1. The part of the lens-shaped contour lying in $\{|z|<1\}$ is denoted $\gamma _+$, and the part lying in $\{|z|>1\}$ is denoted $\gamma _{-}$. We require that $\gamma _+,\gamma _{-} \subset U$. The transformation $T \mapsto S$ is defined by

(3.15)\begin{align} S(z) & = T(z)\nonumber\\ & \quad\times \begin{cases} I, & \mbox{if } z \mbox{ is outside the lenses,} \\ \begin{pmatrix} 1 & 0 \\ e^{{-}W(z)}\omega(z)^{{-}1}e^{{-}2n\xi(z)} & 1 \end{pmatrix}, & \mbox{if } |z| > 1 \mbox{ and inside the lenses,} \\ \begin{pmatrix} 1 & 0 - e^{{-}W(z)}\omega(z)^{{-}1}e^{{-}2n\xi(z)} & 1 \end{pmatrix}, & \mbox{if } |z| < 1 \mbox{ and inside the lenses.} \end{cases} \end{align}

Note from (3.11)–(3.12) that $e^{-2n\xi (z)}$ is analytic in $U \cap ((-\infty,-1)\cup (-1,0))$. It can be verified using the RH problem for $T$ and (3.15) that $S$ satisfies the following RH problem.

Figure 1. The jump contour for $S$ with $m=2$.

RH problem for $S$

1. (a) $S : {\mathbb {C}} \setminus (\gamma _+ \cup \gamma _{-} \cup \mathbb {T}) \to \mathbb {C}^{2 \times 2}$ is analytic, where $\gamma _+, \gamma _-$ are the contours in figure 1 lying inside and outside $\mathbb {T}$, respectively.

2. (b) The jumps for $S$ are as follows.

   \begin{align*} & S_+(z) = S_-(z)\begin{pmatrix} 0 & e^{W(z)}\omega(z) - e^{{-}W(z)}\omega(z)^{{-}1} & 0 \end{pmatrix}, & & z \in \mathbb{T}\setminus \{t_{0},\ldots,t_{m}\}, \\ & S_+(z) = S_-(z)\begin{pmatrix} 1 & 0 \\ e^{{-}W(z)}\omega(z)^{{-}1}e^{{-}2n\xi(z)} & 1 \end{pmatrix}, & & z \in \gamma_+{\cup} \gamma_{-}. \end{align*}

3. (c) As $z \rightarrow \infty$, $S(z) = I + {\mathcal {O}}(z^{-1})$.

4. (d) As $z \to t_{k}$, $k = 0,\ldots,m$, we have

   \begin{align*} & \displaystyle S(z) = \left\{\begin{array}{@{}ll} \begin{pmatrix} {\mathcal{O}}(1) & {\mathcal{O}}(\log (z-t_{k})) \\ {\mathcal{O}}(1) & {\mathcal{O}}(\log (z-t_{k})) \end{pmatrix}, & \mbox{if } z \mbox{ is outside the lenses}, \\ \begin{pmatrix} {\mathcal{O}}(\log (z-t_{k})) & {\mathcal{O}}(\log (z-t_{k})) \\ {\mathcal{O}}(\log (z-t_{k})) & {\mathcal{O}}(\log (z-t_{k})) \end{pmatrix}, & \mbox{if } z \mbox{ is inside the lenses}, \end{array}\right.\\ & \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \displaystyle \mbox{ if } {\rm Re\,} \alpha_{k} = 0, \\ & \displaystyle S(z) = \left\{\begin{array}{@{}ll} \begin{pmatrix} {\mathcal{O}}(1) & {\mathcal{O}}(1) \\ {\mathcal{O}}(1) & {\mathcal{O}}(1) \end{pmatrix}, & \mbox{if } z \mbox{ is outside the lenses}, \\ \begin{pmatrix} {\mathcal{O}}((z-t_{k})^{-\alpha_{k}}) & {\mathcal{O}}(1) \\ {\mathcal{O}}((z-t_{k})^{-\alpha_{k}}) & {\mathcal{O}}(1) \end{pmatrix}, & \mbox{if } z \mbox{ is inside the lenses}, \end{array} \right.\\ & \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \displaystyle \mbox{ if } {\rm Re\,} \alpha_{k} > 0, \\ & \displaystyle S(z) = \begin{pmatrix} {\mathcal{O}}(1) & {\mathcal{O}}((z-t_{k})^{\alpha_{k}}) \\ {\mathcal{O}}(1) & {\mathcal{O}}((z-t_{k})^{\alpha_{k}}) \end{pmatrix},\\ & \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \displaystyle \mbox{ if } {\rm Re\,} \alpha_{k} < 0. \end{align*}

Since $\gamma _+,\gamma _{-} \subset U$ and ${\rm Re\,} \xi (z) > 0$ for $z \in U\setminus \mathbb {T}$ (recall the discussion below (3.7)), the jump matrices $S_{-}(z)^{-1}S_+(z)$ on $\gamma _+\cup \gamma _{-}$ are exponentially close to $I$ as $n \to + \infty$, and this convergence is uniform outside fixed neighbourhoods of $t_{0},\ldots,t_{m}$.

Our next task is to find suitable approximations (called ‘parametrices’) for $S$ in different regions of the complex plane.

RH problem for $P^{(\infty )}$

1. (a) $P^{(\infty )}:\mathbb {C}\setminus \mathbb {T} \to \mathbb {C}^{2\times 2}$ is analytic.

2. (b) The jumps are given by

   (3.16)\begin{align} P^{(\infty)}_+(z) = P^{(\infty)}_-(z) \begin{pmatrix} 0 & e^{W(z)}\omega(z) - e^{{-}W(z)}\omega(z)^{{-}1} & 0 \end{pmatrix},\nonumber\\ z \in \mathbb{T}\setminus \{t_{0},\ldots,t_{m}\}. \end{align}

3. (c) As $z \to \infty$, we have $P^{(\infty )}(z) = I + {\mathcal {O}}(z^{-1})$.

4. (d) As $z \to t_{k}$ from $|z| \lessgtr 1$, $k\in \{0,\ldots,m\}$, we have $P^{(\infty )}(z) = {\mathcal {O}}(1)(z-t_{k})^{-(\frac {\alpha _{k}}{2}\pm \beta _{k})\sigma _{3}}$.

The unique solution to the above RH problem is given by

(3.17)\begin{equation} P^{(\infty)}(z) = \begin{cases} D(z)^{\sigma_3}\begin{pmatrix} 0 & 1 - 1 & 0 \end{pmatrix}, & \mbox{ if } |z| < 1, \\ D(z)^{\sigma_3}, & \mbox{ if } |z| > 1, \end{cases} \end{equation}

where $D(z)$ is the Szegő function defined by

(3.18)\begin{align} & D(z) = D_{W}(z) \prod_{k=0}^{m}D_{\alpha_{k}}(z)D_{\beta_{k}}(z), & & D_{W}(z)=\exp{\left(\frac{1}{2\pi i}\int_{\mathbb{T}} \frac{W(s)}{s-z}{\rm d}s\right)}, \end{align}

(3.19)\begin{align} & D_{\alpha_{k}}(z) = \exp{\left(\frac{1}{2\pi i}\int_{\mathbb{T}} \frac{\log \omega_{\alpha_{k}}(s)}{s-z}{\rm d}s\right)}, & & D_{\beta_{k}}(z) = \exp{\left(\frac{1}{2\pi i}\int_{\mathbb{T}} \frac{\log \omega_{\beta_{k}}(s)}{s-z}{\rm d}s\right)}. \end{align}

The branches of the logarithms in (3.19) can be arbitrarily chosen as long as $\log \omega _{\alpha _k}(s)$ and $\log \omega _{\beta _k}(s)$ are continuous on $\mathbb {T} \setminus t_k$. The function $D$ is analytic on $\mathbb {C}\setminus \mathbb {T}$ and satisfies the jump condition $D_+(z) = D_-(z)e^{W(z)}\omega (z)$ on $\mathbb {T} \setminus \{t_0, \ldots, t_m\}$. The expressions for $D_{\alpha _{k}}$ and $D_{\beta _{k}}$ can be simplified as in [Reference Deift, Its and Krasovsky24, eqs. (4.9)–(4.10)]; we have

(3.20)\begin{equation} D_{\alpha_{k}}(z) D_{\beta_{k}}(z) = \begin{cases} \displaystyle \left( \frac{z-t_{k}}{t_{k}e^{i\pi}} \right)^{\frac{\alpha_{k}}{2}+\beta_{k}} = \frac{e^{(\frac{\alpha_{k}}{2}+\beta_{k})(\log |z-t_{k}| + i \hat{\arg}_{k}(z-t_{k}))}}{e^{(\frac{\alpha_{k}}{2}+\beta_{k})(i\theta_{k}+i\pi)}}, & \mbox{if } |z|<1, \\ \displaystyle \left( \frac{z-t_{k}}{z} \right)^{-\frac{\alpha_{k}}{2}+\beta_{k}} = \frac{e^{(\beta_{k}-\frac{\alpha_{k}}{2})(\log |z-t_{k}| + i \hat{\arg}_{k}(z-t_{k}))}}{e^{(\beta_{k}-\frac{\alpha_{k}}{2})(\log |z| + i \hat{\arg}_{k} z)}}, & \mbox{if } |z|>1, \end{cases} \end{equation}

where $\hat {\arg }_{k}$ was defined below (3.14). Using (1.4), we can also simplify $D_{W}$ as

(3.21)\begin{equation} D_{W}(z) = \begin{cases} e^{W_{0}+W_+(z)}, & |z|<1, \\ e^{{-}W_{-}(z)}, & |z|>1. \end{cases} \end{equation}

RH problem for $P^{(t_{k})}$

1. (a) $P^{(t_k)}: \mathcal {D}_{t_{k}}\setminus (\mathbb {T} \cup \gamma _+ \cup \gamma _{-}) \to \mathbb {C}^{2\times 2}$ is analytic.

2. (b) For $z\in (\mathbb {T} \cup \gamma _+ \cup \gamma _{-})\cap \mathcal {D}_{t_{k}}$, $P_{-}^{(t_k)}(z)^{-1}P_+^{(t_k)}(z)=S_{-}(z)^{-1}S_+(z)$.

3. (c) As $n\to +\infty$, $P^{(t_k)}(z)=(I+{\mathcal {O}}(n^{-1+2|{\rm Re\,} \beta _{k}|}))P^{(\infty )}(z)$ uniformly for $z\in \partial \mathcal {D}_{t_k}$.

4. (d) As $z\to t_k$, $S(z)P^{(t_k)}(z)^{-1}={\mathcal {O}}(1)$.

A solution to the above RH problem can be constructed using hypergeometric functions as in [Reference Deift, Its and Krasovsky24, Reference Foulquié Moreno, Martinez-Finkelshtein and Sousa35]. Consider the function

\[ f_{t_{k}}(z):= 2\pi i \int_{t_{k}}^{z} \psi(s) \frac{{\rm d}s}{is}, \qquad z \in \mathcal{D}_{t_{k}}, \]

where the path is a straight line segment from $t_{k}$ to $z$. This is a conformal map from $\mathcal {D}_{t_k}$ to a neighbourhood of $0$, which satisfies

(3.22)\begin{equation} f_{t_{k}}(z) = 2\pi t_{k}^{{-}1} \psi(t_{k}) (z-t_{k}) \left( 1+{\mathcal{O}}(z-t_{k}) \right), \qquad \mbox{ as } z \to t_{k}. \end{equation}

If $\mathcal {D}_{t_{k}} \cap (-\infty,0] = \emptyset$, $f_{t_{k}}$ can also be expressed as

\[ f_{t_{k}}(z) ={-} 2 \times \begin{cases} \xi(z)-\xi_+(t_{k}), & |z|<1,\\ - (\xi(z)-\xi_{-}(t_{k})), & |z|>1. \end{cases} \]

If $\mathcal {D}_{t_{k}} \cap (-\infty,0] \neq \emptyset$, then instead we have

(3.23)\begin{align} & f_{t_{k}}(z) ={-} 2 \times \begin{cases} \xi(z)-\xi_+(t_{k}), & |z|<1, \; {\rm Im\,} z >0, \\ \xi(z)-\xi_+(t_{k})-\pi i, & |z|<1, \; {\rm Im\,} z <0,\\ - (\xi(z)-\xi_{-}(t_{k})), & |z|>1, \; {\rm Im\,} z >0,\\ - (\xi(z)-\xi_{-}(t_{k})+\pi i), & |z|>1, \; {\rm Im\,} z <0, \end{cases} \quad \mbox{if } {\rm Im\,} t_{k}>0, \\ & f_{t_{k}}(z) ={-} 2 \times \begin{cases} \xi(z)-\xi_+(t_{k})+\pi i, & |z|<1, \; {\rm Im\,} z >0, \\ \xi(z)-\xi_+(t_{k}), & |z|<1, \; {\rm Im\,} z <0,\\ -(\xi(z)-\xi_{-}(t_{k})-\pi i), & |z|>1, \; {\rm Im\,} z >0,\\ -(\xi(z)-\xi_{-}(t_{k})), & |z|>1, \; {\rm Im\,} z <0, \end{cases} \quad \mbox{if } {\rm Im\,} t_{k}<0. \nonumber \end{align}

If $t_{k}=-1$, (3.23) also holds with $\xi _{\pm }(t_k) := \lim _{\epsilon \to 0^+}\xi _{\pm }(e^{(\pi - \epsilon )i})=0$. We define $\omega _{k}$ and $\widetilde {W}_{k}$ by

\begin{align*} \omega_{k}(z)& = e^{{-}2\pi i \beta_{k}\hat{\theta}(z;k)}z^{\beta_{k}}t_{k}^{-\beta_{k}} \prod_{j\neq k} \omega_{\alpha_{j}}(z)\omega_{\beta_{j}}(z),\\ \widetilde{W}_{k}(z) & = \check{\omega}_{\alpha_{k}}(z)^{\frac{1}{2}} \times \begin{cases} e^{- \frac{i\pi\alpha_{k}}{2}}, & z \in Q_{+,k}^{R}\cup Q_{-,k}^{L}, \\ e^{ \frac{i\pi\alpha_{k}}{2}}, & z \in Q_{-,k}^{R} \cup Q_{+,k}^{L}, \end{cases} \end{align*}

where $\hat {\theta }(z;k)=1$ if ${\rm Im\,} z <0$ and $k=0$ and $\hat {\theta }(z;k)=0$ otherwise, $z^{\beta _{k}} := |z|^{\beta _{k}}e^{i\beta _{k} \arg _{0} z}$,

(3.24)\begin{equation} \check{\omega}_{\alpha_{k}}(z)^{1/2} := \frac{(z-t_{k})^{\frac{\alpha_{k}}{2}}}{(z t_{k} e^{i \ell_{k}(z)})^{\alpha_{k}/4}} := \frac{e^{\frac{\alpha_{k}}{2}(\log |z-t_{k}|+i \check{\arg}_{k}(z-t_{k}))}}{e^{\frac{\alpha_{k}}{4}(\log |z| + i \arg_{0}(z) + i \theta_{k} + i \ell_{k}(z))}}, \end{equation}

and (see figure 2)

\begin{align*} Q_{{\pm},k}^{R}& = \{ z \in \mathcal{D}_{t_{k}}: \mp {\rm Re\,} f_{t_{k}}(z) > 0 \mbox{, } {\rm Im\,} f_{t_{k}}(z) >0 \}, \\ Q_{{\pm},k}^{L}& = \{ z \in \mathcal{D}_{t_{k}}: \mp {\rm Re\,} f_{t_{k}}(z) > 0 \mbox{, } {\rm Im\,} f_{t_{k}}(z) <0 \}. \end{align*}

Figure 2. The four quadrants $Q_{\pm,k}^{R}$, $Q_{\pm,k}^{L}$ near $t_k$ and their images under the map $f_{t_k}$.

The argument $\check {\arg }_{k}(z-t_{k})$ in (3.24) is defined to have a discontinuity for $z \in (\overline {Q_{-,k}^{L}} \cap \overline {Q_{-,k}^{R}}) \cup [z_{\star,k},t_{k}\infty )$, $z_{\star,k}:=\overline {Q_{-,k}^{L}} \cap \overline {Q_{-,k}^{R}}\cap \partial \mathcal {D}_{t_{k}}$, and such that $\check {\arg }_{k}((1-0_+)t_{k}-t_{k})=\theta _{k}+\pi$. Note that $\check {\arg }_{k}(z-t_{k})$ is merely a small deformation of the argument $\hat {\arg }_{k}(z-t_{k})$ defined below (3.14). This small deformation is needed to ensure that $E_{t_{k}}$ in (3.26) below is analytic in $\mathcal {D}_{t_{k}}$.

Note that $\omega _{k}$ is analytic in $\mathcal {D}_{t_{k}}$. We now use the confluent hypergeometric model RH problem, whose solution is denoted $\Phi _{\mathrm {HG}}(z;\alpha _{k},\beta _{k})$ (see appendix B for the definition and properties of $\Phi _{\mathrm {HG}}$). If $k \neq 0$ and $\mathcal {D}_{t_{k}} \cap (-\infty,0] = \emptyset$, we define

(3.25)\begin{equation} P^{(t_{k})}(z) = E_{t_{k}}(z)\Phi_{\mathrm{HG}}(nf_{t_{k}}(z);\alpha_{k},\beta_{k})\widetilde{W}_{k}(z)^{-\sigma_{3}}e^{{-}n\xi(z)\sigma_{3}}e^{-\frac{W(z)}{2}\sigma_{3}}\omega_{k}(z)^{-\frac{\sigma_{3}}{2}}, \end{equation}

where $E_{t_{k}}$ is given by

(3.26)\begin{align} E_{t_{k}}(z) & = P^{(\infty)}(z) \omega_{k}(z)^{\frac{\sigma_{3}}{2}}e^{\frac{W(z)}{2}\sigma_{3}} \widetilde{W}_{k}(z)^{\sigma_{3}}\nonumber\\ & \quad \times \left\{ \hspace{-0.18cm} \begin{array}{l l} e^{ \dfrac{i\pi\alpha_{k}}{4}\sigma_{3}}e^{{-}i\pi\beta_{k} \sigma_{3}}, \hspace{-0.2cm} & z \in Q_{+,k}^{R} \\ e^{-\dfrac{i\pi\alpha_{k}}{4}\sigma_{3}}e^{{-}i\pi\beta_{k}\sigma_{3}}, \hspace{-0.2cm} & z \in Q_{+,k}^{L} \\ e^{\dfrac{i\pi\alpha_{k}}{4}\sigma_{3}}\begin{pmatrix} 0 & 1 - 1 & 0 \end{pmatrix} , \quad & z \in Q_{-,k}^{L} \\ e^{-\dfrac{i\pi\alpha_{k}}{4}\sigma_{3}}\begin{pmatrix} 0 & 1 - 1 & 0 \end{pmatrix} , \quad & z \in Q_{-,k}^{R} \\ \end{array}\right\} \quad e^{n\xi_+(t_{k})\sigma_{3}} (nf_{t_{k}}(z))^{\beta_{k}\sigma_{3}}. \end{align}

Here the branch of $f_{t_k}(z)^{\beta _k}$ is such that $f_{t_k}(z)^{\beta _k} = |f_{t_{k}}(z)|^{\beta _k} e^{\beta _k i\arg f_{t_{k}}(z)}$ with $\arg f_{t_k}(z) \in (-\frac {\pi }{2}, \frac {3\pi }{2})$, and the branch for the square root of $\omega _{k}(z)$ can be chosen arbitrarily as long as $\omega _{k}(z)^{1/2}$ is analytic in $\mathcal {D}_{t_{k}}$ (note that $P^{(t_{k})}(z)$ is invariant under a sign change of $\omega _{k}(z)^{1/2}$). If $k \neq 0$, $\mathcal {D}_{t_{k}} \cap (-\infty,0] \neq \emptyset$ and ${\rm Im\,} t_{k} \geq 0$ (resp. ${\rm Im\,} t_{k} < 0$), then we define $P^{(t_{k})}(z)$ as in (3.25) but with $\xi (z)$ replaced by $\xi (z)+\pi i \theta _{-}(z)$ (resp. $\xi (z)+\pi i \theta _+(z)$), where

\[ \theta_{-}(z):= \begin{cases} 1, & \mbox{if } {\rm Im\,} z <0, \; |z|>1, \\ - 1, & \mbox{if } {\rm Im\,} z <0, \; |z|<1, \\ 0, & \mbox{otherwise,} \end{cases} \quad \theta_+(z):= \begin{cases} -1, & \mbox{if } {\rm Im\,} z >0, \; |z|>1, \\ 1, & \mbox{if } {\rm Im\,} z >0, \; |z|<1, \\ 0, & \mbox{otherwise.} \end{cases} \]

Using the definition of $\widetilde {W}_{k}$ and the jumps (3.16) of $P^{(\infty )}$, we verify that $E_{t_{k}}$ has no jumps in $\mathcal {D}_{t_{k}}$. Moreover, since $P^{(\infty )}(z) = {\mathcal {O}}(1)(z-t_{k})^{-(\frac {\alpha _{k}}{2}\pm \beta _{k})\sigma _{3}}$ as $z\to t_{k}$, $\pm (1-|z|) > 0$, we infer from (3.26) that $E_{t_{k}}(z) = {\mathcal {O}}(1)$ as $z \to t_{k}$, and therefore $E_{t_{k}}$ is analytic in $\mathcal {D}_{t_{k}}$. Using (3.26), we see that $E_{t_{k}}(z) = {\mathcal {O}}(1)n^{\beta _{k}\sigma _{3}}$ as $n \to \infty$, uniformly for $z \in \mathcal {D}_{t_{k}}$. Since $P^{(t_{k})}$ and $S$ have the same jumps on $(\mathbb {T}\cup \gamma _+\cup \gamma _{-})\cap \mathcal {D}_{t_{k}}$, $S(z)P^{(t_{k})}(z)^{-1}$ is analytic in $\mathcal {D}_{t_{k}} \setminus \{t_{k}\}$. Furthermore, by (B.5) and condition (d) in the RH problem for $S$, as $z \to t_{k}$ from outside the lenses we have that $S(z)P^{(t_{k})}(z)^{-1}$ is ${\mathcal {O}}(\log (z-t_{k}))$ if ${\rm Re\,} \alpha _{k} = 0$, is ${\mathcal {O}}(1)$ if ${\rm Re\,} \alpha _{k} > 0$, and is ${\mathcal {O}}((z-t_{k})^{\alpha _{k}})$ if ${\rm Re\,} \alpha _{k} < 0$. In all cases, the singularity of $S(z)P^{(t_{k})}(z)^{-1}$ at $z = t_{k}$ is removable and therefore $P^{(t_{k})}$ in (3.25) satisfies condition (d) of the RH problem for $P^{(t_{k})}$.

The value of $E_{t_{k}}(t_{k})$ can be obtained by taking the limit $z \to t_{k}$ in (3.26) (e.g. from the quadrant $Q_{+,k}^{R}$). Using (3.17), (3.18), (3.20), (3.22) and (3.26), we obtain

(3.27)\begin{equation} E_{t_{k}}(t_{k}) = \begin{pmatrix} 0 & 1 - 1 & 0 \end{pmatrix} \Lambda_{k}^{\sigma_{3}}, \end{equation}

where

(3.28)\begin{align} \Lambda_{k} & = e^{\frac{W(t_{k})}{2}}D_{W,+}(t_{k})^{{-}1}\nonumber\\& \quad \times\Bigg[\prod_{j \neq k}D_{\alpha_{j},+}(t_{k})^{{-}1}D_{\beta_{j},+}(t_{k})^{{-}1} \omega_{\alpha_{j}}^{\frac{1}{2}}(t_{k})\omega_{\beta_{j}}^{\frac{1}{2}}(t_{k})\Bigg](2\pi \psi(t_{k})n)^{\beta_{k}}e^{n \xi_+(t_{k})}. \end{align}

In (3.28), the branch of $\omega _{\alpha _{j}}^{\frac {1}{2}}(t_{k})$ is as in (3.24) and $\omega _{\beta _{j}}^{\frac {1}{2}}(t_{k})$ is defined by

\begin{align*} \omega_{\beta_{j}}^{\frac{1}{2}}(t_k) := e^{i\frac{\beta_j}{2}(\theta_{k} - \theta_{j})} \times \begin{cases} e^{\frac{i \pi}{2} \beta_{j}}, & \mbox{if } 0 \leq \theta_{k} < \theta_{j}, \\ e^{-\frac{i \pi}{2} \beta_{j}}, & \mbox{if } \theta_{j} \leq \theta_{k} < 2 \pi. \end{cases} \end{align*}

The expression for $\Lambda _k$ can be further simplified as follows. A simple computation shows that

\begin{align*} D_{\alpha_{j},+}(z) & = |z-t_{j}|^{\frac{\alpha_{j}}{2}}\exp \left( \frac{i \alpha_{j}}{2}\left[ \hat{\arg}_{j}(z-t_{j}) - \theta_{j} - \pi \right] \right) \\ & = |z-t_{j}|^{\frac{\alpha_{j}}{2}}\exp \left( \frac{i \alpha_{j}}{2}\left[ \frac{\ell_{j}(z)}{2}+ \frac{\arg_{0} z - \theta_{j}}{2} - \pi \right] \right), \\ D_{\beta_{j},+}(z) & = |z-t_{j}|^{\beta_{j}}\exp \left( i \beta_{j} \left[ \hat{\arg}_{j}(z-t_{j}) - \theta_{j} - \pi \right] \right) \\ & = |z-t_{j}|^{\beta_{j}}\exp \left( i \beta_{j}\left[ \frac{\ell_{j}(z)}{2}+ \frac{\arg_{0} z - \theta_{j}}{2} - \pi \right] \right) \end{align*}

for $z \in \mathbb {T}$. Therefore, the product in brackets in (3.28) can be rewritten as

\[ \prod_{j \neq k} |t_{k}-t_{j}|^{-\beta_{j}}\exp \left( - \frac{i\alpha_{j}}{2} \frac{\theta_{k} - \theta_{j}}{2} \right)\prod_{j=0}^{k-1} \exp \left( \frac{\pi i \alpha_{j}}{4} \right) \prod_{j=k+1}^{m} \exp \left( -\frac{\pi i \alpha_{j}}{4} \right), \]

and thus

\[ \Lambda_{k} = e^{\frac{W(t_{k})}{2}}D_{W,+}(t_{k})^{{-}1}e^{\frac{i \lambda_{k}}{2}}(2\pi \psi(t_{k})n)^{\beta_{k}}\prod_{j \neq k} |t_{k}-t_{j}|^{-\beta_{j}}, \]

where

(3.29)\begin{equation} \lambda_{k} = \sum_{j=0}^{k-1} \frac{\pi \alpha_{j}}{2} - \sum_{j=k+1}^{m} \frac{\pi \alpha_{j}}{2} - \sum_{j \neq k} \frac{\alpha_{j}(\theta_{k}-\theta_{j})}{2} + 2\pi n \int_{t_{k}}^{{-}1} \psi(s)\frac{{\rm d}s}{is}.\end{equation}

Using (3.25) and (B.2), we obtain

(3.30)\begin{align} & P^{(t_{k})}(z)P^{(\infty)}(z)^{{-}1}\nonumber\\ & \quad = I + \frac{\beta_{k}^{2}-\frac{\alpha_{k}^{2}}{4}}{n f_{t_{k}}(z)} E_{t_{k}}(z) \begin{pmatrix} -1 & \tau(\alpha_{k},\beta_{k}) - \tau(\alpha_{k},-\beta_{k}) & 1 \end{pmatrix}E_{t_{k}}(z)^{{-}1}\nonumber\\ & \qquad + {\mathcal{O}} (n^{{-}2+2|{\rm Re\,} \beta_{k}|}), \end{align}

as $n \to \infty$ uniformly for $z \in \partial \mathcal {D}_{t_{k}}$, where $\tau (\alpha _{k},\beta _{k})$ is defined in (B.3).

RH problem for $R$

1. (a) $R : {\mathbb {C}} \setminus \Gamma _{R} \to \mathbb {C}^{2 \times 2}$ is analytic, where $\Gamma _{R} = \cup _{k=0}^{m} \partial \mathcal {D}_{t_{k}} \cup ((\gamma _+ \cup \gamma _{-}) \setminus \cup _{k=0}^{m} \mathcal {D}_{t_{k}})$ and the circles $\partial \mathcal {D}_{t_{k}}$ are oriented in the clockwise direction.

2. (b) The jumps are given by

   \begin{align*} & R_+(z) = R_{-}(z) P^{(\infty)}(z)& \\ & \quad \times\!\begin{pmatrix} 1 & 0 \\ e^{{-}W(z)}\omega(z)^{{-}1}e^{{-}2n\xi(z)} & 1 \end{pmatrix} P^{(\infty)}(z)^{{-}1}, & \!\!\!z \in (\gamma_+{\cup} \gamma_{-}) {\setminus} \cup_{k=0}^{m} \overline{\mathcal{D}_{t_{k}}},\\ & R_+(z) = R_{-}(z) P^{(t_{k})}(z)P^{(\infty)}(z)^{{-}1}, & z \in \partial \mathcal{D}_{t_{k}}, \, k=0,\ldots,m. \end{align*}

3. (c) As $z \to \infty$, $R(z) = I + {\mathcal {O}}(z^{-1})$.

4. (d) As $z \to z^{*}\in \Gamma _{R}^{*}$, where $\Gamma _{R}^{*}$ is the set of self-intersecting points of $\Gamma _{R}$, we have $R(z) = {\mathcal {O}}(1)$.

Recall that ${\rm Re\,} \xi (z) \geq c > 0$ for $z \in (\gamma _+ \cup \gamma _{-}) \setminus \cup _{k=0}^{m} \mathcal {D}_{t_{k}}$. Moreover, we see from (3.17) that $P^{(\infty )}(z)$ is bounded for $z$ away from the points $t_{0},\ldots,t_{m}$. Using also (3.30), we conclude that as $n \to + \infty$

(3.32)\begin{align} & J_{R}(z) = I + {\mathcal{O}}(e^{{-}cn}), \quad\mbox{uniformly for } z \in (\gamma_+{\cup} \gamma_{-}) \setminus \cup_{k=0}^{m} \overline{\mathcal{D}_{t_{k}}}, \end{align}

(3.33)\begin{align} & J_{R}(z) = I + J_{R}^{(1)}(z)n^{{-}1} + {\mathcal{O}}(n^{{-}2+2\beta_{\max}}), \quad\mbox{uniformly for } z \in \cup_{k=0}^{m} \partial \mathcal{D}_{t_{k}}, \end{align}

where $J_{R}(z):=R_{-}^{-1}(z)R_+(z)$ and

\[ J_{R}^{(1)}(z) = \frac{\beta_{k}^{2}-\frac{\alpha_{k}^{2}}{4}}{f_{t_{k}}(z)} E_{t_{k}}(z) \begin{pmatrix} -1 & \tau(\alpha_{k},\beta_{k}) - \tau(\alpha_{k},-\beta_{k}) & 1 \end{pmatrix}E_{t_{k}}(z)^{{-}1}, \quad z \in \partial\mathcal{D}_{t_{k}}. \]

Furthermore, it is easy to see that the ${\mathcal {O}}$-terms in (3.32)–(3.33) are uniform for $(\theta _{1},\ldots,\theta _{m})$ in any given compact subset $\Theta \subset (0,2\pi )_{\mathrm {ord}}^{m}$, for $\alpha _{0},\ldots,\alpha _{m}$ in any given compact subset $\mathfrak {A}\subset \{z \in \mathbb {C}: {\rm Re\,} z >-1\}$, and for $\beta _{0},\ldots,\beta _{m}$ in any given compact subset $\mathfrak {B}\subset \{z \in \mathbb {C}: {\rm Re\,} z \in (-\frac {1}{2},\frac {1}{2})\}$. Therefore, $R$ satisfies a small norm RH problem, and the existence of $R$ for all sufficiently large $n$ can be proved using standard theory [Reference Deift, Kriecherbauer, McLaughlin, Venakides and Zhou27, Reference Deift and Zhou28] as follows. Define the operator $\mathcal {C}:L^{2}(\Gamma _{R})\to L^{2}(\Gamma _{R})$ by $\mathcal {C}f(z) = \frac {1}{2\pi i}\int _{\Gamma _{R}}\frac {f(s)}{s-z}dz$, and denote $\mathcal {C}_+f$ and $\mathcal {C}_{-}f$ for the left and right non-tangential limits of $\mathcal {C}f$. Since $\Gamma _{R}$ is a compact set, by (3.32)–(3.33) we have $J_{R}-I \in L^{2}(\Gamma _{R})\cap L^{\infty }(\Gamma _{R})$, and we can define

\begin{align*} \mathcal{C}_{J_{R}}: L^{2}(\Gamma_{R})+L^{\infty}(\Gamma_{R}) \to L^{2}(\Gamma_{R}), \\ \mathcal{C}_{J_{R}}f=\mathcal{C}_{-}(f(J_{R}-I)), \\ f \in L^{2}(\Gamma_{R})+L^{\infty}(\Gamma_{R}). \end{align*}

Using $\| \mathcal {C}_{J_{R}} \|_{L^{2}(\Gamma _{R}) \to L^{2}(\Gamma _{R})} \leq C \|J_{R}-I\|_{L^{\infty }(\Gamma _{R})}$ and (3.32)–(3.33), we infer that there exists $n_{0}=n_{0}(\Theta,\mathfrak {A},\mathfrak {B})$ such that $\| \mathcal {C}_{J_{R}} \|_{L^{2}(\Gamma _{R}) \to L^{2}(\Gamma _{R})} <1$ for all $n \geq n_{0}$, all $(\theta _{1},\ldots,\theta _{m})\in \Theta$, all $\alpha _{0},\ldots,\alpha _{m} \in \mathfrak {A}$ and all $\beta _{0},\ldots,\beta _{m}\in \mathfrak {B}$. Hence, for $n \geq n_{0}$, $I-\mathcal {C}_{J_{R}}:L^{2}(\Gamma _{R}) \to L^{2}(\Gamma _{R})$ can be inverted as a Neumann series and thus $R$ exists and is given by

(3.34)\begin{equation} R=I+\mathcal{C}(\mu_{R}(J_{R}-I)), \qquad \mbox{where } \quad \mu_{R}:= I + (I-\mathcal{C}_{J_{R}})^{{-}1}\mathcal{C}_{J_{R}}(I). \end{equation}

Using (3.34), (3.32) and (3.33), we obtain

(3.35)\begin{equation} R(z) = I+R^{(1)}(z)n^{{-}1} + {\mathcal{O}}(n^{{-}2+2\beta_{\max}}), \qquad \mbox{as } n \to + \infty, \end{equation}

uniformly for $(\theta _{1},\ldots,\theta _{m})\in \Theta$, $\alpha _{0},\ldots,\alpha _{m} \in \mathfrak {A}$ and $\beta _{0},\ldots,\beta _{m}\in \mathfrak {B}$, where $R^{(1)}$ is given by

\[ R^{(1)}(z) = \sum_{k=0}^{m}\frac{1}{2\pi i} \int_{\partial \mathcal{D}_{t_{k}}} \frac{J_{R}^{(1)}(s)}{s-z}{\rm d}s. \]

Since the jumps $J_{R}$ are analytic in a neighbourhood of $\Gamma _{R}$, expansion (3.35) holds uniformly for $z \in \mathbb {C}\setminus \Gamma _{R}$. It also follows from (3.34) that (3.35) can be differentiated with respect to $z$ without increasing the error term. For $z \in \mathbb {C}\setminus \cup _{k=0}^{m} \mathcal {D}_{t_{k}}$, a residue calculation using (3.22), (3.27) and (3.30) shows that (recall that $\partial \mathcal {D}_{t_{k}}$ is oriented in the clockwise direction)

(3.36)\begin{equation} R^{(1)}(z) = \sum_{k=0}^{m} \frac{1}{z-t_{k}} \frac{(\beta_{k}^{2}-\frac{\alpha_{k}^{2}}{4})t_{k}}{2\pi \psi(t_{k})} \begin{pmatrix} 1 & \Lambda_{k}^{{-}2} \tau(\alpha_{k},-\beta_{k}) - \Lambda_{k}^{2} \tau(\alpha_{k},\beta_{k}) & -1 \end{pmatrix}. \end{equation}

Remark 3.3 If $k_{0},\ldots,k_{2m+1}\in \mathbb {N}$, $k_{0}+\ldots +k_{2m+1}\geq 1$ and $\partial ^{\vec {k}}:=\partial _{\alpha _{0}}^{k_{0}}\ldots \partial _{\alpha _{m}}^{k_{m}}\partial _{\beta _{0}}^{k_{m+1}}\ldots \partial _{\beta _{m}}^{k_{2m+1}}$, then by (3.17) we have

\[ \partial^{\vec{k}}J_{R}(z) = {\mathcal{O}}(e^{{-}cn}), \quad \mbox{uniformly for } z \in (\gamma_+{\cup} \gamma_{-}) \setminus \cup_{k=0}^{m} \overline{\mathcal{D}_{t_{k}}}, \]

and by the same type of arguments that led to (3.30) we have

\begin{align*} \partial^{\vec{k}}J_{R}(z) = \partial^{\vec{k}}(J_{R}^{(1)}(z))n^{{-}1} + {\mathcal{O}}\left(\frac{(\log n)^{k_{m+1}+\ldots+k_{2m+1}}}{n^{2-2\beta_{\max}}}\right),\\ \mbox{uniformly for } z \in \cup_{k=0}^{m} \partial \mathcal{D}_{t_{k}}. \end{align*}

It follows that

\[ \partial^{\vec{k}}R(z) = \partial^{\vec{k}}(R^{(1)}(z))n^{{-}1} + {\mathcal{O}}\left(\frac{(\log n)^{k_{m+1}+\ldots+k_{2m+1}}}{n^{2-2\beta_{\max}}}\right), \qquad \mbox{as } n \to + \infty. \]

If $W$ is replaced by $tW$, $t \in [0,1]$, then the asymptotics (3.32), (3.33) and (3.35) are uniform with respect to $t$ and can also be differentiated any number of times with respect to $t$ without worsening the error term.