# Asymptotics of bordered Toeplitz determinants and Next-to-Diagonal Ising
Correlations

###### Abstract.

We prove the analogue of the strong Szegő limit theorem for a large class of
bordered Toeplitz determinants. In particular, by applying our results to the
formula of Au-Yang and Perk [AYP87] for the next-to-diagonal correlations
$\langle\sigma_{0,0}\sigma_{N-1,N}\rangle$ in the square lattice Ising model,
we rigorously justify that the next-to-diagonal long-range order is the same
as the diagonal and horizontal ones in the low temperature regime. The
anisotropy-dependence of the subleading term in the asymptotics of the next-
to-diagonal correlations is also established. We use Riemann-Hilbert and
operator theory techniques, independently and in parallel, to prove these
results.

Estelle Basor***American Institute of Mathematics, San Jose, CA 95112, USA,
E-mail<EMAIL_ADDRESS>Torsten Ehrhardt†††Mathematics Department,
University of California, Santa Cruz, CA 95064, USA, E-mail:
<EMAIL_ADDRESS>Roozbeh Gharakhloo‡‡‡Department of Mathematics, Colorado
State University, Fort Collins, CO 80521, USA, E-mail:
<EMAIL_ADDRESS>
Alexander Its§§§Department of Mathematical Sciences, Indiana University-Purdue
University Indianapolis, 402 N. Blackford St., Indianapolis, IN 46202,
Blackford St., Indianapolis, IN 46202, USA. e-mail<EMAIL_ADDRESS>St.
Petersburg State University, Universitetskaya emb. 7/9, 199034, St.
Petersburg, Russia. and Yuqi Li¶¶¶East China Normal University, Shanghai,
China, E-mail<EMAIL_ADDRESS>

##  1\. Introduction

Starting from the seminal works of Szegő, Kaufman and Onsager, Toeplitz
determinants have played a very important role in many areas of analysis and
mathematical physics. Indeed, an extraordinary variety of problems in
mathematics, physics, and engineering can be expressed in terms of Toeplitz
matrices and determinants. We refer to the monograph [BS06] and the more
recent survey paper [DIK13] for the details of the theory and applications of
Toeplitz determinants.

A growing interest has recently developed in the study of certain
generalizations of Toeplitz determinants. Among those are the determinants of
Toeplitz + Hankel matrices - see [DIK11], [BE17], [GI], integrable Fredholm
determinants [IIKS90], [Dei99], $2j-k$ and $j-2k$ determinants [GW]. These
determinants appear in the study of the Ising model in the zig-zag layered
half-plane [CHM19], in the spectral analysis of the Hankel matrices, in the
theory of exactly solvable quantum models [FA06], and in asymptotic analysis
of moments of derivatives of characteristic polynomials
$\Lambda_{A}(s)=\mathrm{det}(I-As)$, where $A\in USp(2N),SO(2N),O^{-}(2N)$
[ABP+2014].

In this paper we are concerned with yet another deformation of Toeplitz
determinants - the so called ”bordered Toeplitz determinants”. The latter also
arise in applications, for example, in the the next-to-diagonal correlation
functions for the Ising model. The goal of this paper is to launch a new
research project devoted to the asymptotics of these determinants and to
discuss this application in particular.

Let $\phi$ and $\psi$ be the $L^{1}$-functions on the (positively oriented)
unit circle,

$\mathbb{T}=\\{z\in\mathbb{C}:|z|=1\\}.$

The bordered Toeplitz determinant, $D^{B}_{N}[\phi;\psi]$, is defined as

(1.1)
$D^{B}_{N}[\phi;\psi]:=\mathrm{det}\begin{pmatrix}\phi_{0}&\cdots&\phi_{N-2}&\psi_{N-1}\\\
\phi_{-1}&\cdots&\phi_{N-3}&\psi_{N-2}\\\ \vdots&\vdots&\vdots&\vdots\\\
\phi_{1-N}&\cdots&\phi_{-1}&\psi_{0}\end{pmatrix},\qquad N>1,$

where

(1.2)
$\phi_{n}=\int_{\mathbb{T}}z^{-n}\phi(z)\frac{\mathrm{d}z}{2\pi\mathrm{i}z},\qquad\psi_{n}=\int_{\mathbb{T}}z^{-n}\psi(z)\frac{\mathrm{d}z}{2\pi\mathrm{i}z},$

are respectively the $n$-th Fourier coefficients of $\phi$ and $\psi$. To fix
the notation, we let

(1.3) $D_{N}[\phi]:=\underset{0\leq j,k\leq
N-1}{\mathrm{det}}\\{\phi_{j-k}\\},$

denote the $N\times N$ (pure) Toeplitz determinant corresponding to the symbol
$\phi$. As with the Toeplitz determinants, the principal analytic question is
the asymptotic behavior of $D^{B}_{N}[\phi;\psi]$ as $N\to\infty.$

The asymptotics of the Toeplitz determinants are well known and given by the
Szegő-Widom theorem [Wid76, Sze52, BS06]

(1.4) $D_{N}[\phi]\sim G[\phi]^{N}E[\phi],\qquad N\to\infty,$

where

(1.5) $G[\phi]=\exp\left([\log\phi]_{0}\right)\quad\text{and}\quad
E[\phi]=\exp\left(\sum_{n\geq 1}n[\log\phi]_{n}[\log\phi]_{-n}\right).$

This holds if the function $\phi$ is sufficiently smooth (e.g., in Hölder
class $C^{1+\epsilon}$), does not vanish on $\mathbb{T}$, and has zero winding
number. Note that the constants involve the $n$-th Fourier coefficients
$[\log\phi]_{n}$ of the continuous logarithm of the function $\phi$.

In this paper, we will show that for the bordered determinants a similar
theorem holds,

(1.6) $D^{B}_{N}[\phi;\psi]\sim G[\phi]^{N}E[\phi]\,F[\phi;\psi],\qquad
N\to\infty,$

and where $F[\phi;\psi]$ is a constant described in Theorems 1.1 and 1.2
below.

Our general result above will become explicit in the case of the next-to-
diagonal Ising correlations. There it happens that $\psi$ is a constant times
something of the form

(1.7) $\psi(z)=\frac{\phi(z)z-d}{z-c},$

where $d$ and $c$ are complex parameters. Hence particular attention will be
paid to such functions. In Theorem 1.1 below, we present the asymptotics of
$D^{B}_{N}[\phi;\psi]$, where $\psi$ is of the more general form

(1.8) $\psi(z)=q_{1}(z)\phi(z)+q_{2}(z),$

where

(1.9)
$q_{1}(z)=a_{0}+a_{1}z+\frac{b_{0}}{z}+\sum^{m}_{j=1}\frac{b_{j}z}{z-c_{j}},\quad\text{and}\quad
q_{2}(z)=\hat{a}_{0}+\hat{a}_{1}z+\frac{\hat{b}_{0}}{z}+\sum_{j=1}^{m}\frac{\hat{b}_{j}}{z-c_{j}},$

where all parameters are complex and the $c_{j}$ are nonzero and do not lie on
the unit circle. Indeed it is straightforward to pass from the rational
functions $q_{1}$ and $q_{2}$ with one simple pole to the ones with multiple
simple poles, as one can use the following elementary properties of bordered
Toeplitz determinants:

(1.10)
$D^{B}_{N}\left[\phi;\sum^{m}_{j=1}a_{j}\psi_{j}\right]=\sum^{m}_{j=1}a_{j}D^{B}_{N}[\phi,\psi_{j}],$
(1.11) $D^{B}_{N}[\phi;\phi]=D_{N}[\phi],$ (1.12)
$D^{B}_{N}[\phi;1]=D_{N-1}[\phi].$

Throughout the the paper, we will refer to a symbol $\phi$ as a Szegő-type
symbol, if it is smooth and nonzero on the unit circle, has winding number
zero, and admits an analytic continuation in a neighborhood of the unit
circle.

######  Theorem 1.1.

Let $D^{B}_{N}[\phi;\psi]$ be the bordered Toeplitz determinant with
$\psi=q_{1}\phi+q_{2}$ given by (1.8) and (1.9), and $\phi$ of Szegő type.
Then, the following asymptotic behavior of $D^{B}_{N}[\phi;\psi]$ as
$N\to\infty$ takes place

(1.13)
$D^{B}_{N}\left[\phi;\psi\right]=G[\phi]^{N}E[\phi]\left(F[\phi;\psi]+O(e^{-\mathfrak{c}N})\right),$

where $G[\phi]$ and $E[\phi]$ are given by (1.5),

(1.14) $F[\phi;\psi]=a_{0}+b_{0}[\log\phi]_{1}+\sum^{m}_{j=1\atop
0<|c_{j}|<1}b_{j}\frac{\alpha(c_{j})}{\alpha(0)}+\frac{1}{\alpha(0)}\left(\hat{a}_{0}-\hat{a}_{1}[\log\phi]_{-1}-\sum^{m}_{j=1\atop|c_{j}|>1}\frac{\hat{b}_{j}}{c_{j}}\alpha(c_{j})\right),$
(1.15) $\alpha(z):=\exp\left[\frac{1}{2\pi
i}\int_{\mathbb{T}}\frac{\ln(\phi(\tau))}{\tau-z}d\tau\right],$

and $\mathfrak{c}$ is some positive constant.

It is occasionally convenient to use different notation related to the
function $\alpha(z)$,

(1.16) $\alpha(z)=\begin{cases}\phi_{+}(z),&|z|<1,\\\
\phi_{-}^{-1}(z),&|z|>1,\end{cases}$

with

(1.17)
$\phi_{+}(z):=\exp\left(\sum_{n=0}^{\infty}[\log\phi]_{n}z^{n}\right),\qquad\phi_{-}(z):=\exp\left(\sum_{n=1}^{\infty}[\log\phi]_{-n}z^{-n}\right).$

In fact, these functions are the factors of a canonical Wiener-Hopf
factorization of the symbol $\phi$, $\phi(z)=\phi_{-}(z)\phi_{+}(z)$, $|z|=1$.
Factors in a Wiener-Hopf factorization are unique up to a multiplicative
constant. With the factors as given above, we have the normalization,

(1.18)
$\phi_{+}(0)=\alpha(0)=G[\phi],\qquad\phi_{-}(\infty)=1=\alpha(\infty).$

More generally we can find the constant $F[\phi;\psi]$ in (1.6) as described
in the following theorem, which is proven using operator theory and Riemann-
Hilbert methods respectively in Sections 3.1 and 5.2.

######  Theorem 1.2.

Let $\psi(z)$ be a function which admits an analytic continuation in a
neighborhood of the unit circle, and let $\phi$ be of Szegő type. Denote by
$\phi_{\pm}(z)$ the factors of a canonical Wiener-Hopf factorization of the
symbol $\phi(z)$, i.e., $\phi=\phi_{-}\phi_{+}$. Then

(1.19)
$D^{B}_{N}\left[\phi;\psi\right]=G[\phi]^{N}E[\phi]\left(F[\phi;\psi]+O(e^{-\mathfrak{c}N})\right),$

where $G[\phi]$ and $E[\phi]$ are given by (1.5),

(1.20) $F[\phi;\psi]=\frac{[\phi_{-}^{-1}\psi]_{0}}{[\phi_{+}]_{0}},$

and $\mathfrak{c}$ is some positive constant.

######  Remark 1.3.

As it is shown in Section 3, with the change to $o(1)$ in the error term, the
asymptotics (1.13) and (1.19) are valid for all $\psi\in L^{2}(\mathbb{T})$
and $\phi$ satisfying the assumptions of the strong Szegő theorem, i.e.,
$\phi(z)$ belongs to a Hölder class $C^{1+\epsilon}$, is nonzero on the unit
circle, and has zero winding number.

In this paper, we also apply our general results mentioned above to the
problem of rigorous evaluation of the next-to-diagonal two-point correlation
function in the Ising model. To that end, let us first recall more precisely
the situation in the two-dimensional Ising model, solved by Onsager (see,
e.g., [MW73]). In this model a $2\mathcal{M}\times 2\mathcal{N}$ rectangular
lattice is considered with an associated spin variable $\sigma_{jk}$ taking
the values $1$ and $-1$ at each vertex $(j,k)$, $-{\mathcal{M}}\leq
j\leq{\mathcal{M}}-1$, $-{\mathcal{N}}\leq k\leq{\mathcal{N}}-1$. There are
$2^{4{\mathcal{M}}{\mathcal{N}}}$ possible spin configurations $\\{\sigma\\}$
of the lattice (a configuration corresponds to values of all $\sigma_{jk}$
fixed). By $J_{h}$ and $J_{v}$ we respectively denote the horizontal and
vertical nearest neighbor coupling constants and with each configuration we
associate its nearest-neighbor coupling energy given by

(1.21)
$E(\\{\sigma\\})=-\sum_{j=-{\mathcal{M}}}^{{\mathcal{M}}-1}\sum_{k=-{\mathcal{N}}}^{{\mathcal{N}}-1}\left(J_{h}\sigma_{jk}\sigma_{j\,k+1}+J_{v}\sigma_{jk}\sigma_{j+1\,k}\right),\qquad
J_{h},J_{v}>0.$

The partition function at a temperature $T>0$ is equal to

(1.22) $Z(T)=\sum_{\\{\sigma\\}}e^{-E(\\{\sigma\\})/k_{B}T},$

where the sum is over all configurations and $k_{B}$ is the Boltzmann
constant. A remarkable feature of this model is the presence of a
thermodynamic phase transition (in the limit of the infinite lattice,
$\mathcal{M},\mathcal{N}\to\infty$) at a certain temperature $T_{c}$ whose
dependence on $J_{h}$, $J_{v}$ is described by the equation,

(1.23)
$\sinh\left(\frac{2J_{h}}{k_{B}T_{c}}\right)\sinh\left(\frac{2J_{v}}{k_{B}T_{c}}\right)=1.$

Define a 2-spin correlation function by the expression

(1.24)
$\langle\sigma_{0,0}\sigma_{N,M}\rangle=\lim_{{\mathcal{M}},{\mathcal{N}}\to\infty}{\frac{1}{Z(T)}}\sum_{\\{\sigma\\}}\sigma_{0,0}\sigma_{N,M}e^{-E(\\{\sigma\\})/k_{B}T}.$

Let us introduce the notations,

(1.25) $\begin{split}\begin{aligned}
S_{h}&=\sinh\left(\frac{2J_{h}}{k_{B}T}\right),&\;\;S_{v}&=\sinh\left(\frac{2J_{v}}{k_{B}T}\right)\,\,,\\\\[4.30554pt]
C_{h}&=\cosh\left(\frac{2J_{h}}{k_{B}T}\right),&C_{v}&=\cosh\left(\frac{2J_{v}}{k_{B}T}\right)\,\,,\end{aligned}\end{split}$

and

(1.26) $k=S_{h}S_{v}.$

In this paper we shall focus on

(1.27) $k>1,$

which, in view of equation (1.23), corresponds to the low temperature regime
$T<T_{c}$. It is known (see, e.g., [MW73, Chap. VIII]) that the diagonal
correlations $\langle\sigma_{0,0}\sigma_{N,N}\rangle$ and the horizontal
correlations $\langle\sigma_{0,0}\sigma_{0,N}\rangle$ have Toeplitz
determinant representations. Indeed, we have

(1.28) $\displaystyle\langle\sigma_{0,0}\sigma_{N,N}\rangle$
$\displaystyle=D_{N}[\widehat{\phi}\
],\qquad\widehat{\phi}(z)=\sqrt{\frac{1-k^{-1}z^{-1}}{1-k^{-1}z}},$ (1.29)
$\displaystyle\langle\sigma_{0,0}\sigma_{0,N}\rangle$
$\displaystyle=D_{N}[\widehat{\eta}\
],\qquad\,\widehat{\eta}(z)=\sqrt{\frac{(1-\alpha_{1}z)(1-\alpha_{2}z^{-1})}{(1-\alpha_{1}z^{-1})(1-\alpha_{2}z)}},$

where the constants $\alpha_{1}$ and $\alpha_{2}$ are given by

$\alpha_{1}=\frac{z_{h}(1-z_{v})}{1+z_{v}},\quad\alpha_{2}=\frac{1-z_{v}}{z_{h}(1+z_{v})},\qquad
z_{h,v}=\tanh{\left(\frac{J_{h,v}}{k_{B}T}\right)}.$

In the low temperature regime, the symbols $\widehat{\phi}$ and
$\widehat{\eta}$ enjoy the regularity properties required by the strong Szegő
limit theorem and the diagonal and horizontal long-range orders

$M_{D}:=\sqrt{\lim_{N\to\infty}\langle\sigma_{0,0}\sigma_{N,N}\rangle}\quad\text{and}\quad
M_{H}:=\sqrt{\lim_{N\to\infty}\langle\sigma_{0,0}\sigma_{0,N}\rangle},$

both evaluate to $(1-k^{-2})^{1/8}$ (see [MW73, Chap. XI]).

In an interesting development, it was shown by Au-Yang and Perk in [AYP87],
that the next-to-diagonal two point correlation function is given by the
following bordered Toeplitz determinant,

(1.30)
$\langle\sigma_{0,0}\sigma_{N-1,N}\rangle=D^{B}_{N}[\widehat{\phi};\widehat{\psi}],$

where $\widehat{\phi}$ is given in (1.28), and

(1.31)
$\widehat{\psi}(z)=\frac{C_{v}z\widehat{\phi}(z)+C_{h}}{S_{v}(z-c_{*})},\qquad\mbox{with}\qquad
c_{*}=-\frac{S_{h}}{S_{v}}.$

This is straightforward to derive these formulae from the original expressions
in [AYP87], and we have provided it as an appendix in Section 5.3. We would
like to emphasize that in the low-temperature regime ($k>1$) and in the
anisotropic case ($J_{h}\neq J_{v}$), the symbols $\widehat{\phi}$ and
$\widehat{\psi}$ satisfy the corresponding assumptions of Theorem 1.1, in
particular, $\widehat{\phi}$ is of Szegő type. The function
$\widehat{\psi}(z)$ actually does not have a pole at $z=c_{*}$, and therefore
it is analytic on a neighborhood of the unit circle, even in the isotropic
case when $c_{*}=-1$.

Our results being applied to the next-to-diagonal theory for the Ising model
show the following large $N$ behavior of the corresponding correlation
function in the low temperature regime ($k>1$), which is valid in both the
isotropic and anisotropic cases.

######  Theorem 1.4.

Let $\langle\sigma_{0,0}\sigma_{N-1,N}\rangle$ be the next-to-diagonal two
point correlation function in the square lattice Ising model. Then, in the
low-temperature regime, the long-range order in the next-to-diagonal direction
is the same as of the diagonal and horizontal ones, i.e.,

(1.32)
$\lim_{N\to\infty}\langle\sigma_{0,0}\sigma_{N-1,N}\rangle=(1-k^{-2})^{1/4}.$

It is worth noticing that, although the bordered Toeplitz determinant which
defines the correlation function $\langle\sigma_{0,0}\sigma_{N-1,N}\rangle$
depends on the relation between $J_{h}$ and $J_{v}$, its leading order
asymptotics does not. However, the sensitivity to the horizontal and vertical
parameters is reflected in the second-order term of the asymptotic expansion
as our next theorem illustrates.

######  Theorem 1.5.

The next-to-diagonal two point correlation function has, in the low-
temperature regime $k>1$, the $N\to\infty$ asymptotics

(1.33) $\displaystyle\langle\sigma_{0,0}\sigma_{N-1,N}\rangle$
$\displaystyle=(1-k^{-2})^{1/4}\left(1+\frac{1}{2\pi(1-k^{-2})}\Big{(}\frac{1}{C_{v}^{2}}+\frac{1}{k^{2}-1}\Big{)}N^{-2}k^{-2N}\Big{(}1+O(N^{-1})\Big{)}\right).$

For comparison, asymptotics of the diagonal correlation function is given by

(1.34) $\displaystyle\langle\sigma_{0,0}\sigma_{N,N}\rangle$
$\displaystyle=(1-k^{-2})^{1/4}\left(1+\frac{1}{2\pi(1-k^{-2})^{2}k^{2}}N^{-2}k^{-2N}\Big{(}1+O(N^{-1})\Big{)}\right),\qquad
N\to\infty$

(see formula (3.27) in Chap. XI of [MW73]). As part of our computation leading
to (1.33), we reconfirm (1.34) as well.

The critical temperature $(k=1)$ and the high temperature regime $(k<1)$
correspond to the appearance of the Fisher-Hartwig type singularities in the
symbol (1.28) and will be considered in a future publication.

It also should be mentioned that Theorems 1.4 and 1.5 confirm the long-range
behavior of the next-to-diagonal correlation functions of the Ising model that
have already been known in the physical literature [CW]666 In fact, in [CW],
the long-range asymptotics is obtained for the general correlation function
$\langle\sigma_{0,0}\sigma_{M,N}\rangle$..

Finally we remark that the constant $F[\phi;\psi]$ can actually vanish for
certain $\psi$. This happens, for example, if $\psi=\phi\frac{z}{z-c}$ with
$|c|>1$ as can be seen from (1.14)). In this case the second-order term in the
asymptotics becomes important.

In this paper we shall present two different approaches to the general problem
of bordered determinants. One is based on the relatively new Riemann-Hilbert
method of the asymptotic analysis of Toeplitz and Hankel determinants (see
[BDJ99], [FIK92], [DIK11]). The Riemann-Hilbert approach has been inspired by
the work [Wit07] where the connection of bordered Toeplitz determinants of the
type $D^{B}_{N}[\phi;q\phi]$ to the system of biorthogonal polynomials on the
unit circle was found for the first time. Another approach is based on the
operator theoretic techniques, and it has been used in the theory of Toeplitz
and Hankel determinants since the classical works of Szegő and Widom (see
[Wid76], [Sze52], [BS06], [BS99], [BE17], [BE01]). For the last 25 years these
two techniques has been very closely interacting and greatly enhancing each
other. In particular, the asymptotic analysis of the bordered Toeplitz
determinants whose results are presented in this work has been carried out
within constant interaction and information exchanges between the first two
and the last three co-authors. Hence we decided that it would be very proper
to present both the operator and Riemann-Hilbert methods of the solution in
one paper.

###  1.1. Outline.

The paper is organized as follows. In Section 2 we shall present the Riemann-
Hilbert representation of the bordered Toeplitz determinant corresponding to a
symbol pair $(\phi,\psi)$, $\psi$ given by (1.8) and (1.9). In this section we
shall basically follow [Wit07] where the connection with the corresponding
system of bi-orthogonal polynomials on the unit circle was first obtained. We
will then prove Theorems 1.1, 1.4, and 1.5 based on the Riemann-Hilbert
formulation. Theorems 1.1, 1.2, 1.4 and 1.5 will be proven using operator
theory techniques in Section 3. In Section 4 a numerical verification for the
asymptotics of the correlation function in Theorem 1.5 as well as for the
asymptotics of $D_{N}^{B}[\phi;\psi]$ in the case $\psi=\phi\frac{z}{z-c}$ is
done. Finally Section 5 contains four appendices respectively on the solution
of the associated Riemann-Hilbert problem, proof of Theorem 1.2 using the
Riemann-Hilbert approach, derivation of the Ising symbol pair
$(\widehat{\phi},\widehat{\psi})$, and some other auxiliary results.

##  2\. Bordered Toeplitz determinants and the Riemann-Hilbert problem for bi-
orthogonal polynomials on the unit circle

As mentioned in the outline, the goal of this section is to prove Theorem 1.1.
In order to achieve that, we will first establish the relationship between the
bordered Toeplitz determinant $D^{B}_{N}[\phi;\psi]$, $\psi$ given by (1.8),
and the solution of the Riemann-Hilbert problem for the system of bi-
orthogonal polynomials on the unit circle (BOPUC). Let $Q_{n}$ and
$\widehat{Q}_{n}$ be respectively defined by

(2.1)
$Q_{n}(z):=\frac{1}{\sqrt{D_{n}[\phi]D_{n+1}[\phi]}}\mathrm{det}\begin{pmatrix}\phi_{0}&\phi_{-1}&\cdots&\phi_{-n}\\\
\phi_{1}&\phi_{0}&\cdots&\phi_{-n+1}\\\ \vdots&\vdots&\ddots&\vdots\\\
\phi_{n-1}&\phi_{n-2}&\cdots&\phi_{-1}\\\ 1&z&\cdots&z^{n}\end{pmatrix},$

and

(2.2)
$\widehat{Q}_{n}(z):=\frac{1}{\sqrt{D_{n}[\phi]D_{n+1}[\phi]}}\mathrm{det}\begin{pmatrix}\phi_{0}&\phi_{-1}&\cdots&\phi_{-n+1}&1\\\
\phi_{1}&\phi_{0}&\cdots&\phi_{-n+2}&z\\\ \vdots&\vdots&\ddots&\vdots\\\
\phi_{n}&\phi_{n-1}&\cdots&\phi_{1}&z^{n}\end{pmatrix},$

where $\phi_{j}$, $j\in\mathbb{Z}$, are defined by (1.2) and $D_{n}[\phi]$ is
given by (1.3). Note that, from (2.1) and (2.2), we have

(2.3)
$Q_{n}(z)=\kappa_{n}z^{n}+\sum^{n-1}_{\ell=0}c_{\ell}z^{\ell},\qquad\mbox{and}\qquad\widehat{Q}_{n}(z)=\kappa_{n}z^{n}+\sum^{n-1}_{\ell=0}\widehat{c}_{\ell}z^{\ell},$

where

(2.4) $\kappa_{n}=\sqrt{\frac{D_{n}[\phi]}{D_{n+1}[\phi]}}.$

One can readily observe that $\\{Q_{n}\\}^{\infty}_{n=0}$ and
$\\{\widehat{Q}_{n}\\}^{\infty}_{n=0}$ form the bi-orthogonal system of
polynomials on the unit circle with respect to the weight $\phi$ :

(2.5)
$\int_{\mathbb{T}}Q_{n}(z)\widehat{Q}_{n}(z^{-1})\phi(z)\frac{\mathrm{d}z}{2\pi\mathrm{i}z}=\delta_{nk},\qquad
n,k=0,1,2,\cdots.$

It is due to J.Baik, P.Deift and K.Johansson ([BDJ99]) that the following
matrix-valued function constructed out of the polynomials $Q_{n}$ and
$\widehat{Q}_{n}$

(2.6)
$X(z;n):=\begin{pmatrix}\kappa_{n}^{-1}Q_{n}(z)&\displaystyle\kappa^{-1}_{n}\int_{\mathbb{T}}\frac{Q_{n}(\zeta)}{(\zeta-z)}\frac{\phi(\zeta)\mathrm{d}\zeta}{2\pi\mathrm{i}\zeta^{n}}\\\
-\kappa_{n-1}z^{n-1}\widehat{Q}_{n-1}(z^{-1})&\displaystyle-\kappa_{n-1}\int_{\mathbb{T}}\frac{\widehat{Q}_{n-1}(\zeta^{-1})}{(\zeta-z)}\frac{\phi(\zeta)\mathrm{d}\zeta}{2\pi\mathrm{i}\zeta}\end{pmatrix},$

satisfies the following Riemann-Hilbert problem for BOPUC, which in the
subsequent parts of this text will occasionally be referred to as the $X$-RHP:

  * •

RH-X1 $X:\mathbb{C}\setminus\mathbb{T}\to\mathbb{C}^{2\times 2}$ is analytic,

  * •

RH-X2 The limits of $X(\zeta)$ as $\zeta$ tends to $z\in\mathbb{T}$ from the
inside and outside of the unit circle exist, and are denoted $X_{\pm}(z)$
respectively and are related by

(2.7) $X_{+}(z)=X_{-}(z)\begin{pmatrix}1&z^{-n}\phi(z)\\\
0&1\end{pmatrix},\qquad z\in\mathbb{T},$

  * •

RH-X3 As $z\to\infty$

(2.8) $X(z)=\big{(}I+O(z^{-1})\big{)}z^{n\sigma_{3}}.$

For convenience of the reader, in the Appendix 5.1 we have provided the
solution of the $X$-RHP when $\phi$ is of Szegő type.

In the following subsections we will analyze bordered Toeplitz determinants of
the following three types

  * •

$D^{B}_{N}[\phi;z^{k}]$,

  * •

$D^{B}_{N}[\phi;q],$

  * •

$D^{B}_{N}[\phi;q\phi]$,

where $q$ is a rational function with simple poles. The lemmas in the
following subsections, whose proofs are inspired by calculations in [Wit07],
show that the bordered Toeplitz determinants of the above types are encoded
into the solution of the $X$-RHP. In fact, we will show that the bordered
Toeplitz determinants of the first two types are related to the $X_{11}$ and
the bordered Toeplitz determinants of the third type are related to the
$X_{12}$, respectively, the $11$ and $12$ entries of the solution of the
$X$-RHP. Later we will show how these cases are relevant to the next-to-
diagonal correlations in the 2D-Ising model.

###  2.1. Bordered Toeplitz determinants of the type
$D^{B}_{N}[\phi;z^{k}],k\in\mathbb{Z}$

Let us start this subsection with the following elementary lemma.

######  Lemma 2.1.

The following identity holds for the Bordered Toeplitz determinants

(2.9) $D^{B}_{n+1}[\phi;z^{k}]=0,\qquad
k\in\mathbb{Z}\setminus\\{0,1,\cdots,n\\}.$

###### Proof.

It suffices to note that all Fourier coefficients $(z^{k})_{j}=0$ for $0\leq
j\leq n$, $k\in\mathbb{Z}\setminus\\{0,1,\cdots n\\}$. ∎

Note that for $k=0$, we obviously have (1.12). Now, we turn our attention to

(2.10) $D^{B}_{n+1}[\phi;z^{k}],\qquad k\in\\{1,\cdots,n\\}.$

From

(2.11)
$D^{B}_{n+1}[\phi;z^{k}]=\mathrm{det}\begin{pmatrix}\phi_{0}&\phi_{-1}&\cdots&\phi_{k-n+1}&\phi_{k-n}&\phi_{k-n-1}&\cdots&\phi_{-n}\\\
\phi_{1}&\phi_{0}&\cdots&\phi_{k-n+2}&\phi_{k-n+1}&\phi_{k-n}&\cdots&\phi_{-n+1}\\\
\vdots&\vdots&\cdots&\vdots\\\
\phi_{n-1}&\phi_{n-2}&\cdots&\phi_{k}&\phi_{k-1}&\phi_{k-2}&\cdots&\phi_{-1}\\\
0&0&\cdots&0&1&0&\cdots&0\end{pmatrix}.$

we observe that (by (2.1) and (2.4)) the determinant on the right hand side of
(2.11) is exactly the coefficient of $z^{n-k}$ in the polynomial

(2.12) $\kappa^{-1}_{n}D_{n}[\phi]Q_{n}(z).$

Let

(2.13) $Q_{n}(z)\equiv\sum^{n}_{j=0}\kappa^{(n)}_{j}z^{j}.$

where for brevity of notation, throughout this paper we use

(2.14) $\kappa_{n}\equiv\kappa^{(n)}_{n}.$

Therefore

(2.15)
$D^{B}_{n+1}[\phi;z^{k}]=D_{n}[\phi]\frac{\kappa^{(n)}_{n-k}}{\kappa_{n}}.$

We are now in a position to express $D^{B}_{n+1}[\phi;z^{k}]$, $1\leq k\leq
n$, in terms of $X$-RHP data in a recursive way as follows:

(2.16)
$D^{B}_{n+1}[\phi;z]=D_{n}[\phi]\lim_{z\to\infty}\left(\frac{X_{11}(z;n)-z^{n}}{z^{n-1}}\right)\equiv
D_{n}[\phi]\frac{\kappa^{(n)}_{n-1}}{\kappa_{n}},$ (2.17)
$D^{B}_{n+1}[\phi;z^{2}]=D_{n}[\phi]\lim_{z\to\infty}\left(\frac{X_{11}(z;n)-z^{n}-\frac{\kappa^{(n)}_{n-1}}{\kappa_{n}}z^{n-1}}{z^{n-2}}\right)\equiv
D_{n}[\phi]\frac{\kappa^{(n)}_{n-2}}{\kappa_{n}},$

and so on. These formulae are recursive, in the sense that the second and
third members of the equality (2.16) can be regarded as the definition of
$k^{(n)}_{n-1}$ in terms of the $X$-RHP, which one needs in (2.17).

Here, in particular we present how the asymptotics of $D^{B}_{n+1}[\phi;z]$
can be obtained from the Riemann-Hilbert data. In lemma (2.7) we will show
that this is actually related to $D_{n}[\phi;\frac{\phi}{z}]$.

######  Lemma 2.2.

Let $\phi$ be of Szegő type. Then, as $n\to\infty$ we have

(2.18)
$D^{B}_{n+1}[\phi;z]=D_{n}[\phi]\left(-\frac{1}{2\pi\mathrm{i}}\int_{\mathbb{T}}\ln(\phi(\tau))\mathrm{d}\tau+O(e^{-\mathfrak{c}n})\right),$

for some positive constant $\mathfrak{c}$.

###### Proof.

Expanding $\alpha(z)$, given by (5.9), as $z\to\infty$ we get

(2.19)
$\alpha(z)=1-\frac{a_{0}}{2\pi\mathrm{i}z}-\left(\frac{a_{1}}{2\pi\mathrm{i}}+\frac{a^{2}_{0}}{8\pi^{2}}\right)\frac{1}{z^{2}}+\cdots,\qquad
z\to\infty,$

where

(2.20)
$a_{k}:=\int_{\mathbb{T}}\tau^{k}\ln\left(\phi(\tau)\right)\mathrm{d}\tau.$

Also from (5.19), and (5.20) we have

(2.21)
$X_{11}(z;n)=\alpha(z)z^{n}\left(1+\frac{O(e^{-2\mathfrak{c}n})}{1+|z|}\right),\qquad
z\in\Omega_{\infty},\qquad n\to\infty.$

Combining (2.16), (2.19) and (2.21) gives (2.18). ∎

In a similar fashion, and with increasing effort, one can obtain similar
formulae for $D^{B}_{n+1}[\phi;z^{k}]$, $k>1$.

###  2.2. Bordered Toeplitz determinants of the type $D^{B}_{N}[\phi;q]$

Let us define

(2.22) $q_{0}(z):=\frac{1}{z-c}.$

The Fourier coefficients of $q_{0}$ are given by

(2.23) $q_{0,j}=\begin{cases}0,&|c|<1,\\\ -(c)^{-j-1},&|c|>1,\end{cases}\qquad
0\leq j\leq n.$

The following lemma establishes how $D^{B}_{N}[\phi;q_{0}]$ is encoded into
$X$-RHP data.

######  Lemma 2.3.

The bordered Toeplitz determinant
$D^{B}_{n+1}[\phi,\displaystyle\frac{1}{z-c}]$, is encoded into $X$-RHP data
described by

(2.24) $D^{B}_{n+1}[\phi;\frac{1}{z-c}]=\begin{cases}0,&|c|<1,\\\
-c^{-n-1}D_{n}[\phi]X_{11}(c;n),&|c|>1,\end{cases}$

where $D_{n}[\phi]$ is given by (1.3) and $X_{11}$ is the $11$ entry of the
solution to RH-X1 through RH-X3.

###### Proof.

The case of $|c|<1$ is obvious due to (2.23). Consider $|c|>1$. Recalling that
$X_{11}(z;n)=\kappa^{-1}_{n}Q_{n}(z)$, from (2.1) and (2.4) we have

(2.25)
$X_{11}(z;n)=\frac{1}{D_{n}[\phi]}\mathrm{det}\begin{pmatrix}\phi_{0}&\phi_{-1}&\cdots&\phi_{-n}\\\
\phi_{1}&\phi_{0}&\cdots&\phi_{-n+1}\\\ \vdots&\vdots&\cdots&\vdots\\\
\phi_{n-1}&\phi_{n-2}&\cdots&\phi_{-1}\\\ 1&z&\cdots&z^{n}\end{pmatrix}.$

Therefore from (2.23)

(2.26)
$\begin{split}-c^{-n-1}D_{n}[\phi]X_{11}(c;n)&=\mathrm{det}\begin{pmatrix}\phi_{0}&\phi_{-1}&\cdots&\phi_{-n}\\\
\phi_{1}&\phi_{0}&\cdots&\phi_{-n+1}\\\ \vdots&\vdots&\cdots&\vdots\\\
\phi_{n-1}&\phi_{n-2}&\cdots&\phi_{-1}\\\
-c^{-n-1}&-c^{-n}&\cdots&-c\end{pmatrix}\\\
&=\mathrm{det}\begin{pmatrix}\phi_{0}&\phi_{-1}&\cdots&\phi_{-n}\\\
\phi_{1}&\phi_{0}&\cdots&\phi_{-n+1}\\\ \vdots&\vdots&\cdots&\vdots\\\
\phi_{n-1}&\phi_{n-2}&\cdots&\phi_{-1}\\\
q_{0,n}&q_{0,n-1}&\cdots&q_{0,0}\end{pmatrix}\equiv
D^{B}_{n+1}[\phi,q_{0}].\end{split}$

∎

######  Corollary 2.4.

We have

(2.27) $D^{B}_{n+1}\left[\phi;\displaystyle
a+\frac{b_{0}}{z}+\sum^{m}_{j=1}\frac{b_{j}}{z-c_{j}}\right]=D_{n}[\phi]\left(a-\sum^{m}_{j=1\atop|c_{j}|>1}b_{j}c^{-n-1}_{j}X_{11}(c_{j};n)\right),$

and for a Szegő type $\phi$

(2.28) $D^{B}_{n+1}\left[\phi;\displaystyle
a+\frac{b_{0}}{z}+\sum^{m}_{j=1}\frac{b_{j}}{z-c_{j}}\right]=G[\phi]^{n}E[\phi]\left(a-\sum^{m}_{j=1\atop|c_{j}|>1}\frac{b_{j}}{c_{j}}\alpha(c_{j})\right)\left(1+O(e^{-\mathfrak{c}n})\right),$

as $n\to\infty$, where

(2.29) $\alpha(z):=\exp\left[\frac{1}{2\pi
i}\int_{\mathbb{T}}\frac{\ln(\phi(\tau))}{\tau-z}d\tau\right],$

$G[\phi]$ and $E[\phi]$ are given by (1.5) and $\mathfrak{c}$ is some positive
constant.

###### Proof.

Note that (2.27) immediately follows from (2.24), (1.12) and (1.10); and then
we get (2.28) as a direct consequence of (1.4), (1.5), (5.19) and (5.20). ∎

###  2.3. Bordered Toeplitz determinants of the type $D^{B}_{N}[\phi;q\phi]$

Now we turn our attention to the bordered Toeplitz determinants where the
border symbol is given by $q\phi$, $q$ being a rational function with simple
poles. Let us start with proving a fundamental identity relating one such
bordered Toeplitz determinant to the pure Toeplitz Riemann-Hilbert data.

######  Lemma 2.5.

Let $\psi_{0}:=q_{0}\phi$, where $q_{0}$ is defined in (2.22), with $c\neq 0$.
Then the bordered Toeplitz determinant $D^{B}_{n}[\phi;\psi_{0}]$ can be
written in terms of the following data from the solution of the X-RHP:

(2.30)
$D^{B}_{n+1}[\phi;\psi_{0}]=-\frac{1}{c}D_{n+1}[\phi]+\frac{1}{c}D_{n}[\phi]X_{12}(c,n),$

where $D_{n}[\phi]$ is given by (1.3) and $X_{12}$ is the $12$ entry of the
solution to RH-X1 through RH-X3.

###### Proof.

Note that

$\psi_{0,j}=\int_{\mathbb{T}}z^{-j}\psi_{0}(z)\frac{\mathrm{d}z}{2\pi\mathrm{i}z}=\int_{\mathbb{T}}z^{-j}\frac{1}{z(z-c)}\phi(z)\frac{\mathrm{d}z}{2\pi\mathrm{i}},$

thus

(2.31)
$\psi_{0,j}=-\frac{1}{c}\int_{\mathbb{T}}z^{-j}\phi(z)\frac{\mathrm{d}z}{2\pi\mathrm{i}z}+\frac{1}{c}\int_{\mathbb{T}}\frac{z^{-j}\phi(z)}{(z-c)}\frac{\mathrm{d}z}{2\pi\mathrm{i}}=-\frac{1}{c}\phi_{j}+\frac{1}{c}\int_{\mathbb{T}}\frac{z^{-j}\phi(z)}{(z-c)}\frac{\mathrm{d}z}{2\pi\mathrm{i}}.$

Now, observe that

$D^{B}_{n+1}[\phi,\psi_{0}]=\mathrm{det}\begin{pmatrix}\phi_{0}&\phi_{-1}&\cdots&\phi_{-n}\\\
\phi_{1}&\phi_{0}&\cdots&\phi_{-n+1}\\\ \vdots&\vdots&\cdots&\vdots\\\
\phi_{n-1}&\phi_{n-2}&\cdots&\phi_{-1}\\\
\psi_{0,n}&\psi_{0,n-1}&\cdots&\psi_{0,0}\end{pmatrix}$
$=\frac{1}{c}\mathrm{det}\begin{pmatrix}\phi_{0}&\phi_{-1}&\cdots&\phi_{-n}\\\
\phi_{1}&\phi_{0}&\cdots&\phi_{-n+1}\\\ \vdots&\vdots&\cdots&\vdots\\\
\phi_{n-1}&\phi_{n-2}&\cdots&\phi_{-1}\\\
-\phi_{n}+\int_{\mathbb{T}}\frac{z^{-n}\phi(z)}{(z-c)}\frac{\mathrm{d}z}{2\pi\mathrm{i}}&-\phi_{n-1}+\int_{\mathbb{T}}\frac{z^{-n+1}\phi(z)}{(z-c)}\frac{\mathrm{d}z}{2\pi\mathrm{i}}&\cdots&-\phi_{0}+\int_{\mathbb{T}}\frac{\phi(z)}{(z-c)}\frac{\mathrm{d}z}{2\pi\mathrm{i}}\end{pmatrix}=$
(2.32)
$-\frac{1}{c}\mathrm{det}\begin{pmatrix}\phi_{0}&\phi_{-1}&\cdots&\phi_{-n}\\\
\phi_{1}&\phi_{0}&\cdots&\phi_{-n+1}\\\ \vdots&\vdots&\cdots&\vdots\\\
\phi_{n-1}&\phi_{n-2}&\cdots&\phi_{-1}\\\
\phi_{n}&\phi_{n-1}&\cdots&\phi_{0}\end{pmatrix}+\frac{1}{c}\mathrm{det}\begin{pmatrix}\phi_{0}&\phi_{-1}&\cdots&\phi_{-n}\\\
\phi_{1}&\phi_{0}&\cdots&\phi_{-n+1}\\\ \vdots&\vdots&\cdots&\vdots\\\
\phi_{n-1}&\phi_{n-2}&\cdots&\phi_{-1}\\\
\int_{\mathbb{T}}\frac{z^{-n}\phi(z)}{(z-c)}\frac{\mathrm{d}z}{2\pi\mathrm{i}}&\int_{\mathbb{T}}\frac{z^{-n+1}\phi(z)}{(z-c)}\frac{\mathrm{d}z}{2\pi\mathrm{i}}&\cdots&\int_{\mathbb{T}}\frac{\phi(z)}{(z-c)}\frac{\mathrm{d}z}{2\pi\mathrm{i}}\end{pmatrix}.$

Now note that, using (2.1) and (2.4) we have

(2.33)
$\kappa_{n}^{-1}\zeta^{-n}Q_{n}(\zeta)=\frac{1}{D_{n}[\phi]}\mathrm{det}\begin{pmatrix}\phi_{0}&\phi_{-1}&\cdots&\phi_{-n}\\\
\phi_{1}&\phi_{0}&\cdots&\phi_{-n+1}\\\ \vdots&\vdots&\cdots&\vdots\\\
\phi_{n-1}&\phi_{n-2}&\cdots&\phi_{-1}\\\
\zeta^{-n}&\zeta^{-n+1}&\cdots&1\end{pmatrix}.$

Combining this equation with (2.6) yields

(2.34)
$X_{12}(z;n)=\frac{1}{D_{n}[\phi]}\mathrm{det}\begin{pmatrix}\phi_{0}&\phi_{-1}&\cdots&\phi_{-n}\\\
\phi_{1}&\phi_{0}&\cdots&\phi_{-n+1}\\\ \vdots&\vdots&\cdots&\vdots\\\
\phi_{n-1}&\phi_{n-2}&\cdots&\phi_{-1}\\\
\int_{\mathbb{T}}\frac{\zeta^{-n}\phi(\zeta)}{\zeta-z}\frac{\mathrm{d}\zeta}{2\pi\mathrm{i}}&\int_{\mathbb{T}}\frac{\zeta^{-n+1}\phi(\zeta)}{\zeta-z}\frac{\mathrm{d}\zeta}{2\pi\mathrm{i}}&\cdots&\int_{\mathbb{T}}\frac{\phi(\zeta)}{\zeta-z}\frac{\mathrm{d}\zeta}{2\pi\mathrm{i}}\end{pmatrix}.$

Thus, using (2.32) and (2.34) we arrive at (2.30). ∎

######  Corollary 2.6.

We have

(2.35)
$D^{B}_{n+1}\left[\phi;\displaystyle\left(a+\sum^{m}_{j=1}\frac{b_{j}z}{z-c_{j}}\right)\phi\right]=aD_{n+1}[\phi]+D_{n}[\phi]\sum^{m}_{j=1}b_{j}X_{12}(c_{j};n),$

and for a Szegő type $\phi$

(2.36)
$D^{B}_{n+1}\left[\phi;\displaystyle\left(a+\sum^{m}_{j=1}\frac{b_{j}z}{z-c_{j}}\right)\phi\right]=G[\phi]^{n+1}E[\phi]\left(a+\frac{1}{G[\phi]}\sum^{m}_{j=1\atop|c_{j}|<1}b_{j}\alpha(c_{j})\right)\left(1+O(e^{-\mathfrak{c}n})\right),$

as $n\to\infty$, where $\alpha$ is defined in (2.29), $G[\phi]$ and $E[\phi]$
are given by (1.5), and $\mathfrak{c}$ is some positive constant.

###### Proof.

(2.35) directly follows from (2.30), (1.11), and (1.10). For the asymptotic
statement, notice that from (5.16), (5.19) and (5.20) we have

(2.37) $X_{12}(c;n)=\begin{cases}\alpha(c)(1+O(e^{-\mathfrak{c}n})),&|c|<1,\\\
R_{1,12}(c;n)\alpha^{-1}(c)c^{-n}(1+O(e^{-\mathfrak{c}n})),&|c|>1,\end{cases}$

as $n\to\infty$, where $R_{1,12}$ is given by (5.18). Now (2.36) follows from
(1.4), (1.5), (2.35), (2.37), and (5.16). ∎

Now, we prove the analogue of Lemma 2.5 for $c=0$.

######  Lemma 2.7.

We have the following identity777Throughout the paper we occasionally use
$\tilde{f}(z)$, to denote $\displaystyle f(z^{-1})$.

(2.38) $D^{B}_{n}[\phi;\frac{1}{z}\phi]=-D^{B}_{n+1}[\tilde{\phi};z],$

and hence for a Szegő type $\phi$

(2.39)
$D^{B}_{n}[\phi;\frac{1}{z}\phi]=\frac{1}{2\pi\mathrm{i}}G[\phi]^{n}E[\phi]\left(\int_{\mathbb{T}}\ln(\tilde{\phi}(\tau))\mathrm{d}\tau+O(e^{-\mathfrak{c}n})\right),$

as $n\to\infty$, where $G[\phi]$ and $E[\phi]$ are given by (1.5), and
$\mathfrak{c}$ is some positive constant.

###### Proof.

Note that

$\displaystyle D^{B}_{n+1}[\tilde{\phi};z]$
$\displaystyle=\mathrm{det}\begin{pmatrix}\tilde{\phi}_{0}&\tilde{\phi}_{-1}&\cdots&\tilde{\phi}_{-n+2}&\tilde{\phi}_{-n+1}&\tilde{\phi}_{-n}\\\
\tilde{\phi}_{1}&\tilde{\phi}_{0}&\cdots&\tilde{\phi}_{-n+3}&\tilde{\phi}_{-n+2}&\tilde{\phi}_{-n+1}\\\
\vdots&\vdots&\cdots&\vdots&\vdots&\vdots\\\
\tilde{\phi}_{n-1}&\tilde{\phi}_{n-2}&\cdots&\tilde{\phi}_{1}&\tilde{\phi}_{0}&\tilde{\phi}_{-1}\\\
0&0&\cdots&0&1&0\end{pmatrix}$
$\displaystyle=\mathrm{det}\begin{pmatrix}\phi_{0}&\phi_{1}&\cdots&\phi_{n-2}&\phi_{n-1}&\phi_{n}\\\
\phi_{-1}&\phi_{0}&\cdots&\phi_{n-3}&\phi_{n-2}&\phi_{n-1}\\\
\vdots&\vdots&\cdots&\vdots&\vdots&\vdots\\\
\phi_{-n+1}&\phi_{-n+2}&\cdots&\phi_{-1}&\phi_{0}&\phi_{1}\\\
0&0&\cdots&0&1&0\end{pmatrix}$
$\displaystyle=-\mathrm{det}\begin{pmatrix}\phi_{0}&\phi_{1}&\cdots&\phi_{n-2}&\phi_{n}\\\
\phi_{-1}&\phi_{0}&\cdots&\phi_{n-3}&\phi_{n-1}\\\
\vdots&\vdots&\cdots&\vdots&\vdots\\\
\phi_{-n+1}&\phi_{-n+2}&\cdots&\phi_{-1}&\phi_{1}\end{pmatrix}$
$\displaystyle=-\mathrm{det}\begin{pmatrix}\phi_{0}&\phi_{-1}&\cdots&\phi_{-n+1}\\\
\phi_{1}&\phi_{0}&\cdots&\phi_{-n+2}\\\ \vdots&\vdots&\cdots&\vdots\\\
\phi_{n-2}&\phi_{n-3}&\cdots&\phi_{-1}\\\
\phi_{n}&\phi_{n-1}&\cdots&\phi_{1}\end{pmatrix}=-D^{B}_{n}[\phi;\frac{1}{z}\phi],$

because the $j$-th Fourier coefficient of $z^{-1}\phi(z)$ is $\phi_{j+1}$. Now
(2.39) immediately follows from (2.18) and the fact that
$D_{n}[\phi]=D_{n}[\tilde{\phi}]$. ∎

######  Lemma 2.8.

For $\psi=z^{k}\phi$, $k=0,1,\cdots,n$, we have

(2.40) $D^{B}_{n}[\phi,z^{k}\phi]=\begin{cases}D_{n}[\phi],&k=0,\\\
0,&k=1,\cdots,n-1,\\\ (-1)^{n-1}D_{n}[z\phi],&k=n.\end{cases}$

###### Proof.

Note that $[z^{k}\phi]_{j}=\phi_{j-k}$ and thus, $D^{B}_{n}[\phi,z^{k}\phi]$
has two identical columns for $k=1,\cdots,n-1$, and (2.40) is obvious for
$k=0$. For $k=n$, (2.40) follows immediately if one moves the border column to
the first column by making $n-1$ swaps of adjacent columns. ∎

Theorem 1.1 is now proven by combining lemmas 2.2, 2.7, 2.8 and the
corollaries 2.4 and 2.6 via (1.10).

###  2.4. Ising model next-to-diagonal correlations.

In this section we focus on the specific symbols $\widehat{\phi}$ and
$\widehat{\psi}$, respectively given by (1.28) and (1.31), corresponding to
the next-to-diagonal correlations in the Ising model via (1.30). For a
derivation of (1.31) from the formulae in [AYP87] see Section 5.3. This is
clear that in the low temperature regime ($k>1$) the symbols $\widehat{\phi}$
and $\widehat{\psi}$ fit the class of symbols considered in Theorem 1.1.
Indeed, comparing (1.31) with (1.8) we can find the corresponding parameters:

(2.41) $\begin{split}\begin{aligned}
&m=1,&a&=b_{0}=\hat{b}_{0}=0,\qquad&b_{1}&=\frac{C_{v}}{S_{v}},\\\
&c_{1}=c_{*}\equiv-\frac{S_{h}}{S_{v}},\qquad&\hat{a}&=\frac{C_{h}}{S_{h}},&\hat{b}_{1}&=-\frac{C_{h}}{S_{h}}.\end{aligned}\end{split}$

Therefore the constant $F[\widehat{\phi};\widehat{\psi}]$ given by (1.14)
simplifies to

(2.42)
$F[\widehat{\phi};\widehat{\psi}]=\begin{cases}\displaystyle\frac{C_{v}}{S_{v}}\alpha(c_{*}),&J_{v}>J_{h},\\\\[11.0pt]
\displaystyle\frac{C_{h}}{S_{h}}\alpha(c_{*}),&J_{v}<J_{h}.\end{cases}$

where we have used

$|c_{*}|\lessgtr 1\iff J_{h}\lessgtr J_{v}.$

Now let us compute $\alpha(c_{*})$. We observe that

(2.43)
$\widehat{\phi}(z)=e^{i\pi/2}k^{1/2}(z-k)^{-1/2}(z-k^{-1})^{1/2}z^{-1/2},$

where the branches of the roots all have arguments from $0$ to $2\pi$.
Recalling the expression (2.29) for $\alpha$, we can compute $\alpha(c_{*})$
by a simple contour integration (deform the integral on $\mathbb{T}$ to the
interval $[0,k^{-1}]$, and note that we get a residue term when $-1<c_{*}<0$
). We eventually arrive at

(2.44)
$\alpha(c_{*})=\begin{cases}\displaystyle\frac{S_{v}}{C_{v}},&J_{v}>J_{h},\\\\[11.0pt]
\displaystyle\frac{S_{h}}{C_{h}},&J_{v}<J_{h}.\end{cases}$

This can also be seen in a more straightforward way by recalling (1.16), which
in the Ising case amounts to:

(2.45)
$\alpha(z)=\begin{cases}\displaystyle\frac{1}{\sqrt{1-k^{-1}z}},&|z|<1,\\\\[11.0pt]
\displaystyle\frac{1}{\sqrt{1-k^{-1}z^{-1}}},&|z|>1.\end{cases}$

Combining (2.42) and (2.44) yields

(2.46) $F[\widehat{\phi};\widehat{\psi}]=1,\qquad J_{h}\neq J_{v}.$

This concludes the proof of Theorem 1.4 in the anisotropic case by recalling
that $G[\widehat{\phi}\ ]=1$ and $E[\widehat{\phi}\ ]=(1-k^{-2})^{1/4}$.

######  Remark 2.9.

Notice that the next-to-diagonal long range order in the isotropic case
$J_{h}=J_{v}$ deserves special attention as $|c_{*}|=1$. It is important to
notice that while it seems that the function $\widehat{\psi}(z)$ has a pole at
$z=c_{*}$ it actually has a removable singularity there. In other words,
$\widehat{\psi}(z)$ (as well as $\widehat{\phi}(z)$) are analytic in a
neighborhood of the unit circle $\mathbb{T}$ irrespective of the value of
$c_{*}$. To be precise both functions are analytic in $z$ for $k^{-1}<|z|<k$.
However, the splitting of $\widehat{\psi}(z)$ done in the proof of Theorem 1.1
would re-introduce this pole at $z=c_{*}$ in both terms and render the proof
invalid when $|c_{*}|=1$. To circumvent this problem we resort to a
“deformation trick”. For a fixed
$0<\varepsilon<\displaystyle\frac{k-k^{-1}}{k+k^{-1}}$, let
$\displaystyle\frac{k^{-1}}{1-\varepsilon}<\rho<\displaystyle\frac{k}{1+\varepsilon}$
and introduce the functions

$\widehat{\phi}_{\rho}(z):=\widehat{\phi}(\rho
z),\qquad\widehat{\psi}_{\rho}(z):=\widehat{\psi}(\rho z),\qquad$

where the condition on $\rho$ ensures that both functions are analytic in the
$\varepsilon$-neighborhood of $\mathbb{T}$:
$\\{z:1-\varepsilon<|z|<1+\varepsilon\\}$. The $n$-th Fourier coefficient of
these new functions differs from $n$-the Fourier coefficient of the original
functions by a factor $\rho^{n}$. For this reason, we have that

$D_{N}[\widehat{\phi}]=D_{N}[\widehat{\phi}_{\rho}],\qquad
D_{N}^{B}[\widehat{\phi};\widehat{\psi}]=D_{N}^{B}[\widehat{\phi}_{\rho};\widehat{\psi}_{\rho}].$

Indeed, the underlying matrices are related to each other by appropriate
multiplication of diagonal matrices. Thus the derivation of the determinant
asymptotics can be based on the pair
$(\widehat{\phi}_{\rho};\widehat{\psi}_{\rho})$ rather than the pair
$(\widehat{\phi};\widehat{\psi})$. These functions are clearly of a similar
form as the original ones. The crucial point however is that when we consider
$\widehat{\psi}_{\rho}$ and split it into two parts, each part has a pole at
$z=c_{*}/\rho$ rather than at $z=c_{*}$. Thus Theorem 1.1 is applicable to
$(\widehat{\phi}_{\rho};\widehat{\psi}_{\rho})$ whenever $|c_{*}|\neq\rho$. As
we can choose this $\rho$ within at least a small range, $k^{-1}<\rho<k$, the
asymptotic results concerning these functions remains true also when
$|c_{*}|=1$. Notice that since the determinant
$D_{N}^{B}[\widehat{\phi}_{\rho};\widehat{\psi}_{\rho}]$ is
$\rho$-independent, in particular, its leading order asymptotics given by
Theorem 1.1 is also $\rho$-independent. This can also be checked directly by
looking at the terms on the right hand side of (1.13). To this end, we
obviously have from (1.5) that $G[\widehat{\phi}_{\rho}]=G[\widehat{\phi}]$
and $E[\widehat{\phi}_{\rho}]=E[\widehat{\phi}]$. Also the role of $\alpha$ on
the right hand side of (1.14) is now played by $\alpha_{\rho}$ which satisfies
$\alpha_{\rho,+}(z)=\alpha_{\rho,-}(z)\widehat{\phi}_{\rho}(z)$ and is
explicitly given by

$\alpha_{\rho}(z)=\begin{cases}\displaystyle\frac{1}{\sqrt{1-k^{-1}\rho
z}},&|z|<1,\\\\[11.0pt]
\displaystyle\frac{1}{\sqrt{1-k^{-1}\rho^{-1}z^{-1}}},&|z|>1.\end{cases}$

Therefore from (1.14) it can be directly checked that
$F[\widehat{\phi}_{\rho};\widehat{\psi}_{\rho}]=1$.

###  2.5. Proof of Theorem 1.5

Based on (1.10), (1.31), (2.27), and (2.35), the bordered Toeplitz determinant
representing the Ising correlation function
$\langle\sigma_{0,0}\sigma_{N-1,N}\rangle$ satisfies the following relation

(2.47)
$\frac{D^{B}_{N}[\widehat{\phi};\widehat{\psi}]}{D_{N-1}[\widehat{\phi}\
]}=\displaystyle\frac{C_{v}}{S_{v}}X_{12}(c_{*};N-1)+\begin{cases}c^{-N+1}_{*}\displaystyle\frac{C_{h}}{S_{h}}X_{11}(c_{*};N-1),&|c_{*}|>1,\\\\[11.0pt]
0,&|c_{*}|<1.\end{cases}$

Also, for the Toeplitz determinant $D_{N-1}[\widehat{\phi}\ ]$ we have that

(2.48) $\ln D_{N}[\widehat{\phi}\ ]=\ln E[\widehat{\phi}\
]-\sum_{n=N}^{\infty}\ln\kappa^{-2}_{n}=\frac{1}{4}\ln\Bigl{(}1-k^{-2}\Bigr{)}-\sum_{n=N}^{\infty}\ln\kappa^{-2}_{n},$

where

(2.49) $\kappa^{-2}_{n}=X_{12}(0),$

and $X(z)\equiv X(z;N-1)$ is the solution of the $X$ \- Riemann-Hilbert
problem, RH-X1 \- RH-X3 generated by the weight $\widehat{\phi}(z)$ and
corresponding to $n=N-1$. Equations (2.47) - (2.49) show us that in order to
calculate the correction to the leading term, i.e.,
$(1-k^{-2})^{\frac{1}{4}}$, to the determinant
$D^{B}_{N}[\widehat{\phi};\widehat{\psi}]$ we need the high terms in the
estimation of the solution $X(z)$ of the Riemann-Hilbert problem.

The asymptotic analysis of the Riemann-Hilbert problem RH-X1 \- RH-X3 is
presented in detail in Section 5.1 and for its solution $X(z;n)$ we have
formula (5.19) where $R(z;n)$ is the solution of the small norm Riemann-
Hilbert problem RH-R1 \- RH-R3. We have already used this formula and the
estimate (5.20) in the proof of Theorem 1.1. Now, we need more terms in
(5.20). These are given by the second term, $R_{2}(z;n)$ in the iterative
series (5.14). From (5.15), (5.17), and (5.18) that

(2.50)
$R_{2}(z;n)=\begin{pmatrix}-\displaystyle\frac{1}{4\pi^{2}}\int_{\Gamma_{1}}b(\mu)\left[\int_{\Gamma_{0}}a(\tau)\frac{d\tau}{\tau-\mu}\right]\frac{d\mu}{\mu-z}&0\\\
\\\
0&-\displaystyle\frac{1}{4\pi^{2}}\int_{\Gamma_{0}}a(\mu)\left[\int_{\Gamma_{1}}b(\tau)\frac{d\tau}{\tau-\mu}\right]\frac{d\mu}{\mu-z}\end{pmatrix},\quad
z\in\mathbb{C}\setminus\Sigma_{R},$

where we have introduced the notations,

$a(z;n)=-z^{n}\widehat{\phi}^{-1}(z)\alpha^{2}(z),\quad\mbox{and}\quad
b(z;n)=z^{-n}\widehat{\phi}^{-1}(z)\alpha^{-2}(z),$

and we also remind that

$\alpha(z)=\exp\left[\frac{1}{2\pi
i}\int_{\mathbb{T}}\frac{\ln(\widehat{\phi}(\tau))}{\tau-z}d\tau\right]=\begin{cases}\displaystyle\frac{1}{\sqrt{1-k^{-1}z}},&|z|<1,\\\\[11.0pt]
\displaystyle\frac{1}{\sqrt{1-k^{-1}z^{-1}}},&|z|>1.\end{cases}$

Formula (2.50) in conjunction with the equations (5.18) and (5.19) yield the
following estimations for the relevant matrix entries of $X(z)$ in the regions
$\Omega_{0}$, $\Omega_{1}$, $\Omega_{2}$, and $\Omega_{\infty}$, (see Figure
5):

(2.51)
$X_{12}(z;n)=\alpha(z)\left(1-\displaystyle\frac{1}{4\pi^{2}}\int_{\Gamma_{1}}b(\mu;n)\left[\int_{\Gamma_{0}}a(\tau;n)\frac{d\tau}{\tau-\mu}\right]\frac{d\mu}{\mu-z}+O(\rho^{-4n})\right),\quad
z\in\Omega_{0}\cup\Omega_{1},$

(2.52) $X_{12}(z;n)=z^{-n}\alpha^{-1}(z)\left(\frac{1}{2\pi
i}\int_{\Gamma_{0}}a(\tau;n)\frac{d\tau}{\tau-z}+O\left(\frac{\rho^{-3n}}{1+|z|}\right)\right),\quad
z\in\Omega_{2}\cup\Omega_{\infty},$

(2.53) $X_{11}(z;n)=-\alpha^{-1}(z)\left(\frac{1}{2\pi
i}\int_{\Gamma_{0}}a(\tau;n)\frac{d\tau}{\tau-z}+O(\rho^{-3n})\right),\quad
z\in\Omega_{0},$

$X_{11}(z;n)=z^{n}\alpha(z)\widehat{\phi}^{-1}(z)\left(1-\displaystyle\frac{1}{4\pi^{2}}\int_{\Gamma_{1}}b(\mu;n)\left[\int_{\Gamma_{0}}a(\tau;n)\frac{d\tau}{\tau-\mu}\right]\frac{d\mu}{\mu-z}+O(\rho^{-4n})\right)$
(2.54) $-\alpha^{-1}(z)\left(\frac{1}{2\pi
i}\int_{\Gamma_{0}}a(\tau;n)\frac{d\tau}{\tau-z}+O(\rho^{-3n})\right),\quad
z\in\Omega_{1},$

$X_{11}(z;n)=z^{n}\alpha(z)\left(1-\displaystyle\frac{1}{4\pi^{2}}\int_{\Gamma_{1}}b(\mu;n)\left[\int_{\Gamma_{0}}a(\tau;n)\frac{d\tau}{\tau-\mu}\right]\frac{d\mu}{\mu-z}+O(\rho^{-4n})\right)$
(2.55) $-\alpha^{-1}(z)\widehat{\phi}^{-1}(z)\left(\frac{1}{2\pi
i}\int_{\Gamma_{0}}a(\tau;n)\frac{d\tau}{\tau-z}+O(\rho^{-3n})\right),\quad
z\in\Omega_{2},$

(2.56)
$X_{11}(z;n)=z^{n}\alpha(z)\left(1-\displaystyle\frac{1}{4\pi^{2}}\int_{\Gamma_{1}}b(\mu;n)\left[\int_{\Gamma_{0}}a(\tau;n)\frac{d\tau}{\tau-\mu}\right]\frac{d\mu}{\mu-z}+O\left(\frac{\rho^{-4n}}{1+|z|}\right)\right),\quad
z\in\Omega_{\infty}.$

In the above equations, it is assumed that the circle $\Gamma_{1}$ is centered
at $z=0$ and has radius $\rho$, the circle $\Gamma_{0}$ is centered at $z=0$
and has radius $\rho^{-1}$, and the inequality

(2.57) $k^{\frac{2}{3}}<\rho<k$

holds. Note that the last equation implies that

(2.58) $\rho^{-3n}=k^{-(2+\delta)n},$

where

$\delta=\frac{3\ln\rho-2\ln k}{\ln k}>0.$

Let us first apply these formulae to the evolution of the higher term in the
asymptotics of the pure Toeplitz determinant $D_{N}[\widehat{\phi}\ ]$. To
this end, taking into account (2.48), (2.49) we need first to estimate the
value $X_{12}(0)$. According to (2.51), we have that

(2.59)
$X_{12}(0;n)=\alpha(0)\left(1-\displaystyle\frac{1}{4\pi^{2}}\int_{\Gamma_{1}}b(\mu;n)\left[\int_{\Gamma_{0}}a(\tau;n)\frac{d\tau}{\tau-\mu}\right]\frac{d\mu}{\mu}+O(\rho^{-4n})\right).$

######  Proposition 2.10.

The following estimates take place,

(2.60) $\begin{split}\frac{1}{2\pi
i}\int_{\Gamma_{0}}a(\tau;n)\frac{d\tau}{\tau-z}=-\frac{1}{\sqrt{\pi}}\alpha^{2}(k^{-1})\frac{\sqrt{k-k^{-1}}}{k}\frac{1}{k^{-1}-z}k^{-n-1/2}n^{-1/2}\left(1+O\left(\frac{1}{n}\right)\right),\quad
n\to\infty,\end{split}$

for all $|z|>\rho^{-1}$, and

(2.61)
$\displaystyle\frac{1}{4\pi^{2}}\int_{\Gamma_{1}}b(\mu;n)\left[\int_{\Gamma_{0}}a(\tau;n)\frac{d\tau}{\tau-\mu}\right]\frac{d\mu}{\mu-z}=-\frac{1}{2\pi}\frac{1}{k^{-1}-k}\frac{1}{k-z}k^{-2n}n^{-2}\left(1+O\left(\frac{1}{n}\right)\right),\quad
n\to\infty,$

for all $|z|<\rho$.

###### Proof.

Consider first the single, $a$ \- integral. It can be deformed to the integral
over the segment $[0,k^{-1}]$ so that we would have,

(2.62) $\frac{1}{2\pi
i}\int_{\Gamma_{0}}a(\tau;n)\frac{d\tau}{\tau-z}=-\frac{1}{\pi}\int_{0}^{k^{-1}}\tau^{n}(k^{-1}-\tau)^{-1/2}\Phi(\tau)d\tau,$

where

$\Phi(\tau)=\sqrt{\tau k^{-1}}\sqrt{k-\tau}\frac{\alpha^{2}(\tau)}{\tau-z}$

is holomorphic at $\tau=k^{-1}$. Let

(2.63) $\Phi(\tau)=\sum_{l=0}^{\infty}d_{l}(\tau-k^{-1})^{l},\quad
d_{0}=\Phi(k^{-1})=\frac{1}{k}\sqrt{k-k^{-1}}\frac{\alpha^{2}(k^{-1})}{k^{-1}-z},$

be the Taylor series of $\Phi(\tau)$ at $\tau=k^{-1}$. Then, according to the
standard Watson Lemma type arguments, we arrive at the asymptotic formula,

(2.64) $\frac{1}{2\pi
i}\int_{\Gamma_{0}}a(\tau;n)\frac{d\tau}{\tau-z}\sim-\frac{1}{\pi}\sum_{l=0}^{\infty}d_{l}\int_{0}^{k^{-1}}\tau^{n}(k^{-1}-\tau)^{l-1/2}d\tau,$

and this asymptotic is uniform in any compact subset of
$\\{z:|z|>\rho^{-1}\\}$ and, in particular, for $z\in\Gamma_{1}$. For the
integrals in the right hand side of (2.64), we have,

(2.65)
$\begin{split}\int_{0}^{k^{-1}}\tau^{n}(k^{-1}-\tau)^{l-1/2}d\tau&=k^{-n-l-1/2}\int_{0}^{1}t^{n}(1-t)^{l-1/2}dt=k^{-n-l-1/2}B(n+1,l+1/2)\\\
&=k^{-n-l-1/2}\frac{\Gamma(n+1)\Gamma(l+1/2)}{\Gamma(n+l+3/2)}=k^{-n-l-1/2}\Gamma(l+1/2)n^{-l-1/2}\left(1+O\left(\frac{1}{n}\right)\right),\end{split}$

as $n\to\infty$, and hence

(2.66) $\frac{1}{2\pi
i}\int_{\Gamma_{0}}a(\tau;n)\frac{d\tau}{\tau-z}=-\frac{1}{\sqrt{\pi}}\alpha^{2}(k^{-1})\frac{\sqrt{k-k^{-1}}}{k}\frac{1}{k^{-1}-z}k^{-n-1/2}n^{-1/2}\left(1+O\left(\frac{1}{n}\right)\right),$

as $n\to\infty$, uniformly in any compact subset of $\\{z:|z|>\rho^{-1}\\}$.
This is the estimate (2.60).

Consider now the double integral (2.61). In view of (2.66) we have at once
that888 We are also taking into account that
$\int_{\Gamma_{1}}b(\mu)f(\mu)d\mu=O(k^{-n})$ for any bounded $f(\mu)$
analytic in the annulus, $\rho-\epsilon<|\mu|<k+\epsilon$.

(2.67)
$\begin{split}\displaystyle\frac{1}{4\pi^{2}}\int_{\Gamma_{1}}b(\mu;n)\left[\int_{\Gamma_{0}}a(\tau;n)\frac{d\tau}{\tau-\mu}\right]\frac{d\mu}{\mu-z}&=-\frac{i}{2\pi^{3/2}}\alpha^{2}(k^{-1})\frac{\sqrt{k-k^{-1}}}{k}k^{-n-1/2}n^{-1/2}\\\
&\times\left[\int_{\Gamma_{1}}b(\mu;n)\frac{d\mu}{(k^{-1}-\mu)(\mu-z)}+O\left(\frac{k^{-n}}{n}\right)\right],\quad
n\to\infty.\end{split}$

Applying to the $b$ -integral in the right hand side of the last equation the
same arguments as we used for derivation of the estimate (2.66), we obtain
that

(2.68)
$\int_{\Gamma_{1}}b(\mu;n)\frac{d\mu}{(k^{-1}-\mu)(\mu-z)}=-i\sqrt{\pi}\alpha^{-2}(k)\frac{1}{(k-z)\sqrt{k-k^{-1}}}\frac{1}{k^{-1}-k}k^{-n+3/2}n^{-3/2}\left(1+O\left(\frac{1}{n}\right)\right),$

as $n\to\infty$, uniformly in any compact subset of $\\{z:|z|<\rho\\}$. This
in turns yields the estimate,

$\displaystyle\frac{1}{4\pi^{2}}\int_{\Gamma_{1}}b(\mu;n)\left[\int_{\Gamma_{0}}a(\tau;n)\frac{d\tau}{\tau-\mu}\right]\frac{d\mu}{\mu-z}=-\frac{1}{2\pi}\frac{\alpha^{2}(k^{-1})\alpha^{-2}(k)}{k^{-1}-k}\frac{1}{k-z}k^{-2n}n^{-2}\left(1+O\left(\frac{1}{n}\right)\right),\quad
n\to\infty,$

which, taking into account that

$\alpha^{-1}(z)\alpha(z^{-1})=1,$

implies (2.61) ∎

Using (2.59), (2.61) and taking into account that $\alpha(0)=1$ we conclude
that

(2.69)
$\kappa^{-2}=X_{12}(0;n)=1+\frac{1}{2\pi}\frac{1}{1-k^{2}}k^{-2n}n^{-2}\left(1+O\left(\frac{1}{n}\right)\right),\quad
n\to\infty,$

and, also,

(2.70)
$\ln\kappa^{-2}_{n}=\frac{1}{2\pi}\frac{1}{1-k^{2}}k^{-2n}n^{-2}\left(1+O\left(\frac{1}{n}\right)\right),\quad
n\to\infty.$

In view of (2.48), we need now the asymptotics of the sum

$\sum_{n=N}^{\infty}\frac{k^{-2n}}{n^{p}},\quad p=2,3.$

This can be easily done by the summation by parts. Indeed, put

$S_{n}=\sum_{l=n}^{\infty}k^{-2l}\equiv k^{-2n}\frac{1}{1-k^{-2}}.$

Then we have,

$\sum_{n=N}^{\infty}\frac{k^{-2n}}{n^{p}}=\sum_{n=N}^{\infty}(S_{n}-S_{n+1})\frac{1}{n^{p}}=\sum_{n=N}^{\infty}S_{n}\frac{1}{n^{p}}-\sum_{n=N}^{\infty}S_{n+1}\frac{1}{n^{p}}$
$=S_{N}\frac{1}{N^{p}}+\sum_{n=N+1}^{\infty}S_{n}\frac{1}{n^{p}}-\sum_{n=N}^{\infty}S_{n+1}\frac{1}{n^{p}}=S_{N}\frac{1}{N^{p}}+\sum_{n=N}^{\infty}S_{n+1}\left(\frac{1}{(n+1)^{p}}-\frac{1}{n^{p}}\right)$
(2.71) $=\frac{1}{1-k^{-2}}N^{-p}k^{-2N}+O\Bigl{(}N^{-p-1}k^{-2N}\Bigr{)}.$

Combining (2.71) with (2.70) and (2.48) we arrive at the final formula for the
asymptotics of the Toeplitz determinant $D_{N}[\widehat{\phi}\ ]$ with the
explicit second term,

(2.72) $D_{N}[\widehat{\phi}\
]=(1-k^{-2})^{1/4}\left(1+\frac{1}{2\pi(1-k^{-2})^{2}}N^{-2}k^{-2N-2}\Bigl{(}1+O(N^{-1})\Bigr{)}\right),\quad
N\to\infty.$

In the next section this formula will be also proven by the operator
technique.

Let us move now to the bordered Topelitz determinant
$D^{B}_{N}[\widehat{\phi};\widehat{\psi}]$ and consider first the case when

$|c_{*}|<1.$

In this case, according to (2.47), we will only need the asymptotics for
$X_{12}(z)$ for $z\in\Omega_{0}\cup\Omega_{1}$, i.e. formula (2.51) with
$z=c_{*}$ and $n=N-1$. Moreover, we can use the estimate (2.61) for the double
integral involved and get at once that

$X_{12}(c_{*};N-1)=\alpha(c_{*})\left(1+\frac{1}{2\pi}\frac{1}{(k^{-1}-k)(k-c_{*})}k^{-2N+2}N^{-2}\left(1+O\left(\frac{1}{N}\right)\right)\right),\quad
N\to\infty.$

The last equation in conjunction with (2.47) yields the formula

(2.73)
$D^{B}_{N}[\widehat{\phi};\widehat{\psi}]=\frac{C_{v}}{S_{v}}\alpha(c_{*})(1-k^{-2})^{1/4}\left(1+\frac{1}{2\pi}\frac{1}{1-k^{-2}}\left(-\frac{k}{k-c_{*}}+\frac{1}{1-k^{-2}}\right)N^{-2}k^{-2N}\Bigl{(}1+O(N^{-1})\Bigr{)}\right).$

This formula, taking into account the definitions (1.26) and (1.31) of the
parameters $k$ and $c_{*}$ and the equation (2.44) for $\alpha(c_{*})$ (the
case we consider now is $J_{v}>J_{h}$) we can rewrite (2.73) as

(2.74) $\displaystyle D_{N}^{B}[\widehat{\phi};\widehat{\psi}]$
$\displaystyle=(1-k^{-2})^{1/4}\left(1+\frac{1}{2\pi(1-k^{-2})}\Big{(}\frac{1}{C_{v}^{2}}+\frac{1}{k^{2}-1}\Big{)}N^{-2}k^{-2N}\Big{(}1+O(N^{-1})\Big{)}\right).$

This proves Theorem 1.5 for the case $|c_{*}|<1$. The proof of Theorem 1.5 for
the case $|c_{*}|>1$ (i.e. if $c_{*}\in\Omega_{2}$ or
$c_{*}\in\Omega_{\infty}$, see Figure 5) follows from almost identical
considerations employed in the case $|c_{*}|<1$. Let us first discuss the case
when $c_{*}\in\Omega_{2}$. In this case we have

(2.75)
$\frac{C_{v}}{S_{v}}X_{12}(c_{*};N-1)=\frac{C_{v}}{S_{v}}c^{-N+1}_{*}\alpha^{-1}(c_{*})\left(\frac{1}{2\pi
i}\int_{\Gamma_{0}}a(\tau;N-1)\frac{d\tau}{\tau-
c_{*}}+O\left(\rho^{-3N}\right)\right),$

and

(2.76)
$\begin{split}c^{-N+1}_{*}\displaystyle\frac{C_{h}}{S_{h}}X_{11}(c_{*};N-1)&=\displaystyle\frac{C_{h}}{S_{h}}\alpha(c_{*})\left(1-\displaystyle\frac{1}{4\pi^{2}}\int_{\Gamma_{1}}b(\mu;N-1)\left[\int_{\Gamma_{0}}a(\tau;N-1)\frac{d\tau}{\tau-\mu}\right]\frac{d\mu}{\mu-
c_{*}}+O(\rho^{-4N})\right)\\\
&-c^{-N+1}_{*}\displaystyle\frac{C_{h}}{S_{h}}\alpha^{-1}(c_{*})\widehat{\phi}^{-1}(c_{*})\left(\frac{1}{2\pi
i}\int_{\Gamma_{0}}a(\tau;N-1)\frac{d\tau}{\tau-
c_{*}}+O(\rho^{-3N})\right).\end{split}$

These are the terms needed to compute
$D^{B}_{N}[\widehat{\phi};\widehat{\psi}]$ in view of (2.47). Notice that the
contribution from (2.75) cancels the contribution from the second term on the
right hand side of (2.76), as one can simply check that

(2.77) $\frac{C_{h}}{S_{h}}\widehat{\phi}^{-1}(c_{*})=\frac{C_{v}}{S_{v}}.$

Now, from (2.44), (2.47), (2.61), (2.72), (2.76) we can easily show that
(1.33) holds when $c_{*}\in\Omega_{2}$. Finally we discuss the case
$c_{*}\in\Omega_{\infty}$. Equation (2.75) still holds in this case (see
(2.52)). Using (2.44) and (2.60) we can write

(2.78)
$\frac{C_{v}}{S_{v}}X_{12}(c_{*};N-1)=\frac{C_{v}C_{h}}{S_{v}S_{h}}c^{-N+1}_{*}\left(-\frac{1}{\sqrt{\pi}}\alpha^{2}(k^{-1})\frac{\sqrt{k-k^{-1}}}{k}\frac{1}{k^{-1}-c_{*}}k^{-N+1/2}N^{-1/2}\right)\left(1+O\left(\frac{1}{N}\right)\right),$

as $N\to\infty$. Using (2.56) and (2.44) we have

(2.79)
$c^{-N+1}_{*}\displaystyle\frac{C_{h}}{S_{h}}X_{11}(c_{*};N-1)=1-\displaystyle\frac{1}{4\pi^{2}}\int_{\Gamma_{1}}b(\mu;N-1)\left[\int_{\Gamma_{0}}a(\tau;N-1)\frac{d\tau}{\tau-\mu}\right]\frac{d\mu}{\mu-
c_{*}}+O(\rho^{-4N})$

The asymptotics of the integral on the right hand side can be computed from
(2.67), however, we can not use (2.68) directly, because when
$z\in\Omega_{\infty}$ we also get a residue term. To that end by a
straightforward calculation when $z\in\Omega_{\infty}$ we find

(2.80) $\int_{\Gamma_{1}}b(\mu;n)\frac{d\mu}{(k^{-1}-\mu)(\mu-
c_{*})}=2\pi\mathrm{i}z^{-n-1}\frac{\sqrt{k-z}}{\sqrt{k-z^{-1}}}-2\mathrm{i}\frac{\sqrt{k}}{z}\int^{k^{-1}}_{0}\frac{t^{n}\sqrt{k^{-1}-t}}{(t-z^{-1})\sqrt{k-t}\sqrt{t}}\mathrm{d}t.$

We notice that the residue term (combined with the prefactors coming from
(2.67)) exactly cancels out the contribution from (2.78). Finally, the
asymptotic expansion of the second term in (2.80) can be written as a series
involving Beta functions similar to what is shown in equations (2.62) through
(2.65). Finding the asymptotics of the first term in that series using
Stirling’s formula and then combining this with (2.47), (2.67), (2.72) and
(2.79) finishes the proof of Theorem 1.5 for $c_{*}\in\Omega_{\infty}$.

##  3\. Asymptotics of Bordered Toeplitz determinants: Operator Theory
approach

###  3.1. General results

For $\phi\in L^{1}(\mathbb{T})$ we define the $N\times N$ Toeplitz matrix,

(3.1) $T_{N}(\phi):=\begin{pmatrix}\phi_{0}&\phi_{-1}&\cdots&\phi_{-N+1}\\\
\phi_{1}&\phi_{0}&\ddots&\vdots\\\ \vdots&\ddots&\ddots&\phi_{-1}\\\
\phi_{N-1}&\cdots&\phi_{1}&\phi_{0}\end{pmatrix},$

where, as before, $\phi_{n}$ are the Fourier coefficients of $\phi$.
Occasionally, the notation

$[\phi]_{n}=\frac{1}{2\pi}\int_{0}^{2\pi}\phi(e^{i\theta})e^{-in\theta}\,d\theta$

will be used as well. Clearly, $\mathrm{det}\,T_{N}(\phi)=D_{N}[\phi]$.

In what follows, $e_{0}$ stands for the column vector $(1,0,0,\dots,0)^{T}$ in
$\mathbb{C}^{N}$, and $e_{0}^{T}$ signals its transpose, the row vector
$(1,0,0,\dots,0)$.

######  Proposition 3.1.

Let $\phi,\psi\in L^{1}(\mathbb{T})$, and assume that $T_{N}(\phi)$ is
invertible. Then

(3.2) $D_{N}^{B}[\phi;\psi]=D_{N}[\phi]\cdot
e_{0}^{T}T_{N}^{-1}(\phi)T_{N}(\psi)e_{0}.$

###### Proof.

Via a rearrangement of rows and columns in (1.1) (or, more formally, by
multiplying with a “flip matrix” from both sides) we see that

(3.3)
$D^{B}_{N}[\phi;\psi]=\mathrm{det}\begin{pmatrix}\psi_{0}&\phi_{-1}&\cdots&\phi_{-N-1}\\\
\psi_{1}&\phi_{0}&\ddots&\vdots\\\ \vdots&\vdots&\ddots&\phi_{-1}\\\
\psi_{N-1}&\phi_{N-2}&\cdots&\phi_{0}\end{pmatrix}.$

By observing that $T_{N}(\psi)e_{0}$ is the column vector
$(\psi_{0},\dots,\psi_{N-1})^{T}$, the statement follows from Cramer’s rule. ∎

In view of computing the asymptotics, the above formula reduces bordered
Toeplitz determinants to usual Toeplitz determinants $D_{N}[\phi]$ and the
scalar quantity

(3.4) $\displaystyle
F_{N}[\phi;\psi]:=e_{0}^{T}T_{N}^{-1}(\phi)T_{N}(\psi)e_{0}.$

Under appropriate assumptions, the asymptotics of $D_{N}[\phi]$ is given by
the Szegő-Widom limit theorem, whereas the asymptotics of the scalar quantity
follows from the asymptotics of the inverse of the Toeplitz matrix
$T_{N}(\phi)$.

We proceed to give some operator-theoretic background; for details we refer to
[BS06] or [BS99]. For $\phi\in L^{1}(\mathbb{T})$ we define the infinite
Toeplitz and Hankel matrices

$T(\phi)\cong(\phi_{j-k}),\qquad H(\phi)\cong(\phi_{j+k+1}),\qquad 0\leq
j,k<\infty.$

In case $\phi\in L^{\infty}(\mathbb{T})$ these represent bounded linear
operators acting on $\ell^{2}(\mathbb{Z}_{\geq 0})$. Note that Toeplitz and
Hankel operator satisfy the identity

(3.5) $\displaystyle T(\phi\psi)$ $\displaystyle=$ $\displaystyle
T(\phi)T(\psi)+H(\phi)H(\tilde{\psi}),$

where $\phi,\psi\in L^{\infty}(\mathbb{T})$ and
$\tilde{\psi}(z):=\psi(z^{-1})$. In particular,

(3.6) $T(\psi_{-}\phi\psi_{+})=T(\psi_{-})T(\phi)T(\psi_{+})$

if $\psi_{\pm}\in H^{\infty}_{\pm}$, where

$H^{\infty}_{\pm}:=\left\\{\,f\in L^{\infty}(\mathbb{T})\,:\,f_{n}=0\mbox{ for
all }\mp n>0\right\\}$

are the usual Hardy spaces. We will identify functions in
$H^{\infty}_{\pm}(\mathbb{T})$ with their analytic extensions onto the inside
or outside, resp., of $\mathbb{T}$.

Among the various versions of Wiener-Hopf factorization we are going to use
the following one. We say that a function $\phi\in C(\mathbb{T})$ has a
continuous canonical Wiener-Hopf factorization if it can be written as

$\phi(z)=\phi_{-}(z)\phi_{+}(z),\qquad|z|=1,$

where

$\phi_{\pm},\phi_{\pm}^{-1}\in H^{\infty}_{\pm}\cap C(\mathbb{T}).$

A sufficient criterium for the existence of such a factorization is that
$\phi$ belongs to the Hölder class $C^{{\varepsilon}}(\mathbb{T})$ for some
${\varepsilon}>0$, is nonvanishing on $\mathbb{T}$ and has winding number zero
(see, e.g., [BS06, Sect. 10.2]). In this case, the Wiener-Hopf factorization
is given by

(3.7) $\displaystyle\phi_{+}(z)$
$\displaystyle=\exp\left(\sum_{n=0}^{\infty}z^{n}[\log\phi]_{n}\right),\quad$
$\displaystyle\phi_{-}(z)$
$\displaystyle=\exp\left(\sum_{n=1}^{\infty}z^{-n}[\log\phi]_{-n}\right).$

On the other hand, a necessary condition is that $\phi$ is continuous and
nonvaninshing on $\mathbb{T}$ and has winding number zero.

For $\phi\in C(\mathbb{T})$ the Toeplitz operator $T(\phi)$ on
$\ell^{2}(\mathbb{Z}_{\geq 0})$ is invertible if and only if $\phi$ does not
vanish on $\mathbb{T}$ and has winding number zero. If $\phi$ admits a
continuous canonical Wiener-Hopf factorization then the inverse of $T(\phi)$
is given by

(3.8) $T^{-1}(\phi)=T(\phi_{+}^{-1})T(\phi_{-}^{-1}).$

as can be seen from (3.6).

Finally, let us introduce the finite section projection

$P_{N}:(f_{0},f_{1},\dots)^{T}\mapsto(f_{0},f_{1},\dots,f_{N-1},0,0,\dots)^{T}$

acting on $\ell^{2}(\mathbb{Z}_{\geq 0})$. As usual, we will identify
$\mathbb{C}^{N}$ with the image of $P_{N}$. Correspondingly we have
$P_{N}T(\phi)P_{N}=T_{N}(\phi)$. The complementary projection is
$Q_{N}=I-P_{N}$, and we remark that $Q_{N}=V_{N}V_{-N}$ where $V_{N}=T(z^{N})$
and $V_{-N}=T(z^{-N})$ are forward and backward shift operators. Despite
having used the notation $e_{0}$ for finite vectors already, we will also use
it to refer to the infinite column vector,

$e_{0}=(1,0,0,\dots)^{T}\in\ell^{2}(\mathbb{Z}_{\geq 0}).$

Correspondingly, $e_{0}^{T}$ stands for infinite row vector $(1,0,0,\dots)$ or
the respective linear functional on $\ell^{2}(\mathbb{Z}_{\geq 0})$.

######  Proposition 3.2.

Let $\psi\in L^{2}(\mathbb{T})$, and assume that $\phi\in C(\mathbb{T})$ does
not vanish on $\mathbb{T}$ and has winding number zero. Then

$F_{N}[\phi;\psi]\to F[\phi;\psi]\qquad\mbox{as}\quad N\to\infty,$

where the constant

(3.9) $F[\phi;\psi]:=e_{0}^{T}T^{-1}(\phi)T(\psi)e_{0}.$

If, in addition, $\phi$ has a continuous canonical Wiener-Hopf factorization
$\phi=\phi_{-}\phi_{+}$, then

$F[\phi;\psi]=\frac{[\phi_{-}^{-1}\psi]_{0}}{[\phi_{+}]_{0}}.$

###### Proof.

Under above the assumptions on $\phi$, it is well-known (see, e.g., [BS99,
Sect. 1.5 and 2.3]) that the Toeplitz operator $T(\phi)$ is invertible, that
the matrix $T_{N}(\phi)$ is invertible for sufficiently large $N$, and that
$T_{N}^{-1}(\phi)$ converges to $T^{-1}(\phi)$ strongly on
$\ell^{2}(\mathbb{Z}_{\geq 0})$ as $N\to\infty$. Here we use the afore-
mentioned identification of $\mathbb{C}^{N}$ with a subspace of
$\ell^{2}(\mathbb{Z}_{\geq 0})$ and the corresponding identification of an
$N\times N$ matrix with an operator on $\ell^{2}(\mathbb{Z}_{\geq 0})$.
Obviously, $T_{N}(\psi)e_{0}\to T(\psi)e_{0}$ in the norm of
$\ell^{2}(\mathbb{Z}_{\geq 0})$. Therefore, again in the norm

$T_{N}^{-1}(\phi)T_{N}(\psi)e_{0}\to
T^{-1}(\phi)T(\psi)e_{0}\qquad\mbox{as}\quad N\to\infty.$

This proves the first assertion. As to the evaluation of the constant we use
(3.6) and (3.8) to see that

$T^{-1}(\phi)T(\psi)=T(\phi_{+}^{-1})T(\phi_{-}^{-1})T(\psi)=T(\phi_{+}^{-1})T(\phi_{-}^{-1}\psi)$

and

$F[\phi;\psi]=e_{0}^{T}T^{-1}(\phi)T(\psi)e_{0}=e_{0}^{T}T(\phi_{+}^{-1})T(\phi_{-}^{-1}\psi)e_{0},$

where in the last expression we interpret the operators on
$\ell^{2}(\mathbb{Z}_{\geq 0})$ as infinite matrices. Since $T(\phi_{+}^{-1})$
is lower triangular it follows that
$F[\phi;\psi]=[\phi_{+}^{-1}]_{0}\cdot[\phi_{-}^{-1}\psi]_{0}$. Observe that
$[\phi_{+}^{-1}]_{0}=\phi_{+}^{-1}(0)=1/[\phi_{+}]_{0}$. ∎

The previous results combined with the Szegő-Widom limit theorem (1.4)
establishes the first order (or leading order) asymptotics for bordered
Toeplitz determinants. In fact, if we assume that $\phi$ is in Hölder class
$C^{1+\epsilon}$, does not vanish on the unit circle $\mathbb{T}$, and has
winding number zero, then

(3.10)
$D_{N}^{B}[\phi;\psi]=G[\phi]^{N}E[\phi]\left(F[\phi;\psi]+o(1)\right),\qquad
N\to\infty,$

where

$G[\phi]=\exp([\log\phi]_{0}),\quad
E[\phi]=\mathrm{det}\,T(\phi)T(\phi^{-1})=\exp\left(\sum_{n\geq
1}n[\log\phi]_{n}[\log\phi]_{-n}\right).$

Thus, what has been stated in Remark 1.3 regarding (1.19) is proved, which is
Theorem 1.2 except for the claim that the error term is decaying
exponentially.

Let us emphasize at this point that it can happen that $F[\phi;\psi]$ is zero.
In this case, the subleading terms in the asymptotics might be of interest as
well. Later in this section will take up this question.

Let $\phi$ be a function with a continuous canonical Wiener-Hopf factorization
$\phi=\phi_{-}\phi_{+}$. Each function $\psi\in L^{2}(\mathbb{T})$ has a
unique representation of the form

(3.11) $\psi=\phi p_{+}+p_{-}\quad\mbox{ with }p_{+}\in
H^{2}(\mathbb{T}),\quad p_{-}\in H^{2}_{-}(\mathbb{T}),$

where

$\displaystyle H^{2}(\mathbb{T})$ $\displaystyle=\Big{\\{}f\in
L^{2}(\mathbb{T})\,:\,f_{n}=0\mbox{ for all }n<0\Big{\\}},$ $\displaystyle
H^{2}_{-}(\mathbb{T})$ $\displaystyle=\Big{\\{}f\in
L^{2}(\mathbb{T})\,:\,f_{n}=0\mbox{ for all }n\geq 0\Big{\\}}$

are the corresponding Hardy spaces. Indeed, (3.11) is equivalent to

(3.12) $\phi_{-}^{-1}\psi=\phi_{+}p_{+}+\phi_{-}^{-1}p_{-},$

from which it can be seen that the terms $p_{+}$ and $p_{-}$ are uniquely
given by

$p_{+}=\phi_{+}^{-1}P[\phi_{-}^{-1}\psi],\qquad
p_{-}=\phi_{-}(I-P)[\phi_{-}^{-1}\psi].$

Here $P$ is the Riesz projection (i.e., the orthogonal projection on
$L^{2}(\mathbb{T})$ with range equal to $H^{2}(\mathbb{T})$). We remark that
if we consider the Toeplitz operator $T(\phi)$ on $H^{2}(\mathbb{T})$ (rather
than on $\ell^{2}(\mathbb{Z}_{\geq 0})$), then

(3.13) $\displaystyle p_{+}=T^{-1}(\phi)P[\psi].$

######  Theorem 3.3.

Let $\phi\in C(\mathbb{T})$ have a continuous canonical Wiener-Hopf
factorization $\phi=\phi_{-}\phi_{+}$. Assume that $\psi=\phi p_{+}+p_{-}$
with $p_{+}\in H^{2}(\mathbb{T})$ and $p_{-}\in H^{2}_{-}(\mathbb{T})$. Then

(3.14) $F[\phi;\psi]=[p_{+}]_{0}.$

In particular,

(3.15) $\displaystyle
F[\phi;\left(a_{0}+a_{1}z+\frac{b_{0}}{z}+\sum_{j=1}^{m}\frac{b_{j}z}{z-c_{j}}\right)\phi]$
$\displaystyle=a_{0}+b_{0}[\log\phi]_{1}+\sum_{|c_{j}|<1}b_{j}\frac{\phi_{+}(c_{j})}{\phi_{+}(0)},$
(3.16) $\displaystyle
F[\phi;\left(a_{0}+a_{1}z+\frac{b_{0}}{z}+\sum_{j=1}^{m}\frac{b_{j}}{z-c_{j}}\right)]$
$\displaystyle=\frac{a_{0}}{\phi_{+}(0)\phi_{-}(\infty)}-a_{1}\frac{[\log\phi]_{-1}}{\phi_{+}(0)\phi_{-}(\infty)}-\sum_{|c_{j}|>1}\frac{b_{j}}{c_{j}\phi_{+}(0)\phi_{-}(c_{j})}.$

###### Proof.

To prove (3.14) we note that identity (3.12) implies that

$[\phi_{-}^{-1}\psi]_{0}=[\phi_{+}p_{+}]_{0}=[\phi_{+}]_{0}[p_{+}]_{0},$

and thus $F[\phi;\psi]=[p_{+}]_{0}$. We remark that this can also be obtained
from (3.13).

For the evaluations of $F[\phi;\psi]$ for concrete $\psi$ we compute the
corresponding function $p_{+}(z)$ and then obtain $[p_{+}]_{0}=p_{+}(0)$. This
function can be obtained most conveniently by writing down the decomposition
(3.12), $\phi_{-}^{-1}\psi=\phi_{+}p_{+}+\phi_{-}^{-1}p_{-}$, explicitly.

We start with considering the cases related to (3.15). For
$\psi=\frac{z}{z-c}\phi$ with $|c|>1$, we have $p_{-}=0$, i.e.,

$\phi_{-}^{-1}\psi=\frac{\phi_{+}(z)z}{z-c}=\phi_{+}p_{+},\quad
p_{+}(z)=\frac{z}{z-c},\quad[p_{+}]_{0}=p_{+}(0)=0.$

The same conclusion is obtained in the case $\psi=z\phi$. Hence the
corresponding terms $a_{1}$ and $b_{j}$ (whenever $|c_{j}|>1$) do not occur on
the right hand side of (3.15).

For $\psi=\frac{z}{z-c}\phi$ with $|c|<1$, the decomposition is

$\phi_{-}^{-1}\psi=\frac{\phi_{+}(z)z}{z-c}=\frac{\phi_{+}(z)z-\phi_{+}(c)c}{z-c}+\frac{\phi_{+}(c)c}{z-c}.$

Hence

$p_{+}(z)=\frac{z-\phi_{+}^{-1}(z)\phi_{+}(c)c}{z-c},\qquad
p_{+}(0)=\frac{\phi_{+}(c)}{\phi_{+}(0)}.$

The case $c=0$ covers the case $\psi=\phi$ (related with the coefficient
$a_{0}$) as well.

Lastly, if $\psi=\phi/z$ we observe that

$\phi_{-}^{-1}\psi=\frac{\phi_{+}(z)}{z}=\frac{\phi_{+}(z)-\phi_{+}(0)}{z}+\frac{\phi_{+}(0)}{z},$

whence

$p_{+}(z)=\phi_{+}^{-1}(z)\frac{\phi_{+}(z)-\phi_{+}(0)}{z},\qquad[p_{+}]_{0}=p_{+}(0)=\frac{\phi_{+}^{\prime}(0)}{\phi_{+}(0)}=(\log\phi_{+}(z))^{\prime}|_{z=0}=[\log\phi]_{1}.$

Here recall the definition the Wiener-Hopf factor in (3.7).

Now let us turn to the cases related to (3.16). For $\psi=\frac{1}{z-c}$ with
$|c|<1$ or $\psi=\frac{1}{z}$, we will have $p_{+}=0$ and $p_{-}=\psi$. Hence
the terms $b_{0}$ and $b_{j}$ (whenever $|c_{j}|<1$) do not occur on the right
hand side of (3.16).

For $\psi=\frac{1}{z-c}$ with $|c|>1$, the decomposition is

$\phi_{-}^{-1}\psi=\frac{\phi_{-}^{-1}(z)}{z-c}=\frac{\phi_{-}^{-1}(c)}{z-c}+\frac{\phi_{-}^{-1}(z)-\phi_{-}^{-1}(c)}{z-c}.$

Thus

$p_{+}(z)=\frac{\phi_{+}^{-1}(z)\phi_{-}^{-1}(c)}{z-c},\quad\mbox{ and }\quad
p_{+}(0)=-\frac{1}{c\phi_{+}(0)\phi_{-}(c)}.$

The case $\psi=1$ is treated in the same way. Finally, for $\psi=z$ we
decompose

$\phi_{-}^{-1}\psi=\eta_{+}^{\prime}(0)+(\phi_{-}^{-1}(z)-\eta_{+}^{\prime}(0)\frac{1}{z})z,\quad\mbox{
with }\quad\eta_{+}(z)=\phi_{-}^{-1}(z^{-1}).$

Hence $p_{+}(z)=\phi_{+}^{-1}(z)\eta_{+}^{\prime}(0)$ and
$p_{+}(0)=\phi_{+}^{-1}(0)\eta_{+}^{\prime}(0)$, where

$\frac{\eta_{+}^{\prime}(0)}{\eta_{+}(0)}=(\log\eta_{+}(z))^{\prime}|_{z=0}=-[\log\phi]_{-1},\quad\mbox{
and }\quad\eta_{+}(0)=\phi_{-}^{-1}(\infty),$

using (3.7). This complete the proof. ∎

The previous theorem combined with the Szegő-Widom Theorem implies Theorems
1.1, up to the claim that the error terms are exponentially decaying. In fact,
what has been claimed in Remark 1.3 regarding (1.13) is proved. The
identification of the constant $F[\phi;\psi]$ as given in (3.15) and (3.16)
coincides with the expression for (1.14) by taking the relations (1.16) and
(1.18) into account.

###  3.2. Higher order asymptotics

The following “exact formula” is the key to the higher order asymptotics of
$F_{N}[\phi;\psi]$.

######  Proposition 3.4.

Let $\phi\in C^{1/2+{\varepsilon}}(\mathbb{T})$ be a nonvanishing function on
the unit circle with winding number zero. Assume that $\phi=\phi_{-}\phi_{+}$
is its Wiener-Hopf factorization. Let

$\lambda_{N}(z)=z^{-N}\lambda(z),\qquad\lambda(z)=\frac{\phi_{-}(z)}{\phi_{+}(z)},$

and put $K_{N}=H(\lambda_{N})H(\tilde{\lambda}_{N}^{-1})$. Then

  * (i)

$D_{N}[\phi]=G[\phi]^{N}E[\phi]\mathrm{det}(I-K_{N})$.

  * (ii)

$T_{N}(\phi)$ is invertible if and only if $I-K_{N}$ is invertible.

  * (iii)

In this case,

(3.17)
$T_{N}^{-1}(\phi)=T_{N}(\phi_{+}^{-1})P_{N}\left(I-T(\lambda_{N}^{-1})(I-K_{N})^{-1}T(\lambda_{N})\right)P_{N}T_{N}(\phi_{-}^{-1}).$

###### Proof.

We note that (i) is the Borodin-Okounkov-Case-Geronimo (BOCG) identity (see,
e.g., [BS06, Sect. 10.40]), and (ii) is an obvious consequence of it. Formula
(3.17) is basically formula (10.27) or (10.47) in [BS06]. ∎

######  Theorem 3.5.

Let $\phi$ have a continuous canonical Wiener-Hopf factorization
$\phi=\phi_{-}\phi_{+}$, and assume that $\psi=\phi p_{+}+p_{-}$ with
$p_{+}\in H^{2}(\mathbb{T})$ and $p_{-}\in H^{2}_{-}(\mathbb{T})$. Then

(3.18) $F_{N}[\phi;\psi]=F[\phi;\psi]-\frac{1}{[\phi_{+}]_{0}}\cdot
e_{0}^{T}T(\lambda^{-1}_{N})(I-K_{N})^{-1}T(\phi_{-}z^{-N}p_{+})e_{0}.$

###### Proof.

Note that $T_{N}(p_{-})e_{0}=0$. Using (3.17) we consider

$\displaystyle e_{0}^{T}T_{N}^{-1}(\phi)T_{N}(\phi p_{+})e_{0}$
$\displaystyle=e_{0}^{T}T_{N}(\phi_{+}^{-1})P_{N}\left(I-T(\lambda_{N}^{-1})(I-K_{N})^{-1}T(\lambda_{N})\right)P_{N}T_{N}(\phi_{-}^{-1})T_{N}(\phi
p_{+})e_{0}$ $\displaystyle=[\phi_{+}^{-1}]_{0}\cdot
e_{0}^{T}\left(I-T(\lambda_{N}^{-1})(I-K_{N})^{-1}T(\lambda_{N})\right)T(\phi_{-}^{-1})P_{N}T(\phi
p_{+})e_{0}.$

Apart from the factor $[\phi_{+}^{-1}]_{0}=[\phi_{+}]_{0}^{-1}$, this
decomposes into

$\displaystyle e_{0}^{T}T(\phi_{-}^{-1})P_{N}T(\phi
p_{+})e_{0}+e_{0}^{T}T(\lambda_{N}^{-1})(I-K_{N})^{-1}T(\lambda_{N})T(\phi_{-}^{-1})Q_{N}T(\phi
p_{+})e_{0}$ $\displaystyle-
e_{0}^{T}T(\lambda_{N}^{-1})(I-K_{N})^{-1}T(\lambda_{N})T(\phi_{-}^{-1})T(\phi
p_{+})e_{0}.$

The first two terms equal

$\displaystyle e_{0}^{T}T(\phi_{-}^{-1})P_{N}T(\phi
p_{+})e_{0}+e_{0}^{T}T(\lambda_{N}^{-1})(I-K_{N})^{-1}T(\lambda_{N})T(\phi_{-}^{-1})Q_{N}T(\phi
p_{+})e_{0}$ $\displaystyle=e_{0}^{T}T(\phi_{-}^{-1})P_{N}T(\phi
p_{+})e_{0}+e_{0}^{T}T(\lambda_{N}^{-1})(I-K_{N})^{-1}T(\lambda_{N})T(\lambda_{N}^{-1})T(\phi_{+}^{-1})V_{-N}T(\phi
p_{+})e_{0}$ $\displaystyle=e_{0}^{T}T(\phi_{-}^{-1})P_{N}T(\phi
p_{+})e_{0}+e_{0}^{T}T(\lambda_{N}^{-1})T(\phi_{+}^{-1})V_{-N}T(\phi
p_{+})e_{0}$ $\displaystyle=e_{0}^{T}T(\phi_{-}^{-1})P_{N}T(\phi
p_{+})e_{0}+e_{0}^{T}T(\phi_{-}^{-1})V_{N}V_{-N}T(\phi p_{+})e_{0}$
$\displaystyle=e_{0}^{T}T(\phi_{-}^{-1})T(\phi
p_{+})e_{0}=e_{0}^{T}T(\phi_{+}p_{+})e_{0}=[\phi_{+}]_{0}\cdot[p_{+}]_{0},$

which give the first (constant) term in (3.18). The third term from above
equals

$\displaystyle-
e_{0}^{T}T(\lambda_{N}^{-1})(I-K_{N})^{-1}T(\lambda_{N})T(\phi_{-}^{-1})T(\phi
p_{+})e_{0}$
$\displaystyle=-e_{0}^{T}T(\lambda_{N}^{-1})(I-K_{N})^{-1}T(\lambda_{N})T(\phi_{+}p_{+})e_{0}$
$\displaystyle=-e_{0}^{T}T(\lambda_{N}^{-1})(I-K_{N})^{-1}T(\phi_{-}z^{-N}p_{+})e_{0},$

which provides the second term in (3.18). ∎

The previous result allows to obtain improvements of Proposition 3.2 by
expanding the term $(I-K_{N})^{-1}$ in formula (3.18) into the Neumann series.
Notice that $K_{N}=V_{-N}H(\lambda)H(\tilde{\lambda}^{-1})V_{N}$ and
$H(\lambda)H(\tilde{\lambda}^{-1})$ is compact since $\lambda(z)$ is
continuous. Hence $K_{N}$ converges in the operator norm to zero. In
particular, the following conclusions can be drawn.

######  Corollary 3.6.

Under the same assumptions as in the previous theorem,

(3.19) $\displaystyle F_{N}[\psi;\phi]$
$\displaystyle=[p_{+}]_{0}-\frac{1}{[\phi_{+}]_{0}}\cdot
e_{0}^{T}T(\lambda^{-1}_{N})T(\phi_{-}z^{-N}p_{+})e_{0}$
$\displaystyle\qquad+O\left(\|K_{N}\|_{L(\ell^{2}(\mathbb{Z}_{\geq
0}))}\|P[\tilde{\lambda}^{-1}_{N}]\|_{H^{2}}\|P[\phi_{-}z^{-N}p_{+}]\|_{H^{2}}\right)\quad\mbox{
as }N\to\infty.$

Therein, the first term $[p_{+}]_{0}$ is the constant, whereas the second one,
which can be written as the sum

$-\frac{1}{[\phi_{+}]_{0}}\sum_{n=N}^{\infty}\left[\frac{\phi_{+}}{\phi_{-}}\right]_{-n}\Big{[}\phi_{-}p_{+}\Big{]}_{n},$

converges to zero as $N\to\infty$. One should expect that in many cases,
(i.e., unless some “cancellation” occurs in the previous sum), the third (or
error) term converges faster to zero because it contains $\|K_{N}\|$.

In the case that the generating functions $\phi$ and $\psi$ are analytic in a
neighborhood of $\mathbb{T}$, exponentially fast convergence can be derived.

######  Corollary 3.7.

Let $\phi(z)$ be analytic an nonvanishing function on the annulus
$a_{1}<|z|<b_{1}$ with winding number zero, and let $\psi(z)$ be analytic on
the annulus $a_{2}<|z|<b_{2}$, where $a_{i}<1<b_{i}$. Then, for each $\kappa$
with $\kappa>a_{1}\max\\{b_{1}^{-1},b_{2}^{-1}\\}$, we have

(3.20) $\displaystyle F_{N}[\phi;\psi]$
$\displaystyle=F[\phi;\psi]+O(\kappa^{N}),\qquad N\to\infty,$

and

(3.21) $\displaystyle D_{N}^{B}[\phi;\psi]$
$\displaystyle=G[\phi]^{N}E[\phi]\Big{(}F[\phi;\psi]+O(\kappa^{N})\Big{)},\qquad
N\to\infty.$

###### Proof.

The function $\lambda(z)^{-1}=\phi_{+}(z)/\phi_{-}(z)$ is analytic on
$a_{1}<|z|<b_{1}$ as well, and hence the Fourier coefficients
$[\lambda^{-1}]_{-n}=O(\kappa_{1}^{n})$ as $n\to+\infty$ for each
$\kappa_{1}>a_{1}$. As
$\phi_{-}p_{+}=\phi_{-}\phi_{+}^{-1}P[\phi_{-}^{-1}\psi]$, the function
$P[\phi_{-}^{-1}\psi]$ is analytic on the disc $|z|<b_{2}$ and $\phi_{-}p_{+}$
is analytic on the annulus $a_{1}<|z|<\min\\{b_{1},b_{2}\\}$. Thus, for every
$\kappa_{2}>\max\\{b_{1}^{-1},b_{2}^{-1}\\}$, the Fourier coefficients
$[\phi_{-}p_{+}]_{n}=O(\kappa_{2}^{n})$ as $n\to+\infty$. Using this
information about the Fourier coeffcients, it is easily seen that the second
and third term on the right hand side of (3.19) decays as
$O(\kappa_{1}^{N}\kappa_{2}^{N})$ as $N\to\infty$. This implies (3.20). For
(3.21), we notice that this follows from (3.20) combined with
Proposition3.4(ii) since a similar estimate can be made for the
$\mathrm{det}(I-K_{N})$ term. ∎

This together with Theorem 3.3 completes the proofs of Theorems 1.1 and
Theorem 1.2.

###  3.3. Concrete evaluations

The functions that are of interest in the Ising model are
$\phi=\widehat{\phi}$ given by (1.28),

(3.22) $\displaystyle\phi(z)=\sqrt{\frac{1-k^{-1}z^{-1}}{1-k^{-1}z}},\qquad
k>1,$

and the function $\hat{\psi}$ given by (1.31). Apart from a constant factor,
this function can be written as

$\psi(z)=\frac{\phi(z)z-\phi(c)c}{z-c}$

with $c=c_{*}<0$. Notice that $\phi$ is analytic (and nonzero) on $\mathbb{C}$
except on the branch cut

$\Gamma_{k}:=[0,k^{-1}]\cup[k,+\infty).$

Therefore, being more general than necessary for the Ising model, we can also
allow for complex values $c\notin\Gamma_{k}$. Indeed, $\phi(c)$ is well-
defined, and therefore $\psi(z)$ is analytic on
$\mathbb{C}\setminus\Gamma_{k}$.

We can apply the formulas established in Theorem 3.5 and Corollary 3.6
directly to $\psi$, and this is what we will do below. Alternatively, we could
split $\psi$ into two terms

$\psi(z)=\phi(z)\frac{z}{z-c}-\frac{\phi(c)c}{z-c}$

This basically means that we deal with the functions

$\phi(z)\frac{z}{z-c}\quad\mbox{ and }\quad\frac{1}{z-c}.$

We do not have to exclude the values $c\in\Gamma_{k}$, but to exclude $|c|=1$
and distinguish the cases $|c|>1$ and $|c|<1$. In the latter case, the
asymptotics can be gleaned from Theorem 3.9 and it should be noted that
$F_{N}[\phi;\frac{1}{z-c}]=0$. In the former case, the asymptotics of
$F_{N}[\phi;\phi\frac{z}{z-c}]$ is discussed numerically in Section 4.2, but
we refrain from providing the rigorous details.

Note that $\phi$ has Wiener-Hopf factors given by

(3.23)
$\displaystyle\phi_{+}(z)=(1-k^{-1}z)^{-1/2},\qquad\phi_{-}(z)=(1-k^{-1}z^{-1})^{1/2}.$

We see that

(3.24)
$\displaystyle\lambda_{N}(z)=z^{-N}\lambda(z),\quad\lambda(z)=\sqrt{(1-k^{-1}z^{-1})(1-k^{-1}z)}$

We start with the asymptotics of $D_{N}[\phi]$.

######  Theorem 3.8.

For $\phi$ given by (3.22) with $k>1$, we have that

$D_{N}[\phi]=(1-k^{-2})^{1/4}\left(1+\frac{1}{2\pi(1-k^{-2})^{2}}N^{-2}k^{-2N-2}(1+O(N^{-1}))\right),\qquad
N\to\infty.$

###### Proof.

We are going to use the BOCG identity stated in Proposition 3.4(i). A
straightforward evaluation of the constants gives $G[\phi]=1$ and
$E[\phi]=(1-k^{-2})^{1/4}$. Thus we are left with analyzing

$\mathrm{det}(I-K_{N})=1-\mathrm{trace}\,K_{N}+O(\|K_{N}\|_{1}^{2}),\qquad
N\to\infty.$

Let us first estimate the trace norm of the operator
$K_{N}=V_{-N}H(\lambda)H(\tilde{\lambda}^{-1})V_{N}$. Since $\lambda(z)$ is
analytic on the annulus $k^{-1}<|z|<k$, the Fourier coefficients decay as
$[\lambda]_{n}=O(\kappa^{|n|})$ as $|n|\to\infty$ for each fixed
$\kappa>k^{-1}$. A straightforward computation of the Hilbert-Schmidt norm of
the Hankel operators appearing in $K_{N}$ implies that the trace norm of
$K_{N}$ decays exponentially as

$\|K_{N}\|_{1}=O(\kappa^{2N}),\qquad N\to\infty.$

As a consequence the term $O(\|K_{N}\|_{1}^{2})$ is negligible in comparison
to the other expected terms.

Let us finally compute the asymptotics of the trace of $K_{N}$. Obviously,

$\displaystyle\mathrm{trace}\,K_{N}$
$\displaystyle=\sum_{n=N}^{\infty}\sum_{j=0}^{\infty}[\lambda]_{n+j+1}[\lambda^{-1}]_{-n-j-1}=\sum_{n,j=0}^{\infty}[\lambda]_{n+j+1+N}[\lambda^{-1}]_{-n-j-1-N}$
$\displaystyle=\sum_{n=0}^{\infty}(n+1)[\lambda]_{n+1+N}[\lambda^{-1}]_{-n-1-N}.$

In view of (3.24), the asymptotics of the Fourier coefficients of $\lambda$
and $\lambda^{-1}$ is given by

(3.25) $\displaystyle[\lambda]_{n}$
$\displaystyle=\frac{\sqrt{1-k^{-2}}}{\Gamma(-1/2)}n^{-3/2}k^{-n}\Big{(}1+O(n^{-1})\Big{)},$
(3.26) $\displaystyle[\lambda^{-1}]_{-n}=[\lambda^{-1}]_{n}$
$\displaystyle=\frac{1}{\Gamma(1/2)\sqrt{1-k^{-2}}}n^{-1/2}k^{-n}\Big{(}1+O(n^{-1})\Big{)},$

as $n\to\infty$. Here we used Lemma 5.3 with $b=k$, $\zeta_{0}(z)=0$ and
$\omega=1/2$, $\xi(z)=k^{-1/2}(1-k^{-1}z^{-1})^{1/2}$ in the first case and
$\omega=-1/2$, $\xi(z)=k^{1/2}(1-k^{-1}z^{-1})^{-1/2}$ in the second case.
Hence

$\displaystyle\mathrm{trace}\,K_{N}$
$\displaystyle=\sum_{n=0}^{\infty}\frac{(n+1)}{\Gamma(-1/2)\Gamma(1/2)}(n+N+1)^{-2}k^{-2(N+n+1)}\Big{(}1+O((n+N)^{-1})\Big{)}$
$\displaystyle=-\frac{1}{2\pi(1-k^{-2})^{2}}N^{-2}k^{-2N-2}\Big{(}1+O(N^{-1})\Big{)},\qquad
N\to\infty,$

by Lemma 5.2. Combining all this we arrive at

$\mathrm{det}(I-K_{N})=1+\frac{1}{2\pi(1-k^{-2})^{2}}N^{-2}k^{-2N-2}\Big{(}1+O(N^{-1})\Big{)},\qquad
N\to\infty,$

and this proves the assertion. ∎

Let us now turn to the asymptotics of $F_{N}[\phi;\psi]$ in a setting which is
slightly more general than necessary for the Ising model.

######  Theorem 3.9.

For $\phi$ given by (3.22) with $k>1$, $c\in\mathbb{C}\setminus[k,+\infty)$,
let

$\psi(z)=\frac{\phi(z)z-d}{z-c}$

where

$d=\begin{cases}\phi(c)c&\mbox{ if }|c|\geq 1\\\ \mbox{arbitrary}&\mbox{ if
}|c|<1.\end{cases}$

Then, as $N\to\infty$,

$\displaystyle F_{N}[\phi;\psi]$
$\displaystyle=\frac{k^{1/2}}{(k-c)^{1/2}}-\frac{ck^{1/2}}{2\pi(k-c)^{3/2}(1-k^{-2})}N^{-2}k^{-2N}\Big{(}1+O(N^{-1})\Big{)}.$

###### Proof.

We are going to use Corollary 3.6 and start with identifying the functions
therein. To compute $p_{+}$ recall (3.11) and (3.12) to see that the latter
decomposition, $\phi_{-}^{-1}\psi=\phi_{+}p_{+}+\phi_{-}^{-1}p_{-}$ is given
by

$\frac{\phi_{+}(z)z-\phi_{-}^{-1}(z)d}{z-c}=\frac{\phi_{+}(z)z-\phi_{+}(c)c}{z-c}+\frac{\phi_{+}(c)c-\phi_{-}^{-1}(z)d}{z-c}.$

The first term is analytic for $|z|<k$, while second term is analytic for
$|z|>1-{\varepsilon}$ and vanishes at $z=\infty$. Hence

$p_{+}(z)=\frac{z-\phi_{+}^{-1}(z)\phi_{+}(c)c}{z-c},$

and

$[p_{+}]_{0}=p_{+}(0)=\frac{\phi_{+}(c)}{\phi_{+}(0)}=\frac{k^{1/2}}{(k-c)^{1/2}}.$

Furthermore,

$\phi_{-}(z)p_{+}(z)=\phi_{-}(z)\frac{z-\phi_{+}^{-1}(z)\phi_{+}(c)c}{z-c}=(1-k^{-1}z^{-1})^{1/2}\frac{z-(1-k^{-1}z)^{1/2}(1-k^{-1}c)^{-1/2}c}{z-c}.$

Lemma 5.3 with $\omega=1/2$, $b=k$,

$\xi(z)=-\frac{(1-k^{-1}z^{-1})^{1/2}(k-c)^{-1/2}c}{z-c},\qquad\xi(k)=-\frac{(1-k^{-2})^{1/2}c}{(k-c)^{3/2}},$

gives

$[\phi_{-}p_{+}]_{n}=-\frac{(1-k^{-2})^{1/2}c}{\Gamma(-1/2)(k-c)^{3/2}}n^{-3/2}k^{-n+1/2}\Big{(}1+O(n^{-1})\Big{)},\quad
n\to\infty.$

Hence, together with (3.26),

$[\lambda^{-1}]_{-n}[\phi_{-}p_{+}]_{n}=\frac{c}{2\pi(k-c)^{3/2}}n^{-2}k^{-2n+1/2}\Big{(}1+O(n^{-1})\Big{)},\quad
n\to\infty.$

Therefore, we get

$\displaystyle e_{0}^{T}T(\lambda^{-1}_{N})T(\phi_{-}z^{-N}p_{+})e_{0}$
$\displaystyle=\sum_{n=0}^{\infty}[\lambda^{-1}]_{-n-N}\Big{[}\phi_{-}p_{+}\Big{]}_{n+N}$
$\displaystyle=\frac{ck^{1/2}}{2\pi(k-c)^{3/2}(1-k^{-2})}N^{-2}k^{-2N}(1+O(N^{-1}))$

using Lemma 5.1. Noting that $[\phi_{+}]_{0}=\phi_{+}(0)=1$ and that the error
term in Corollary 3.6 decays even as $O(\kappa^{4N})$ (for any fixed
$k^{-1}<\kappa<1$), proves the asymptotics. ∎

######  Corollary 3.10.

Let $\phi=\widehat{\phi}$ be given by (3.22) with $k=S_{h}S_{v}>1$ and

$\widehat{\psi}(z)=r\frac{\phi(z)z-\phi(c_{*})c_{*}}{z-c_{*}}$

with $r=C_{v}/S_{v}$ and $c_{*}=-S_{h}/S_{v}$. Then, as $N\to\infty$,

(3.27)
$\displaystyle\frac{D_{N}^{B}[\widehat{\phi};\widehat{\psi}]}{D_{N}[\widehat{\phi}]}$
$\displaystyle=1+\frac{1}{2\pi
C_{v}^{2}(1-k^{-2})}N^{-2}k^{-2N}\Big{(}1+O(N^{-1})\Big{)},$ (3.28)
$\displaystyle D_{N}^{B}[\widehat{\phi};\widehat{\psi}]$
$\displaystyle=(1-k^{-2})^{1/4}\left(1+\frac{1}{2\pi(1-k^{-2})}\Big{(}\frac{1}{C_{v}^{2}}+\frac{1}{k^{2}-1}\Big{)}N^{-2}k^{-2N}\Big{(}1+O(N^{-1})\Big{)}\right).$

###### Proof.

We notice that

$\frac{k^{1/2}}{(k-c_{*})^{1/2}}=\frac{S_{v}}{C_{v}}=\frac{1}{r},\qquad\frac{-c_{*}}{k-c_{*}}=\frac{1}{C_{v}^{2}}.$

The rest is straightforward computation. ∎

With this computation we have proved the final two theorems stated in the
introduction.

##  4\. Numerical Verifications

In this section, we assume that $\phi\equiv\widehat{\phi}$, the symbol for the
Ising model defined by (1.28). To fix the problem, we set
$\frac{J_{h}}{k_{B}}=\frac{1}{2}$ and $\frac{J_{v}}{k_{B}}=\frac{1}{4}$ for
the $J_{h}>J_{v}$ case and $\frac{J_{h}}{k_{B}}=\frac{1}{4}$ and
$\frac{J_{v}}{k_{B}}=\frac{1}{2}$ for the $J_{h}<J_{v}$ case. Solving (1.23)
numerically in both cases, we get $T_{c}=0.820508964964\cdots$. We thus fix
$T=\frac{4}{5}<T_{c}$ in the following numerical verifications, which ensures
that $\widehat{\phi}$ is of Szegő type. Then, we have
$k=\sinh(\frac{2J_{h}}{k_{B}T})\sinh(\frac{2J_{v}}{k_{B}T})\approx
1.067666675$, which is, as expected, bigger than $1$. In fact, the reason why
we choose $T$ so close to $T_{c}$ is that the error terms often have factors
of the form $N^{-m}k^{-nN}$, so a $k$ slightly greater than $1$ will guarantee
the results being not so small for relatively large $N$.

For computing $D_{N}^{B}[\phi;\psi]$ and $D_{N}[\phi]$, from (1.2) we first
compute $\phi_{j}$, $j=1-N,\cdots,N-1$, and $\psi_{j}$, $j=0,\cdots N-1$, by
the trapezoidal rule up to precision of more than $100$ digits (which is far
more than needed in the following calculations). Then we compute
$D_{N}^{B}[\phi;\psi]$ and $D_{N}[\phi]$ directly from (1.1) and (1.3)
respectively.

###  4.1. Verification of (1.33)

Let us define

$G_{N}^{A}:=\left(\frac{D_{N}^{B}[\widehat{\phi};\widehat{\psi}]}{\sqrt[4]{1-k^{-2}}}-1\right)\frac{2\pi(1-k^{-2})N^{2}k^{2N}}{C_{v}^{-2}+(k^{2}-1)^{-1}}.$

Then formula (1.33) is equivalent to

(4.1) $\displaystyle G_{N}^{A}=1+O(N^{-1}).$

$N$$500$$1000$$0.8$$0.9$$1.0$$G_{N}^{A}$$\frac{J_{h}}{k_{B}}=\frac{1}{2}$$\frac{J_{v}}{k_{B}}=\frac{1}{4}$$T=\frac{4}{5}$
Figure 1. Plot of
$G_{N}^{A}=(\frac{D_{N}^{B}[\widehat{\phi};\widehat{\psi}]}{\sqrt[4]{1-k^{-2}}}-1)\frac{2\pi(1-k^{-2})N^{2}k^{2N}}{C_{v}^{-2}+(k^{2}-1)^{-1}}$.
Fitting the $10$ points by $\sum\limits_{i=0}^{9}g_{-i}N^{-i}$, we get
$g_{0}\approx 1-5.643811*10^{-11}$, $g_{-1}\approx-25.367279$, $g_{-2}\approx
798.967$, $g_{-3}\approx-30863.1$, $g_{-4}\approx 1.42051\times 10^{6}$,
$g_{5}\approx-7.55845\times 10^{7}$, $g_{-6}\approx 4.41553\times 10^{9}$,
$g_{-7}\approx-2.51667\times 10^{11}$, $g_{-8}\approx 1.12018\times 10^{13}$,
$g_{-9}\approx-2.59581\times 10^{14}$. Appending more points such as adding
$G_{N}^{A}$ for $N=250$, $g_{i}$ will change with an obvious pattern: the
smaller $i$ is, the smaller percent change is. For example, the change of
$g_{0}$ is smaller than $10^{-10}$, and $g_{-6}$ will change to $4.55987\times
10^{9}$ while $g_{-9}$ will change to $-7.08571\times 10^{14}$.

Figure 1 is the numerical result for a case $J_{h}>J_{v}$ with
$\frac{J_{h}}{k_{B}}=\frac{1}{2}$ and $\frac{J_{v}}{k_{B}}=\frac{1}{4}$.
$g_{0}=1$ and a finite fixed $g_{-1}$ show asymptotics (4.1) is indeed right.
A more careful look into the numerical values of $g_{i}$ suggests that $\sum
g_{-i}N^{-i}$ is an asymptotic series.

In principle, $G_{N}^{A}$ is only defined on integer $N$. The red line is a
smooth link of the ten points obtained by numerical experiments. The numerical
values of $G_{N}^{A}$ for other integer $N$ will be visually indistinguishable
from the points on the red line.

$N$$500$$1000$$0.8$$0.9$$1.0$$G_{N}^{A}$$\frac{J_{h}}{k_{B}}=\frac{1}{4}$$\frac{J_{v}}{k_{B}}=\frac{1}{2}$$T=\frac{4}{5}$
Figure 2. Plot of
$G_{N}^{A}=(\frac{D_{N}^{B}[\widehat{\phi};\widehat{\psi}]}{\sqrt[4]{1-k^{-2}}}-1)\frac{2\pi(1-k^{-2})N^{2}k^{2N}}{C_{v}^{-2}+(k^{2}-1)^{-1}}$.
Fitting the $10$ points by $\sum\limits_{i=0}^{9}g_{-i}N^{-i}$, we get
$g_{0}\approx 1-5.92566\times 10^{-11}$, $g_{-1}\approx-26.191288$,
$g_{-2}\approx 830.85573$, $g_{-3}\approx-32206.7$, $g_{-4}\approx
1.48546\times 10^{6}$, $g_{-5}\approx-7.91507\times 10^{7}$, $g_{-6}\approx
4.62828\times 10^{9}$, $g_{-7}\approx-2.63956\times 10^{11}$, $g_{-8}\approx
1.1753\times 10^{13}$, $g_{-9}\approx-2.72407\times 10^{14}$.

Figure 2 is the numerical result for a case $J_{h}<J_{v}$ with
$\frac{J_{h}}{k_{B}}=\frac{1}{4}$ and $\frac{J_{v}}{k_{B}}=\frac{1}{2}$.
Numerical values of $g_{-i}$ also show the series $\sum g_{-i}N^{-i}$ is an
asymptotic one. $g_{0}=1$ and a finite fixed $g_{-1}$ show (4.1) is also true
for this case.

###  4.2. The sensitivity for the case $\psi=\widehat{\phi}\frac{z}{z-c}$ with
$c$ near $1$

If $c<1$, $F_{N}[\widehat{\phi};\psi]$ is given by Theorem 3.9. Also recall
from (3.2) and (3.4) that

$F_{N}[\phi;\psi]=\frac{D_{N}^{B}[\phi;\psi]}{D_{N}[\phi]}.$

So Theorem 3.9 means that

(4.2)
$G_{N}^{B}:=-\left(\frac{D_{N}^{B}[\phi;\psi]}{D_{N}[\phi]}-\frac{k^{1/2}}{(k-c)^{1/2}}\right)\frac{2\pi(k-c)^{3/2}(1-k^{-2})}{ck^{1/2}}N^{2}k^{2N}=1+O(N^{-1}).$

Figure 3 is the plot of $G_{N}^{B}$ with $c=\frac{975}{1000}$. $g_{0}=1$ and a
finite fixed $g_{-1}$ verify Theorem 3.9 numerically.

$N$$500$$1000$$0.8$$0.9$$1.0$$G_{N}^{B}$$\frac{J_{h}}{k_{B}}=\frac{1}{2}$$\frac{J_{v}}{k_{B}}=\frac{1}{4}$$T=\frac{4}{5}$$c=\frac{975}{1000}$
Figure 3. Plot of
$G_{N}^{B}=-\left(\frac{D_{N}^{B}[\widehat{\phi};\widehat{\psi}]}{D_{N}[\widehat{\phi}]}-\frac{k^{1/2}}{(k-c)^{1/2}}\right)\frac{2\pi(k-c)^{3/2}(1-k^{-2})}{ck^{1/2}}N^{2}k^{2N}$.
Fitting the $10$ points by $\sum\limits_{i=0}^{9}g_{-i}N^{-i}$, we get
$g_{0}\approx 1-1.5286\times 10^{-10}$, $g_{-1}\approx-27.7534$,
$g_{-2}\approx 938.882$, $g_{-3}\approx-39575.1$, $g_{-4}\approx 2.02757\times
10^{6}$, $g_{-5}\approx-1.22089\times 10^{8}$, $g_{-6}\approx 8.089\times
10^{9}$, $g_{-7}\approx-5.12614\times 10^{11}$, $g_{-8}\approx 2.45747\times
10^{13}$, $g_{-9}\approx-5.96334\times 10^{14}$.

It is not surprising that Figures 1, 2 and 3 look so similar since they all
have the same $g_{0}=1$ and similar $g_{-2}$ and $g_{-3}$.

Now, let us consider the case $c>1$. For $c>1$, we recall that
$p_{+}(z)=\frac{z}{z-c}$. Now, let us compute $F_{N}[\psi;\widehat{\phi}]$ by
Corollary 3.6. First, $[p_{+}]_{0}=p_{+}(0)=0$. Next,
$\widehat{\phi}_{-}p_{+}=\sqrt{1-k^{-1}z^{-1}}\frac{z}{z-c}$. Therefore,

(4.3)
$\displaystyle[\widehat{\phi}_{-}p_{+}]_{n}=\int_{\mathbb{T}}\sqrt{1-\frac{1}{kz}}\frac{z^{-n}}{z-c}\frac{dz}{2\pi
i}=-\sqrt{1-\frac{1}{kc}}c^{-n},\quad\text{for }n>1.$

Recall that

$[\lambda^{-1}]_{-n}=\int_{\mathbb{T}}\frac{z^{n}}{\sqrt{(1-k^{-1}z^{-1})(1-k^{-1}z)}}\frac{dz}{2\pi
iz}=\frac{1}{\pi}\int_{0}^{\frac{1}{k}}\frac{z^{n-1}}{\sqrt{(k^{-1}z^{-1}-1)(1-k^{-1}z)}}dz.$

We get

(4.4)
$\displaystyle\sum_{n=N}^{\infty}-[\lambda^{-1}]_{-n}[\widehat{\phi}_{-}p_{+}]_{n}$
$\displaystyle=$
$\displaystyle\frac{1}{\pi}\sqrt{1-\frac{1}{kc}}\int_{0}^{\frac{1}{k}}\frac{\sum\limits_{n=N}^{\infty}z^{n-1}c^{-n}}{\sqrt{(k^{-1}z^{-1}-1)(1-k^{-1}z)}}dz$
$\displaystyle=$
$\displaystyle\frac{1}{\pi}\sqrt{1-\frac{1}{kc}}\int_{0}^{\frac{1}{k}}\frac{\left(\frac{z}{c}\right)^{N-1}\frac{1}{c}}{\left(1-\frac{z}{c}\right)\sqrt{(k^{-1}z^{-1}-1)(1-k^{-1}z)}}dz$
$\displaystyle=$
$\displaystyle\frac{1}{\pi}c^{-N}k^{-N}\sqrt{1-\frac{1}{kc}}\int_{0}^{1}\frac{t^{N-\frac{1}{2}}}{\left(1-\frac{t}{kc}\right)\sqrt{(1-t)(1-k^{-2}t)}}dt$
(4.5) $\displaystyle\approx$ $\displaystyle\frac{1}{\sqrt{\pi
N}}\frac{c^{-N}k^{-N}}{\sqrt{1-\frac{1}{kc}}\sqrt{1-k^{-2}}}.$

(4.4) is the exact value of the second term in Corollary 3.6. Actually, we do
not use (4.5) since (4.4) itself can be calculated directly. Let us define
$\Delta_{N}$ as

(4.6)
$\Delta_{N}:=\frac{1}{\pi}c^{-N}k^{-N}\sqrt{1-\frac{1}{kc}}\int_{0}^{1}\frac{t^{N-\frac{1}{2}}}{\left(1-\frac{t}{kc}\right)\sqrt{(1-t)(1-k^{-2}t)}}dt-\frac{D_{N}^{B}[\widehat{\phi};\psi]}{D_{N}[\widehat{\phi}]}.$

Then $\Delta_{N}$ is the negative of the third term in Corollary 3.6.

$N$$500$$1000$$-100$$-200$$\ln\Delta_{N}$$\frac{J_{h}}{k_{B}}=\frac{1}{2}$$\frac{J_{v}}{k_{B}}=\frac{1}{4}$$T=\frac{4}{5}$$c=\frac{1025}{1000}$
Figure 4. Plot of $\ln\Delta_{N}$, where
$\Delta_{N}=\frac{1}{\pi}c^{-N}k^{-N}\sqrt{1-\frac{1}{kc}}\int_{0}^{1}\frac{t^{N-\frac{1}{2}}}{\left(1-\frac{t}{kc}\right)\sqrt{(1-t)(1-k^{-2}t)}}dt-\frac{D_{N}^{B}[\widehat{\phi};\psi]}{D_{N}[\widehat{\phi}]}$.
Fitting the $10$ points by $g_{1}N+g_{L}\ln
N+\sum\limits_{i=0}^{7}g_{-i}N^{-i}$, we get $g_{1}\approx-0.2211193827735$,
$g_{L}\approx-2.499999847$ and $g_{0}\approx 3.928447182$. Notice that the
numerical $g_{1}$ and $g_{L}$ obtained by the fitting of the $10$ points are
very close to $3\ln k^{-1}+\ln c^{-1}\approx-0.22111938274235$ and
$-\frac{5}{2}$. This suggests that $g_{1}=3\ln k^{-1}+\ln c^{-1}$ and
$g_{L}=-\frac{5}{2}$ are exact.

Figure 4 is the plot of $\Delta_{N}$ with $c=\frac{1025}{1000}$. The numerical
results mean

(4.7)
$\frac{D_{N}^{B}[\widehat{\phi};\psi]}{D_{N}[\widehat{\phi}]}=\frac{1}{\pi}c^{-N}k^{-N}\sqrt{1-\frac{1}{kc}}\int_{0}^{1}\frac{t^{N-\frac{1}{2}}}{\left(1-\frac{t}{kc}\right)\sqrt{(1-t)(1-k^{-2}t)}}dt+O(N^{-\frac{5}{2}}c^{-N}k^{-3N}),$

in this case.

##  5\. Appendices

###  5.1. Solution of the Riemann-Hilbert problem for BOPUC with Szegő-type
symbols

The following Riemann-Hilbert problem for BOPUC is due to J.Baik, P.Deift and
K.Johansson.

  * •

RH-X1 $X:\mathbb{C}\setminus\mathbb{T}\to\mathbb{C}^{2\times 2}$ is analytic,

  * •

RH-X2 The limits of $X(\zeta)$ as $\zeta$ tends to $z\in\mathbb{T}$ from the
inside and outside of the unit circle exist, and are denoted $X_{\pm}(z)$
respectively and are related by

(5.1) $X_{+}(z)=X_{-}(z)\begin{pmatrix}1&z^{-n}\phi(z)\\\
0&1\end{pmatrix},\qquad z\in\mathbb{T},$

  * •

RH-X3 As $z\to\infty$

(5.2) $X(z)=\big{(}I+O(z^{-1})\big{)}z^{n\sigma_{3}},$

(see [Dei99],[DIK11],[CIK11]). Below we show the standard steepest descent
analysis to asymptotically solve this problem, in the case where $\phi$ is a
symbol analytic in a neighborhood of the unit circle and with zero winding
number. Note that the symbol $\phi$ associated to the 2D Ising model in the
low temperature regime enjoys these properties. We first normalize the
behavior at $\infty$ by defining

(5.3) $T(z;n):=\begin{cases}X(z;n)z^{-n\sigma_{3}},&|z|>1,\\\
X(z;n),&|z|<1.\end{cases}$

The function $T$ defined above satisfies the following RH problem

  * •

RH-T1 $T(\cdot;n):\mathbb{C}\setminus\mathbb{T}\to\mathbb{C}^{2\times 2}$ is
analytic,

  * •

RH-T2 $T_{+}(z;n)=T_{-}(z;n)\begin{pmatrix}z^{n}&\phi(z)\\\
0&z^{-n}\end{pmatrix},\qquad z\in\mathbb{T}$,

  * •

RH-T3 $T(z;n)=I+O(1/z),\qquad z\to\infty,$

So $T$ has a highly-oscillatory jump matrix as $n\to\infty$. The next
transformation yields a Riemann Hilbert problem, normalized at infinity,
having an exponentially decaying jump matrix on the lenses. Note that we have
the following factorization of the jump matrix of the $T$-RHP:

(5.4) $\begin{pmatrix}z^{n}&\phi(z)\\\
0&z^{-n}\end{pmatrix}=\begin{pmatrix}1&0\\\
z^{-n}\phi(z)^{-1}&1\end{pmatrix}\begin{pmatrix}0&\phi(z)\\\
-\phi(z)^{-1}&0\end{pmatrix}\begin{pmatrix}1&0\\\
z^{n}\phi(z)^{-1}&1\end{pmatrix}\equiv J_{0}(z;n)J^{(\infty)}(z)J_{1}(z;n).$

Now, we define the following function :

(5.5) $S(z;n):=\begin{cases}T(z;n)J^{-1}_{1}(z;n),&z\in\Omega_{1},\\\
T(z;n)J_{0}(z;n),&z\in\Omega_{2},\\\
T(z;n),&z\in\Omega_{0}\cup\Omega_{\infty}.\end{cases}$

Also introduce the following function on
$\Gamma_{S}:=\Gamma_{0}\cup\Gamma_{1}\cup\mathbb{T}$

(5.6) $J_{S}(z;n)=\begin{cases}J_{1}(z;n),&z\in\Gamma_{0},\\\
J^{(\infty)}(z),&z\in\mathbb{T},\\\ J_{0}(z;n),&z\in\Gamma_{1}.\\\
\end{cases}$

$\Omega_{1}$$\Omega_{2}$$\Gamma_{1}$$\mathbb{T}$$\Gamma_{0}$$\Omega_{0}$$\Omega_{\infty}$$\Gamma_{S}\
:$ Figure 5. Opening of lenses: the jump contour for the $S$-RHP.

We have the following Riemann-Hilbert problem for $S(z;n)$

  * •

RH-S1 $S(\cdot;n):\mathbb{C}\setminus\Gamma_{S}\to\mathbb{C}^{2\times 2}$ is
analytic.

  * •

RH-S2 $S_{+}(z;n)=S_{-}(z;n)J_{S}(z;n),\qquad z\in\Gamma_{S}.$

  * •

RH-S3 $S(z;n)=I+O(1/z),\qquad\text{as}\ z\to\infty$.

Note that the matrices $J_{0}(z;n)$ and $J_{1}(z;n)$ tend to the identity
matrix uniformly on their respective contours, exponentially fast as
$n\to\infty$.

####  5.1.1. Global parametrix RHP

We are looking for a piecewise analytic function
$P^{(\infty)}(z):\mathbb{C}\setminus\mathbb{T}:\to\mathbb{C}^{2\times 2}$ such
that

  * •

RH-Global1 $P^{(\infty)}$ is holomorphic in $\mathbb{C}\setminus\mathbb{T}$.

  * •

RH-Global2 for $z\in\mathbb{T}$ we have

(5.7) $P_{+}^{(\infty)}(z)=P_{-}^{(\infty)}(z)\begin{pmatrix}0&\phi(z)\\\
-\phi^{-1}(z)&0\end{pmatrix}.$

  * •

RH-Global3 $P^{(\infty)}(z)=I+O(1/z),\qquad\text{as}\ z\to\infty$.

We can find a piecewise analytic function $\alpha$ which solves the following
scalar multiplicative Riemann-Hilbert problem

(5.8) $\alpha_{+}(z)=\alpha_{-}(z)\phi(z)\qquad z\in\mathbb{T}.$

By Plemelj-Sokhotski formula we have

(5.9) $\alpha(z)=\exp\left[\frac{1}{2\pi
i}\int_{\mathbb{T}}\frac{\ln(\phi(\tau))}{\tau-z}d\tau\right],$

Now, using (5.8) we have the following factorization

(5.10) $\begin{pmatrix}0&\phi(z)\\\
-\phi^{-1}(z)&0\end{pmatrix}=\begin{pmatrix}\alpha_{-}^{-1}(z)&0\\\
0&\alpha_{-}(z)\end{pmatrix}\begin{pmatrix}0&1\\\
-1&0\end{pmatrix}\begin{pmatrix}\alpha^{-1}_{+}(z)&0\\\
0&\alpha_{+}(z)\end{pmatrix}.$

So, the function

(5.11) $P^{(\infty)}(z):=\begin{cases}\begin{pmatrix}0&\alpha(z)\\\
-\alpha^{-1}(z)&0\end{pmatrix},&|z|<1,\\\ \begin{pmatrix}\alpha(z)&0\\\
0&\alpha^{-1}(z)\end{pmatrix},&|z|>1,\end{cases}$

satisfies (5.7). Also, by the properties of the Cauchy integral,
$P^{(\infty)}(z)$ is holomorphic in $\mathbb{C}\setminus\mathbb{T}$. Moreover,
$\alpha(z)=1+O(z^{-1})$, as $z\to\infty$ and hence

(5.12) $P^{(\infty)}(z)=I+O(1/z),\qquad z\to\infty.$

Therefore $P^{(\infty)}$ given by (5.11) is the unique solution of the global
parametrix Riemann-Hilbert problem.

####  5.1.2. Small-norm RHP

Let us consider the ratio

(5.13) $R(z;n):=S(z;n)\left[P^{(\infty)}(z)\right]^{-1}.$

We have the following Riemann-Hilbert problem for $R(z;n)$

  * •

RH-R1 $R$ is holomorphic in $\mathbb{C}\setminus(\Gamma_{0}\cup\Gamma_{1})$.

  * •

RH-R2 $R_{+}(z;n)=R_{-}(z;n)J_{R}(z;n),\qquad
z\in\Gamma_{0}\cup\Gamma_{1}=:\Sigma_{R}$,

  * •

RH-R3 $R(z;n)=I+O(1/z)\qquad\text{as}\ z\to\infty$.

This Riemann Hilbert problem is solvable for large $n$ ([DKM+99a],[DKM+99b])
and $R(z;n)$ can be written as

(5.14) $R(z;n)=I+R_{1}(z;n)+R_{2}(z;n)+R_{3}(z;n)+\cdots,\ \ \ \ \ \ n\geq
n_{0}$

where $R_{k}$ can be found recursively. Indeed

(5.15)
$R_{k}(z;n)=\frac{1}{2\pi\mathrm{i}}\int_{\Sigma_{R}}\frac{\left[R_{k-1}(\mu;n)\right]_{-}\left(J_{R}(\mu;n)-I\right)}{\mu-z}\mathrm{d}\mu,\qquad
z\in\mathbb{C}\setminus\Sigma_{R},\qquad k\geq 1.$

It is easy to check that $R_{2\ell}(z;n)$ is diagonal and $R_{2\ell+1}(z;n)$
is off-diagonal; $\ell\in\mathbb{N}\cup\\{0\\}$, and that

(5.16) $R_{k,ij}(z;n)=\frac{O(\rho^{-kn})}{1+|z|},\qquad n\to\infty,\qquad
k\geq 1,\qquad z\in\mathbb{C}\setminus\Sigma_{R},$

where $\rho$ (resp. $\rho^{-1}$) is the radius of $\Gamma_{1}$(resp.
$\Gamma_{0}$). Let us compute $R_{1}(z;n)$; we have

(5.17) $J_{R}(z)-I=\begin{cases}P^{(\infty)}(z)\begin{pmatrix}0&0\\\
z^{n}\phi^{-1}(z)&0\end{pmatrix}\left[P^{(\infty)}(z)\right]^{-1},&z\in\Gamma_{0},\\\
P^{(\infty)}(z)\begin{pmatrix}0&0\\\
z^{-n}\phi^{-1}(z)&0\end{pmatrix}\left[P^{(\infty)}(z)\right]^{-1},&z\in\Gamma_{1},\end{cases}=\begin{cases}\begin{pmatrix}0&-z^{n}\phi^{-1}(z)\alpha^{2}(z)\\\
0&0\end{pmatrix},&z\in\Gamma_{0},\\\ \begin{pmatrix}0&0\\\
z^{-n}\phi^{-1}(z)\alpha^{-2}(z)&0\end{pmatrix},&z\in\Gamma_{1}.\end{cases}$

Therefore

(5.18) $R_{1}(z;n)=\begin{pmatrix}0&-\displaystyle\frac{1}{2\pi
i}\int_{\Gamma_{0}}\frac{\tau^{n}\phi^{-1}(\tau)\alpha^{2}(\tau)}{\tau-z}d\tau\\\
\displaystyle\frac{1}{2\pi
i}\int_{\Gamma_{1}}\frac{\tau^{-n}\phi^{-1}(\tau)\alpha^{-2}(\tau)}{\tau-z}d\tau&0\end{pmatrix}.$

####  5.1.3. Tracing back Riemann-Hilbert transformations

If we trace back the Riemann-Hilbert problems $R\mapsto S\mapsto T\mapsto Y$
we will obtain

(5.19) $X(z;n)=R(z;n)\begin{cases}\begin{pmatrix}\alpha(z)&0\\\
0&\alpha^{-1}(z)\end{pmatrix}z^{n\sigma_{3}},&z\in\Omega_{\infty},\\\
\begin{pmatrix}\alpha(z)&0\\\
-z^{-n}\alpha^{-1}(z)\phi^{-1}(z)&\alpha^{-1}(z)\end{pmatrix}z^{n\sigma_{3}},&z\in\Omega_{2},\\\
\begin{pmatrix}z^{n}\alpha(z)\phi^{-1}(z)&\alpha(z)\\\
-\alpha^{-1}(z)&0\end{pmatrix},&z\in\Omega_{1},\\\
\begin{pmatrix}0&\alpha(z)\\\
-\alpha^{-1}(z)&0\end{pmatrix},&z\in\Omega_{0},\\\ \end{cases}$

where for $z\in\mathbb{C}\setminus\Sigma_{R}$, as $n\to\infty$, we have

(5.20)
$R(z;n)=\displaystyle\begin{pmatrix}1+\frac{O(\rho^{-2n})}{1+|z|}&R_{1,12}(z;n)+\frac{O(\rho^{-3n})}{1+|z|}\\\
R_{1,21}(z;n)+\frac{O(\rho^{-3n})}{1+|z|}&1+\frac{O(\rho^{-2n})}{1+|z|}\end{pmatrix}.$

###  5.2. Proof of Theorem 1.2 using the Riemann-Hilbert approach

For the inverse of a Toeplitz matrix
$T_{n}[\phi]=\\{\phi_{j-k}\\}^{n-1}_{j,k=0}$, we have

(5.21)
$\left(T^{-1}_{n}[\phi]\right)_{j,k}=\delta_{jk}+\langle\boldsymbol{\mathcal{R}^{(\phi)}_{n}}[z^{k}],z^{j}\rangle,\qquad
0\leq j,k\leq n-1,$

where $\delta_{jk}$ is the Kronecker delta function,

(5.22) $\langle
f(z),g(z)\rangle=\int_{\mathbb{T}}f(z)\overline{g(z)}\frac{\mathrm{d}z}{2\pi\mathrm{i}z},$

and

(5.23)
$\boldsymbol{\mathcal{R}^{(\phi)}_{n}}:f(z)\mapsto\int_{\mathbb{T}}\mathcal{R}^{(\phi)}_{n}(z,w)f(w)\mathrm{d}w$

is the Resolvent operator with the kernel

(5.24)
$\mathcal{R}^{(\phi)}_{n}(z,w)=\frac{X^{(\phi)}_{11}(z)X^{(\phi)}_{21}(w)-X^{(\phi)}_{11}(w)X^{(\phi)}_{21}(z)}{z-w}\frac{\phi(w)-1}{2\pi\mathrm{i}w^{n}},$

where $X^{(\phi)}_{11}(z)\equiv X^{(\phi)}_{11}(z;n)$ and
$X^{(\phi)}_{12}(z)\equiv X^{(\phi)}_{12}(z;n)$ are the entries of the
solution to the RH-X1 through RH-X3. In terms of the associated biorthogonal
polynomials, in view of (2.6), we can write

(5.25)
$\mathcal{R}^{(\phi)}_{n}(z,w)=\frac{\sqrt{D_{n-1}[\phi]D_{n+1}[\phi]}}{D_{n}[\phi]}\frac{Q_{n}(w)Q^{*}_{n-1}(z)-Q_{n}(z)Q^{*}_{n-1}(w)}{z-w}\frac{\phi(w)-1}{2\pi\mathrm{i}w^{n}},$

where we have used the standard notation

$P^{*}_{n}(z):=z^{n}P_{n}(z^{-1})$

for a polynomial $P_{n}(z)$ of degree $n$.

Let $\vec{x}=(x_{0},x_{1},\cdots,x_{N-1})^{T}$ and
$\vec{\psi}=(\psi_{N-1},\psi_{N-2},\cdots,\psi_{0})^{T}$. Applying the
Cramer’s rule to the linear system $T_{n}[\tilde{\phi}]\vec{x}=\vec{\psi}$
gives

$x_{N-1}=\frac{\mathrm{det}\begin{pmatrix}\tilde{\phi}_{0}&\tilde{\phi}_{-1}&\cdots&\tilde{\phi}_{-N+2}&\psi_{N-1}\\\
\tilde{\phi}_{1}&\tilde{\phi}_{0}&\cdots&\tilde{\phi}_{-N+3}&\psi_{N-2}\\\
\vdots&\vdots&\cdots&\vdots&\vdots\\\
\tilde{\phi}_{N-1}&\tilde{\phi}_{N-2}&\cdots&\tilde{\phi}_{1}&\psi_{0}\end{pmatrix}}{D_{N}[\tilde{\phi}]}=\frac{\mathrm{det}\begin{pmatrix}\phi_{0}&\phi_{1}&\cdots&\phi_{N-2}&\psi_{N-1}\\\
\phi_{-1}&\phi_{0}&\cdots&\phi_{N-3}&\psi_{N-2}\\\
\vdots&\vdots&\cdots&\vdots&\vdots\\\
\phi_{1-N}&\phi_{2-N}&\cdots&\phi_{-1}&\psi_{0}\end{pmatrix}}{D_{N}[\phi]}.$

Comparing this with (1.1) we observe that

(5.26) $D^{B}_{N}[\phi;\psi]=D_{N}[\phi]x_{N-1}$

In view of (??), (5.21), and (5.26) we have

$F_{N}[\phi;\psi]\equiv
x_{N-1}=\sum_{\ell=0}^{N-1}\left(T^{-1}_{N}[\tilde{\phi}]\right)_{N-1,\ell}\psi_{N-1-\ell}=\sum_{\ell=0}^{N-1}\left(\delta_{N-1,\ell}+\left\langle\boldsymbol{\mathcal{R}^{(\tilde{\phi})}_{N}}[z^{\ell}],z^{N-1}\right\rangle\right)\psi_{N-1-\ell}.$

Thus

(5.27)
$F_{N}[\phi;\psi]=\psi_{0}+\sum_{\ell=0}^{N-1}\left\langle\boldsymbol{\mathcal{R}^{(\tilde{\phi})}_{N}}[z^{\ell}],z^{N-1}\right\rangle\psi_{N-1-\ell}$

From (5.23) and (5.24) we have

(5.28)
$\begin{split}&\mathcal{I}_{N}:=\sum_{\ell=0}^{N-1}\left\langle\boldsymbol{\mathcal{R}^{(\tilde{\phi})}_{N}}[z^{\ell}],z^{N-1}\right\rangle\psi_{N-1-\ell}\\\
&\sum_{\ell=0}^{N-1}\left\\{\int_{\mathbb{T}}\left(\int_{\mathbb{T}}\frac{X^{(\tilde{\phi})}_{11}(z)X^{(\tilde{\phi})}_{21}(w)-X^{(\tilde{\phi})}_{11}(w)X^{(\tilde{\phi})}_{21}(z)}{z-w}\frac{\tilde{\phi}(w)-1}{2\pi\mathrm{i}w^{N}}w^{\ell}\mathrm{d}w\right)z^{-N}\frac{\mathrm{d}z}{2\pi\mathrm{i}}\right\\}\psi_{N-1-\ell}.\end{split}$

Note that

(5.29)
$\sum^{N-1}_{\ell=0}w^{\ell-N}\psi_{N-1-\ell}=\frac{1}{w}\sum^{N-1}_{k=0}w^{-k}\psi_{k}=\frac{1}{w}\psi_{i}\left(\frac{1}{w}\right)+\mathcal{O}\left(e^{-c_{0}N}\right),$

where $c_{0}$ is some positive constant and

(5.30)
$\psi(z)=\sum^{\infty}_{k=0}\psi_{k}z^{k}+\sum^{\infty}_{k=1}\psi_{-k}z^{-k}\equiv\psi_{i}(z)+\psi_{o}(z).$

Therefore

(5.31)
$\mathcal{I}_{N}\simeq\int_{\mathbb{T}}\int_{\mathbb{T}}\frac{X^{(\tilde{\phi})}_{11}(z)X^{(\tilde{\phi})}_{21}(w)-X^{(\tilde{\phi})}_{11}(w)X^{(\tilde{\phi})}_{21}(z)}{z-w}\frac{\tilde{\phi}(w)-1}{2\pi\mathrm{i}w}\psi_{i}\left(\frac{1}{w}\right)z^{-N}\frac{\mathrm{d}w\mathrm{d}z}{2\pi\mathrm{i}}$

Let

$\mathcal{D}(z):=\exp\left[\frac{1}{2\pi\mathrm{i}}\int_{\mathbb{T}}\frac{\ln(\tilde{\phi}(\tau))}{\tau-z}\mathrm{d}\tau\right].$

One can easily check that

(5.32) $\mathcal{D}(z)=\frac{\alpha(0)}{\tilde{\alpha}(z)},$

where $\alpha$ is the Szegő function corresponding to the symbol $\phi$, given
by (5.9). For $\mathcal{D}$ we have

(5.33) $\mathcal{D}_{+}(z)=\mathcal{D}_{-}(z)\tilde{\phi}(z),\qquad
z\in\mathbb{T}.$

Recall from (5.16), (5.19), and (5.20) that

(5.34)
$X^{(\tilde{\phi})}_{11}(z)\simeq\mathcal{D}_{+}(z)z^{N}\tilde{\phi}^{-1}(z),\quad\text{and}\quad
X^{(\tilde{\phi})}_{11}(z)\simeq-\mathcal{D}^{-1}_{+}(z).$

Therefore

(5.35)
$\begin{split}\mathcal{I}_{N}&\simeq\int_{\mathbb{T}}\int_{\mathbb{T}}\frac{-\mathcal{D}^{-1}_{+}(w)\mathcal{D}_{+}(z)z^{N}\tilde{\phi}^{-1}(z)+\mathcal{D}^{-1}_{+}(z)\mathcal{D}_{+}(w)w^{N}\tilde{\phi}^{-1}(w)}{z-w}\frac{\tilde{\phi}(w)-1}{2\pi\mathrm{i}w}\psi_{i}\left(\frac{1}{w}\right)z^{-N}\frac{\mathrm{d}w\mathrm{d}z}{2\pi\mathrm{i}}\\\
&=\int_{\mathbb{T}}\int_{\mathbb{T}}\frac{-\mathcal{D}^{-1}_{+}(w)\mathcal{D}_{-}(z)+\mathcal{D}^{-1}_{-}(z)\mathcal{D}_{+}(w)\tilde{\phi}^{-1}(z)\tilde{\phi}^{-1}(w)(w/z)^{N}}{z-w}\frac{\tilde{\phi}(w)-1}{2\pi\mathrm{i}w}\psi_{i}\left(\frac{1}{w}\right)\frac{\mathrm{d}w\mathrm{d}z}{2\pi\mathrm{i}}\\\
\end{split}$

Now we deform the contour of integration for variables $w$ and $z$
respectively to the contours $\mathbb{T}_{+}$ and $\mathbb{T}_{-}$
respectively, where $\mathbb{T}_{+}$ is a circle with radius less than one in
the domain of analyticity of $\tilde{\phi}$ and $\psi$, and $\mathbb{T}_{-}$
is a circle with radius more than one in the domain of analyticity of
$\tilde{\phi}$ and $\psi$. So we have

(5.36)
$\begin{split}\mathcal{I}_{N}&\simeq\int_{\mathbb{T}_{-}}\int_{\mathbb{T}_{+}}\frac{-\mathcal{D}^{-1}(w)\mathcal{D}(z)+\mathcal{D}^{-1}(z)\mathcal{D}(w)\tilde{\phi}^{-1}(z)\tilde{\phi}^{-1}(w)(w/z)^{N}}{z-w}\frac{\tilde{\phi}(w)-1}{2\pi\mathrm{i}w}\psi_{i}\left(\frac{1}{w}\right)\frac{\mathrm{d}w\mathrm{d}z}{2\pi\mathrm{i}}\\\
&\simeq\int_{\mathbb{T}_{-}}\int_{\mathbb{T}_{+}}\frac{-\mathcal{D}^{-1}(w)\mathcal{D}(z)}{z-w}\frac{\tilde{\phi}(w)-1}{(2\pi\mathrm{i})^{2}w}\psi_{i}\left(\frac{1}{w}\right)\mathrm{d}w\mathrm{d}z\\\
&=-\int_{\mathbb{T}_{-}}\int_{\mathbb{T}_{+}}\mathcal{D}^{-1}(w)\mathcal{D}(z)\sum^{\infty}_{k=0}\left(\frac{w}{z}\right)^{k}\frac{\tilde{\phi}(w)-1}{(2\pi\mathrm{i})^{2}wz}\psi_{i}\left(\frac{1}{w}\right)\mathrm{d}w\mathrm{d}z\\\
&=-\sum^{\infty}_{k=0}\left[\int_{\mathbb{T}_{+}}\mathcal{D}^{-1}(w)w^{k}\frac{\tilde{\phi}(w)-1}{2\pi\mathrm{i}w}\psi_{i}\left(\frac{1}{w}\right)\mathrm{d}w\right]\left[\int_{\mathbb{T}_{-}}\mathcal{D}(z)z^{-k}\frac{\mathrm{d}z}{2\pi\mathrm{i}z}\right]\end{split}$

Note that

(5.37)
$\begin{split}\int_{\mathbb{T}_{-}}\mathcal{D}(z)z^{-k}\frac{\mathrm{d}z}{2\pi\mathrm{i}z}&=\int_{\mathbb{T}}\mathcal{D}_{-}(z)z^{-k}\frac{\mathrm{d}z}{2\pi\mathrm{i}z}=\int_{\mathbb{T}}\mathcal{D}_{-}(z^{-1})z^{k}\frac{\mathrm{d}z}{2\pi\mathrm{i}z}=\int_{\mathbb{T}}\tilde{\mathcal{D}}_{+}(z)z^{k}\frac{\mathrm{d}z}{2\pi\mathrm{i}z}\\\
&=\int_{\mathbb{T}_{+}}\tilde{\mathcal{D}}(z)z^{k}\frac{\mathrm{d}z}{2\pi\mathrm{i}z}=\int_{\mathbb{T}_{+}}\left(\tilde{\mathcal{D}}(0)+\sum^{\infty}_{j=1}c_{j}z^{j}\right)z^{k}\frac{\mathrm{d}z}{2\pi\mathrm{i}z}\\\
&=\tilde{\mathcal{D}}(0)\int_{\mathbb{T}_{+}}z^{k}\frac{\mathrm{d}z}{2\pi\mathrm{i}z}=\tilde{\mathcal{D}}(0)\delta_{k0}=\delta_{k0}.\end{split}$

Thus,

(5.38)
$\begin{split}\mathcal{I}_{N}&\simeq-\int_{\mathbb{T}_{+}}\mathcal{D}^{-1}(w)(\tilde{\phi}(w)-1)\psi_{i}\left(\frac{1}{w}\right)\frac{\mathrm{d}w}{2\pi\mathrm{i}w}\\\
&=-\int_{\mathbb{T}}\mathcal{D}_{+}^{-1}(w)(\tilde{\phi}(w)-1)\psi_{i}\left(\frac{1}{w}\right)\frac{\mathrm{d}w}{2\pi\mathrm{i}w}\\\
&=-\int_{\mathbb{T}}\left(\mathcal{D}_{-}^{-1}(w)-\mathcal{D}_{+}^{-1}(w)\right)\psi_{i}\left(\frac{1}{w}\right)\frac{\mathrm{d}w}{2\pi\mathrm{i}w}\end{split}$

Note that

$\int_{\mathbb{T}}\mathcal{D}_{-}^{-1}(w)\psi_{i}\left(\frac{1}{w}\right)\frac{\mathrm{d}w}{2\pi\mathrm{i}w}=\int_{\mathbb{T}}\mathcal{D}_{-}^{-1}(w^{-1})\psi_{i}\left(w\right)\frac{\mathrm{d}w}{2\pi\mathrm{i}w}=$
$\int_{\mathbb{T}}\frac{\alpha_{+}(w)}{\alpha(0)}\psi_{i}\left(w\right)\frac{\mathrm{d}w}{2\pi\mathrm{i}w}=\int_{\mathbb{T}_{+}}\frac{\alpha(w)}{\alpha(0)}\psi_{i}\left(w\right)\frac{\mathrm{d}w}{2\pi\mathrm{i}w}=\psi_{0}.$

Therefore

(5.39)
$\begin{split}\mathcal{I}_{N}&\simeq-\psi_{0}+\int_{\mathbb{T}}\mathcal{D}_{+}^{-1}(w)\psi_{i}\left(\frac{1}{w}\right)\frac{\mathrm{d}w}{2\pi\mathrm{i}w}=-\psi_{0}+\int_{\mathbb{T}}\mathcal{D}_{+}^{-1}(w)\left(\psi\left(\frac{1}{w}\right)-\psi_{o}\left(\frac{1}{w}\right)\right)\frac{\mathrm{d}w}{2\pi\mathrm{i}w}\end{split}$

Note that $\psi_{o}\left(1/w\right)$ is an analytic function inside the unit
circle with $\psi_{o}\left(1/w\right)=O(w)$ as $w\to 0$, and thus

$\int_{\mathbb{T}}\mathcal{D}_{+}^{-1}(w)\psi_{o}\left(\frac{1}{w}\right)\frac{\mathrm{d}w}{2\pi\mathrm{i}w}=0.$

Hence, using this and (5.33) we have

(5.40)
$\begin{split}\mathcal{I}_{N}&\simeq-\psi_{0}+\int_{\mathbb{T}}\frac{\tilde{\psi}(w)}{\mathcal{D}_{-}(w)\tilde{\phi}(w)}\frac{\mathrm{d}w}{2\pi\mathrm{i}w}=-\psi_{0}+\int_{\mathbb{T}}\frac{\psi(w)}{\mathcal{D}_{-}(w^{-1})\phi(w)}\frac{\mathrm{d}w}{2\pi\mathrm{i}w}\end{split}$

Note that

$\mathcal{D}_{-}(w^{-1})=\tilde{\mathcal{D}}_{+}(w)=\frac{\alpha(0)}{\alpha_{+}(w)},\quad\text{and}\quad\mathcal{D}_{+}(w^{-1})=\tilde{\mathcal{D}}_{-}(w)=\frac{\alpha(0)}{\alpha_{-}(w)}.$

Therefore

(5.41)
$\begin{split}\mathcal{I}_{N}&\simeq-\psi_{0}+\int_{\mathbb{T}}\frac{\alpha_{+}(w)\psi(w)}{\alpha(0)\phi(w)}\frac{\mathrm{d}w}{2\pi\mathrm{i}w}=-\psi_{0}+\frac{1}{\alpha(0)}\int_{\mathbb{T}}\alpha_{-}(w)\psi(w)\frac{\mathrm{d}w}{2\pi\mathrm{i}w}\end{split}$

Comparing the Wiener-Hopf factorization $\phi(w)=\phi_{-}(w)\phi_{+}(w)$ with
the scalar Riemann-Hilbert jump condition
$\alpha_{+}(w)=\alpha_{-}(w)\phi(w)$, we can identify $\alpha_{-}$ with
$\phi^{-1}_{-}$ and $\alpha_{+}$ with $\phi_{+}$, and thus

(5.42)
$\mathcal{I}_{N}\simeq-\psi_{0}+\frac{[\phi^{-1}_{-}\psi]_{0}}{[\phi_{+}]_{0}}.$

Finally recalling (5.27), and taking the limit $N\to\infty$ we arrive at the
conclusion of proposition 3.2:

(5.43) $F[\phi;\psi]=\frac{[\phi^{-1}_{-}\psi]_{0}}{[\phi_{+}]_{0}}.$

###  5.3. Derivation of the symbol pair corresponding to the next-to-diagonal
Ising correlations

As it is shown in [AYP87], the next-to-diagonal two point correlation function
is given by the following bordered Toeplitz determinant,

(5.44)
$\langle\sigma_{0,0}\sigma_{N-1,N}\rangle=\mathrm{det}\begin{pmatrix}A_{0}&\cdots&A_{N-2}&B_{N-1}\\\
A_{-1}&\cdots&A_{N-3}&B_{N-2}\\\ \vdots&\vdots&\vdots&\vdots\\\
A_{1-N}&\cdots&A_{-1}&B_{0}\end{pmatrix},\ \ \ \ \ N>1,$

where in the notations of [AYP87],

(5.45)
$A_{n}=\frac{1}{2\pi}\int_{-\pi}^{\pi}e^{-in\theta}\Phi(\theta)d\theta,$
(5.46) $\Phi(\theta)=\frac{S-S^{\prime*}e^{-i\theta}}{\sqrt{\Omega(\theta)}},$
(5.47) $\Omega(\theta)=S^{2}+(S^{\prime*})^{2}-2SS^{\prime*}\cos(\theta),$
(5.48)
$B_{n}=\frac{1}{2\pi}\int^{\pi}_{-\pi}e^{-in\theta}\Psi(\theta)d\theta,$

and

(5.49)
$\Psi(\theta)=\frac{1}{\sqrt{\Omega(\theta)}}\left(SC^{\prime*}-\frac{CS^{\prime*}(SS^{\prime*}+e^{-i\theta})}{CC^{\prime*}+\sqrt{\Omega(\theta)}}\right).$

The quantities $C,S,C^{\prime*},$ and $S^{\prime*}$ are determined by the
physical parameters of the model according to the equations,

(5.50) $C:=\cosh(2K),\quad S:=\sinh(2K),\quad
C^{\prime*}:=\frac{\cosh(2K^{\prime})}{\sinh(2K^{\prime})},\quad
S^{\prime*}:=\frac{1}{\sinh(2K^{\prime})},$

where

(5.51) $K=\frac{J_{h}}{k_{B}T},\qquad\mbox{and}\qquad
K^{\prime}=\frac{J_{v}}{k_{B}T}.$

Using (1.25) and (1.26) we can write (5.46), (5.47) and (5.49) in our
notations as:

(5.52) $\Omega(\theta)=\frac{k^{2}+1-2k\cos(\theta)}{S^{2}_{v}},$ (5.53)
$\Phi(\theta)=\frac{k-e^{-i\theta}}{\sqrt{k^{2}+1-2k\cos(\theta)}},$

and

(5.54)
$\Psi(\theta)=\frac{1}{\sqrt{k^{2}+1-2k\cos(\theta)}}\left(S_{h}C_{v}-\frac{C_{h}(S_{h}+S_{v}e^{-i\theta})}{C_{h}C_{v}+\sqrt{k^{2}+1-2k\cos(\theta)}}\right).$

Recall that $\widehat{\phi}$ is given by (1.28) is

(5.55)
$\widehat{\phi}(z)=\sqrt{\frac{1-k^{-1}z^{-1}}{1-k^{-1}z}}=\frac{k-z^{-1}}{\sqrt{k^{2}+1-k(z+z^{-1})}}.$

This together with (5.53) immediately yields

(5.56) $\Phi(\theta)=\widehat{\phi}(e^{i\theta}).$

Next we want to show that $\Psi(\theta)=\widehat{\psi}(e^{i\theta})$. To that
end note that

(5.57)
$\frac{1}{C_{h}C_{v}+\sqrt{k^{2}+1-k(z+z^{-1})}}=\frac{C_{h}C_{v}-\sqrt{k^{2}+1-k(z+z^{-1})}}{S_{h}^{2}+S_{v}^{2}+k(z+z^{-1})}=\frac{C_{h}C_{v}z-z\sqrt{k^{2}+1-k(z+z^{-1})}}{k(z-c_{*})(z-c^{-1}_{*})},$

where

(5.58) $c_{*}:=-\frac{S_{h}}{S_{v}}.$

Therefore, as $S_{h}+S_{v}z^{-1}=S_{h}z^{-1}(z-c_{*}^{-1})$,

(5.59)
$\frac{S_{h}+S_{v}z^{-1}}{C_{h}C_{v}+\sqrt{k^{2}+1-k(z+z^{-1})}}=\frac{S_{h}C_{h}C_{v}}{k(z-c_{*})}-\frac{S_{h}\sqrt{k^{2}+1-k(z+z^{-1})}}{k(z-c_{*})}.$

Combining this with (5.54) gives

$\displaystyle\Psi(\theta)$
$\displaystyle=\frac{S_{h}C_{v}}{\sqrt{k^{2}+1-k(z+z^{-1})}}-\frac{S_{h}C^{2}_{h}C_{v}}{k(z-c_{*})\sqrt{k^{2}+1-k(z+z^{-1})}}+\frac{S_{h}C_{h}}{k(z-c_{*})}$
$\displaystyle=\frac{S_{h}C_{v}}{\sqrt{k^{2}+1-k(z+z^{-1})}}\left[1-\frac{C_{h}^{2}}{k(z-c_{*})}\right]+\frac{S_{h}C_{h}}{k(z-c_{*})},\qquad\qquad
z=e^{i\theta}.$

and the term in the brackets becomes $(kz-1)/(k(z-c_{*}))$. Now, using (5.55)
we obtain the formula for $\widehat{\psi}$ given by (1.31):

(5.60)
$\Psi(\theta)=\frac{C_{v}z\widehat{\phi}(z)+C_{h}}{S_{v}(z-c_{*})}\equiv\widehat{\psi}(z).$

Let us also remark that

(5.61)
$\widehat{\psi}(z)=\frac{C_{v}}{S_{v}}\cdot\frac{z\widehat{\phi}(z)-c_{*}\widehat{\phi}(c_{*})}{z-c_{*}},$

which can be seen from a straightforward computations as well. To summarize,
we have shown that

(5.62)
$\langle\sigma_{0,0}\sigma_{N-1,N}\rangle=D_{N}[\widehat{\phi};\widehat{\psi}].$

with $\widehat{\phi}$ and $\widehat{\psi}$ given by (1.28) and (1.31).

###  5.4. Auxiliary results

######  Lemma 5.1.

Let $|a|<1$ and $\omega$ be complex parameters. Then

$\sum_{n=N}^{\infty}a^{n}n^{\omega}(1+O(n^{-1}))=\frac{a^{N}N^{\omega}}{1-a}(1+O(N^{-1})),\qquad
N\to\infty.$

###### Proof.

We basically can apply summation by parts,

$\displaystyle(1-a)\sum_{n=N}^{\infty}a^{n}n^{\omega}(1+O(n^{-1}))$
$\displaystyle=\sum_{n=N}^{\infty}a^{n}n^{\omega}(1+O(n^{-1}))-\sum_{n=N}^{\infty}a^{n+1}n^{\omega}(1+O(n^{-1}))$
$\displaystyle=a^{N}N^{\omega}(1+O(N^{-1}))+$
$\displaystyle\qquad\sum_{n=N}^{\infty}a^{n+1}\Big{(}(n+1)^{\omega}(1+O(n^{-1}))-n^{\omega}(1+O(n^{-1}))\Big{)}$
$\displaystyle=a^{N}N^{\omega}(1+O(N^{-1}))+\sum_{n=N}^{\infty}a^{n+1}O(n^{\omega-1}).$

The last term we can split into

$\sum_{n=N}^{2N-1}a^{n}O(n^{\omega-1})=O(a^{N}N^{\omega-1}),\qquad\sum_{n=2N}^{\infty}a^{n}O(n^{\omega-1})=\sum_{n=2N}^{\infty}O(a^{n}q^{n})=O((aq)^{2N}).$

In the latter we choose $1<q<|a|^{-1}$, which guarantees that
$n^{\omega-1}=O(q^{n})$ and $(aq)^{2N}=O(N^{\omega-1})$. ∎

######  Lemma 5.2.

Let $|a|<1$ and $\omega$ be complex parameters. Then

$\sum_{n=0}^{\infty}(n+1)(n+N)^{\omega}a^{n+N}=\frac{a^{N}N^{\omega}}{(1-a)^{2}}(1+O(N^{-1})),\qquad
N\to\infty.$

###### Proof.

After dividing by $a^{N}$, the difference between the series and the leading
term is

$\displaystyle\sum_{n=0}^{\infty}(n+1)(n+N)^{\omega}a^{n}-\frac{N^{\omega}}{(1-a)^{2}}=\sum_{n=0}^{\infty}(n+1)a^{n}\Big{(}(n+N)^{\omega}-N^{\omega}\Big{)}$
$\displaystyle=\sum_{n=0}^{\infty}(n+1)^{2}a^{n}O(\max\\{(n+N)^{\mathrm{Re}(\omega)-1},N^{\mathrm{Re}(\omega)-1}\\})=O(N^{\mathrm{Re}(\omega)-1}).$

This implies the estimate. ∎

######  Lemma 5.3.

Let $\zeta(z)$ be a function holomorphic on
$\\{z\in\mathbb{C}\,:\,1-{\varepsilon}<|z|<b+{\varepsilon}\\}\setminus[b,b+{\varepsilon})$
with $b>1$, ${\varepsilon}>0$. Further assume that in some neighborhood of
$[b,b+{\varepsilon})$ this function is of the form

$\zeta(z)=(b-z)^{\omega}\xi(z)+\zeta_{0}(z)$

with $\xi(z)$ and $\zeta_{0}(z)$ being holomorphic, and
$\mathrm{Re}(\omega)>-1$. Then the Fourier coefficients of $\zeta$ have the
asymptotics

$\zeta_{n}=\Big{(}\frac{\xi(b)}{\Gamma(-\omega)}+O(n^{-1})\Big{)}n^{-\omega-1}b^{-n+\omega}\qquad
n\to+\infty.$

###### Proof.

In the formula for the Fourier coefficients we deform the contour into a
slightly bigger circle with radius $b(1+\delta_{n})$ (where $\delta_{n}=\delta
n^{-1/2}$ and $0<\delta<{\varepsilon}/b$ is fixed) and a line segments along
the branch cut $[b,b+b\delta_{n}]$ on both sides,

$\displaystyle\zeta_{n}$ $\displaystyle=\frac{1}{2\pi
i}\int_{|z|=1}\zeta(z)z^{-n-1}\,dz$ $\displaystyle=\frac{1}{2\pi
i}\int_{|z|=b(1+\delta_{n})}\zeta(z)z^{-n-1}\,dz+\frac{1}{2\pi
i}\int_{b}^{b(1+\delta_{n})}\left((b-t-i0)^{\omega}-(b-t+i0)^{\omega}\right)\xi(t)t^{-n-1}\,dt.$

The first integral being
$O(\delta_{n}^{-|\mathrm{Re}(\omega)|}b^{-n}(1+\delta_{n})^{-n})=O(n^{|\mathrm{Re}(\omega)|/2}b^{-n}e^{-n^{1/2}\delta})$
is negligible. The second one becomes

$-\frac{\sin(\omega\pi)}{\pi}\int_{b}^{b(1+\delta_{n})}(t-b)^{\omega}\xi(t)t^{-n-1}\,dt=-\frac{\sin(\omega\pi)}{\pi}b^{\omega-n}\int_{0}^{\delta_{n}}\xi(b+bs)s^{\omega}(1+s)^{-n-1}\,ds.$

Therein, the integral (without the factors in front of it) equals

$\displaystyle\int_{0}^{\delta_{n}}(\xi(b)+O(s))s^{\omega}e^{-(n+1)(s+O(s^{2}))}\,ds$
$\displaystyle=n^{-\omega-1}\int_{0}^{n^{1/2}\delta}\Big{(}\xi(b)+O({\textstyle\frac{u}{n}})\Big{)}u^{\omega}e^{-u+O(\frac{u+u^{2}}{n})}\,du$
$\displaystyle=\xi(b)n^{-\omega-1}\int_{0}^{n^{1/2}\delta}u^{\omega}e^{-u}\,du+n^{-\omega-2}\int_{0}^{n^{1/2}\delta}u^{\omega}O(u+u^{2})e^{-u}\,du$
$\displaystyle=n^{-\omega-1}\Big{(}\xi(b)\Gamma(1+\omega)+O(n^{-1})\Big{)}.$

Combining all this give the asymptotic formula. ∎

Acknowledgements. The authors would like to thank Pavel Bleher, Barry McCoy,
Vitaly Tarasov, and Nicholas Witte for their interest in this project and for
helpful conversations. EB, TE and RG acknowledge American Institute of
Mathematics for providing excellent working conditions and their support
during the SQuaRE program ”Asymptotic behavior of Toeplitz and Toeplitz+Hankel
determinants” where part of their work was done during the 2019 and 2020
meetings. EB acknowledges support from the NSF grant DMS-2050092. TE
acknowledges support from the Simons Foundation Collaboration Grant # 525111.
AI acknowledges support from the NSF grant DMS-1955265.

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