Exact Solution of the Six-Vertex Model with Domain Wall
Boundary Conditions: Antiferroelectric Phase
PAVEL BLEHER
Indiana University–Purdue University Indianapolis
KARL LIECHTY
Indiana University–Purdue University Indianapolis
Abstract
We obtain the large-n asymptotics of the partition function Zn of the six-vertex
model with domain wall boundary conditions in the antiferroelectric phase region, with the weights a D sinh. t/, b D sinh. C t/, c D sinh.2 /,
jtj < . We prove the conjecture of Zinn-Justin, that as n ! 1, Zn D
2
C #4 .n!/F n Œ1 C O.n1 /, where ! and F are given by explicit expressions
in and t, and #4 .´/ is the Jacobi theta function. The proof is based on the
Riemann-Hilbert approach to the large-n asymptotic expansion of the underlying discrete orthogonal polynomials and on the Deift-Zhou nonlinear steepestdescent method. © 2009 Wiley Periodicals, Inc.
1 Introduction and Formulation of the Main Result
1.1 Definition of the Model
The six-vertex model, or the model of two-dimensional ice, is stated on a square
n n lattice with arrows on the edges. The arrows obey the rule that at every
vertex there are two arrows pointing in and two arrows pointing out. Such a rule is
sometimes called the ice rule. There are only six possible configurations of arrows
at each vertex, hence the name of the model; see Figure 1.1.
We will consider the domain wall boundary conditions (DWBC), in which the
arrows on the upper and lower boundaries point into the square, and the ones on
the left and right boundaries point out. One possible configuration with DWBC on
the 4 4 lattice is shown on Figure 1.2.
For each possible vertex state we assign a weight wi ; i D 1; : : : ; 6, and define,
as usual, the partition function as a sum over all possible arrow configurations of
the product of the vertex weights,
(1.1)
Zn D
X
arrow configurations w. /;
w. / D
Y
x2Vn
w t .xI/ D
6
Y
iD1
Communications on Pure and Applied Mathematics, Vol. LXIII, 0779–0829 (2010)
© 2009 Wiley Periodicals, Inc.
Ni ./
wi
;
780
P. M. BLEHER AND K. LIECHTY
(1)
(2)
(3)
(4)
(5)
(6)
F IGURE 1.1. The six arrow configurations allowed at a vertex.
F IGURE 1.2. An example of a 4 4 configuration with domain wall
boundary conditions (DWBC).
where Vn is the n n set of vertices, t.xI / 2 f1; : : : ; 6g is the vertex type of
configuration at vertex x according to Figure 1.1, and Ni . / is the number of
vertices of type i in the configuration . The sum is taken over all possible configurations obeying the given boundary condition. The Gibbs measure is defined then
as
w. /
:
(1.2)
n . / D
Zn
Our main goal is to obtain the large-n asymptotics of the partition function Zn .
The six-vertex model has six parameters: the weights wi . By using some conservation laws it can be reduced to only two parameters (see, e.g., [1, 8, 14]). Namely,
we have that
(1.3)
Zn .w1 ; w2 ; w3 ; w4 ; w5 ; w6 / D C.n/Zn .a; a; b; b; c; c/
and
(1.4)
n . I w1 ; w2 ; w3 ; w4 ; w5 ; w6 / D n . I a; a; b; b; c; c/;
EXACT SOLUTION OF THE SIX-VERTEX MODEL
where
aD
(1.5)
p
w1 w2 ;
bD
and
p
w3 w4 ;
C.n/ D
(1.6)
w5
w6
:
(1.8)
n2
Zn .a; a; b; b; c; c/ D c Zn
and
n . I a; a; b; b; c; c/ D n
p
w5 w6 ;
n=2
Furthermore,
(1.7)
cD
781
a a b b
; ; ; ; 1; 1
c c c c
a a b b
I ; ; ; ; 1; 1 ;
c c c c
so that a general weight reduces to the two parameters
a
c
and bc .
1.2 Exact Solution of the Six-Vertex Model for a Finite n
Introduce the parameter
a2 C b 2 c 2
:
2ab
The phase diagram of the six-vertex model consists of three phase regions: the
ferroelectric phase region, > 1; the antiferroelectric phase region, < 1;
and the disordered phase region, 1 < < 1. Observe that ja bj > c in the
ferroelectric phase region and c > a Cb in the antiferroelectric phase region, while
in the disordered phase region a; b; c satisfy the triangle inequalities. In the three
phase regions we parametrize the weights in the standard way: for the ferroelectric
phase,
D
(1.9)
(1.10)
a D sinh.t /;
b D sinh.t C /;
c D sinh.2j j/;
0 < j j < tI
for the antiferroelectric phase,
(1.11)
a D sinh. t/;
b D sinh. C t/;
c D sinh.2 /;
jtj < I
and for the disordered phase
(1.12)
a D sin. t/;
b D sin. C t/;
c D sin.2 /;
jtj < :
The phase diagram of the six-vertex model is shown in Figure 1.3. The phase
diagram and the Bethe ansatz solution of the six-vertex model for periodic and
antiperiodic boundary conditions are thoroughly discussed in the works of Lieb
[21, 22, 23, 24], Lieb and Wu [25], Sutherland [28], Baxter [4], and Batchelor,
Baxter, O’Rourke, and Yung [3]. See also the work of Wu and Lin [30], in which
the Pfaffian solution for the six-vertex model with periodic boundary conditions is
obtained on the free fermion line, D 0. Brascamp, Kunz, and Wu [10] prove
the equality of the free energy with periodic and free boundary conditions under
782
P. M. BLEHER AND K. LIECHTY
bc
F
D
1
A(1)
A(2)
AF
A(3)
F
0
1
ac
F IGURE 1.3. The phase diagram of the model, where F, AF, and D mark
ferroelectric, antiferroelectric, and disordered phases, respectively. The
circular arc corresponds to the so-called free fermion line, when D 0,
and the three dots correspond to 1-, 2-, and 3-enumeration of alternating
sign matrices.
various conditions on a; b; c, and they also prove the existence of the spontaneous
staggered polarization for sufficiently small values of the parameters ac and bc .
In this paper we will discuss the antiferroelectric phase region, and we will
use parametrization (1.11). The parameter in the antiferroelectric phase region
reduces to
D cosh.2 /:
(1.13)
The six-vertex model with DWBC was introduced by Korepin [17], who derived
important recursion relations for the partition function of the model. These recursion relations were solved by Izergin [15], and this led to a beautiful determinantal
formula for the partition function with DWBC. A detailed proof of this formula,
usually called the Izergin-Korepin formula, and its generalizations are given in the
paper of Izergin, Coker, and Korepin [16]. When the weights are parametrized
according to (1.11), the Izergin-Korepin formula is
2
(1.14)
Zn D
Œsinh. t/ sinh. C t/n n
;
Qn1 2
j
Š
j D0
where n is the Hankel determinant,
j Ck2 d
(1.15)
n D det
;
j
Ck2
dt
1j;kn
and
(1.16)
.t/ D
sinh.2 /
:
sinh. t/ sinh. C t/
EXACT SOLUTION OF THE SIX-VERTEX MODEL
783
An elegant derivation of the Izergin-Korepin formula from the Yang-Baxter equations is given in the papers of Korepin and Zinn-Justin [18] and Kuperberg [20]
(see also the book of Bressoud [11]).
One of the applications of the determinantal formula is that it implies that the
function n solves the Toda equation
@
I
@t
compare the work of Sogo [27]. The Toda equation was used by Korepin and ZinnJustin [18] to derive the free energy of the six-vertex model with DWBC, assuming
some ansatz on the behavior of subdominant terms in the large-n asymptotics of
the free energy.
Another application of the Izergin-Korepin formula is that n can be expressed
in terms of a partition function of a random matrix model; see the paper [31] of
Zinn-Justin. Namely, let us write .t/ in the form of the Laplace transform of a
discrete measure,
1
X
sinh.2 /
D2
e 2t l2 jlj:
(1.18)
.t/ D
sinh. t/ sinh. C t/
(1.17)
n n00 n0 D nC1 n1 ;
n 1; .0 / D
2
lD1
Then
1
X
2
(1.19)
2n
n D
nŠ
l1 ;:::;ln D1
where
(1.20)
.l/
2
.l/ D
n
Y
e 2t li 2 jli j ;
iD1
Y
.lj li /
1i<j n
is the Vandermonde determinant. For a proof see [31] or appendix A in [6]. We
omit the proof of this and some other formulae in the paper, due to a publishing
limitation on the length of the paper. For the proofs, see [6].
Introduce now discrete monic polynomials Pj .x/ D x j C orthogonal on the
set Z with respect to the weight,
w.l/ D e 2t l2 jlj;
(1.21)
so that
(1.22)
1
X
Pj .l/Pk .l/w.l/ D hk ıjk :
lD1
Then it follows from (1.19) that
(1.23)
2
n D 2n
n1
Y
kD0
see appendix B of [6].
hk I
784
P. M. BLEHER AND K. LIECHTY
1.3 Rescaling of the Weight
Set
2
(1.24)
n D
; x D ln ;
n
wn .x/ D e n.jxj x/ ;
and
Pnk .x/ D
(1.25)
Consider also the lattice
(1.26)
kn Pk
D
t
;
x
:
n
2k
; k2Z :
Ln D x D
n
Then from (1.22) we obtain that the monic polynomials Pnk .x/ satisfy the orthogonality condition,
X
Pnj .x/Pnk .x/wn .x/n D hnk ıjk ; hnk D hk 2kC1
:
(1.27)
n
x2Ln
We can then combine equations (1.14), (1.23), and (1.27) to obtain
2 n1
nab n Y hnk
; a D sinh. t/; b D sinh. C t/:
(1.28)
Zn D
.kŠ/2
kD0
For what follows we will need to extend the weight wn .x/ to the complex plane.
We do so by defining wn .´/ on the complex plane as
wn .´/ D e nV .´/
(1.29)
where
(1.30)
(
V .´/ D
´ ´
when Re ´ 0;
´ ´ when Re ´ 0;
so that V .´/, and thus wn .´/, is two-valued on the imaginary axis.
1.4 Main Result: Asymptotics of the Partition Function
This work is a continuation of the work [5] of the first author with Vladimir
Fokin and [7, 8] by the authors of the present work. In [5] the authors obtain the
large-n asymptotics of the partition function Zn in the disordered phase. They
prove the conjecture of Paul Zinn-Justin [31] that the large-n asymptotics of Zn in
the disordered phase has the following form: For some " > 0,
(1.31)
2
Zn D C n F n Œ1 C O.n" /:
Furthermore, they find the exact value of the exponent ,
(1.32)
D
2 2
1
:
12 3
.
2 /
EXACT SOLUTION OF THE SIX-VERTEX MODEL
785
The value of F in the disordered phase is given by the formula
F D
(1.33)
ab
t ;
2 cos 2
a D sin. t/; b D sin. C t/;
in parametrization (1.12).
In the work [8] we obtain the following large-n asymptotic formula for Zn in
the ferroelectric phase region: For any " > 0,
2
1"
Zn D C G n F n Œ1 C O.e n
(1.34)
/;
where C D 1 e 4 , G D e t , and F D sinh.t C / in parametrization (1.10).
Finally, in the work [7] we obtain the following large-n asymptotic formula for
Zn on the borderline between the ferroelectric and disordered phase regions:
(1.35)
Zn .a; a; a C 1; a C 1; 1; 1/ D C n G
p
n
2
F n Œ1 C O.n1=2 /;
where C > 0,
(1.36)
1
D ;
4
r 3
a
G D exp ;
2
F D a C 1;
and is the Riemann zeta function.
In the present paper we obtain the large-n asymptotic formula for Zn in the antiferroelectric phase region. The formulation of the main result of the present paper
and the proofs involve the Jacobi theta functions. Let us review their definition and
basic properties.
There are four Jacobi theta functions:
#1 .´/ D 2
#2 .´/ D 2
(1.37)
1
X
1 2
.1/n q .nC 2 / sin..2n C 1/´/;
nD0
1
X
1 2
q .nC 2 / cos..2n C 1/´/;
nD0
#3 .´/ D 1 C 2
#4 .´/ D 1 C 2
1
X
nD1
1
X
2
q n cos.2n´/;
2
.1/n q n cos.2n´/;
nD1
where q is the elliptic nome. We will assume that 1 > q > 0. Figure 1.4 shows the
graphs of #1 and #2 (left figure) and #3 and #4 (right figure) on the interval Œ0; for q D 0:5. Observe that #1 and #4 are increasing on Œ0; 2 while #2 and #3 are
decreasing on this interval.
786
P. M. BLEHER AND K. LIECHTY
F IGURE 1.4. The graphs of #1 and #2 (left figure) and #3 and #4 (right
figure) on the interval Œ0; for q D 0:5.
The Jacobi theta functions satisfy the following periodicity conditions:
#1 .´ C / D #1 .´/; #1 .´ C / D e 2i´ q 1 #1 .´/;
(1.38)
#2 .´ C / D #2 .´/; #2 .´ C / D e 2i´ q 1 #2 .´/;
#3 .´ C / D #3 .´/;
#3 .´ C / D e 2i´ q 1 #3 .´/;
#4 .´ C / D #4 .´/;
#4 .´ C / D e 2i´ q 1 #4 .´/;
where is a pure imaginary number related to q by the equation
(1.39)
q D e i :
The theta functions also satisfy the symmetry conditions
(1.40)
and the equations
#1 .´/ D #1 .´/;
#2 .´/ D #2 .´/;
#3 .´/ D #3 .´/;
#4 .´/ D #4 .´/;
#1 .´/ D #2 ´ ;
2
#3 .´/ D #4 ´ C
;
2
(1.41)
i :
#1 .´/ D ie i´C 4 #4 ´ C
2
The only zeroes of the theta functions are
D 0; #3
C
D 0; #4
D 0;
(1.42)
#1 .0/ D 0; #2
2
2
2
2
and their shifts by m
C n
, m; n 2 Z.
In the antiferroelectric phase region we use parametrization (1.11), with two
parameters t and such that jtj < . In what follows we will also use the following
EXACT SOLUTION OF THE SIX-VERTEX MODEL
787
parameters:
t
.1 C /
2 .1; 1/; ! D
2 .0; /:
2
The elliptic nome for all Jacobi theta functions in this paper will be equal to
(1.43)
D
q D e (1.44)
2
=2
:
Our main result in the present paper is the following asymptotic formula for Zn :
T HEOREM 1.1 As n ! 1,
2
Zn D C #4 .n!/F n .1 C O.n1 //;
(1.45)
where C > 0 is a constant, and
sinh. t/ sinh. C t/#10 .0/
:
(1.46)
F D
2 #1 .!/
The asymptotic formula (1.45) proves the conjecture of Zinn-Justin in [31]. The
proof of Theorem 1.1 will be based on the Riemann-Hilbert approach to discrete
orthogonal polynomials. An important first step in this approach is constructing
the equilibrium measure.
2 Equilibrium Measure
2.1 Heuristic Motivation and Definition of the Equilibrium Measure
If we scale the variables in (1.19) as i D 2li =n, then we can rewrite formula
(1.19) as
X
2
n D
(2.1)
2n
nŠ
2 H . /
n e n
;
n
2 2
n Z
where
d .x/ D
(2.2)
n
1X
ı.x j /;
n
j D1
and
(2.3)
“
H./ D
log
1
d.x/d.y/ C
jx yj
Z
.jxj x/d.x/;
x6Dy
where all integrals are over R.
Due to the factor .n2 / in the exponent of (2.1), we expect the sum, in the largen limit, to be focused in a neighborhood of a global minimum of the functional H .
Clearly, we have that is a probability measure and
(2.4)
.a; b/ ba
2
for any 1 < a < b < 1;
788
P. M. BLEHER AND K. LIECHTY
because in (2.2), j 2
mind, we define
2
n Z
and j 6D k if j 6D k. With these constraints in
E0 D inf H./
(2.5)
where the infimum is taken over all probability measures satisfying (2.4). It is
possible to prove that there exists a unique minimizer 0 so that
E0 D H.0 /I
(2.6)
see, e.g., the works of Saff and Totik [26], Dragnev and Saff [13], and Kuijlaars
[19]. Furthermore, 0 has support on a finite number of intervals and is absolutely
continuous with respect to the Lebesgue measure. The minimizer 0 is called the
equilibrium measure.
Denote the density function of the equilibrium measure as .x/ and its resolvent
as !, so we have
Z
d0
.x/dx
(2.7)
D .x/; !.´/ D
;
dx
´x
R
and
1
.!.x i0/ !.x C i0//:
2
i
The structure of the equilibrium measure 0 is studied in the paper of Zinn-Justin
[31], who shows that 0 has support on an interval Œ˛; ˇ, with a saturated region
Œ˛ 0 ; ˇ 0 in which
1
; x 2 Œ˛ 0 ; ˇ 0 ;
(2.9)
.x/ D
2
and two unsaturated regions, Œ˛; ˛ 0 and Œˇ 0 ; ˇ, in which
1
; x 2 .˛; ˛ 0 / [ .ˇ 0 ; ˇ/I
(2.10)
0 < .x/ <
2
see Figure 2.1. We also have that
.x/ D
(2.8)
˛ < ˛ 0 < 0 < ˇ 0 < ˇ;
(2.11)
so that the origin, which is a singular point of the potential V .x/ D jxj x, lies
inside the saturated region Œ˛ 0 ; ˇ 0 .
The measure 0 is uniquely determined by the Euler-Lagrange variational conditions
Z
(2.12)
2
D l
log jx yjd0 .y/ .jxj x/ l
l
for x 2 Œ˛; ˛ 0 [ Œˇ 0 ; ˇ;
for x 2 Œ˛ 0 ; ˇ 0 ;
for x … Œ˛; ˇ;
where l is the Lagrange multiplier. The Euler-Lagrange variational conditions imply
(2.13)
!.x i0/ C !.x C i0/ D C sgn.x/ for x 2 Œ˛; ˛ 0 [ Œˇ 0 ; ˇ;
EXACT SOLUTION OF THE SIX-VERTEX MODEL
789
ρ(x)
α
α’
β’
0
β
F IGURE 2.1. The equilibrium density function .x/.
whereas in the saturated region, we have
(2.14)
.x/ D
1
1
.!.x i0/ !.x C i0// D
2
i
2
for x 2 Œ˛ 0 ; ˇ 0 :
Now we will give a detailed description of the equilibrium measure. We begin
with explicit formulae for the endpoints of the support of the equilibrium measure.
2.2 Explicit Formulae for the Endpoints
P ROPOSITION 2.1 The endpoints of the support of the equilibrium measure 0 are
equal to
˛ D (2.15)
ˇ 0 D #10 . !2 /
;
#1 . !2 /
#30 . !2 /
;
#3 . !2 /
˛ 0 D ˇ D #40 . !2 /
;
#4 . !2 /
#20 . !2 /
:
#2 . !2 /
The differences between the endpoints are equal to
˛ 0 ˛ D #42 .0/
(2.16)
#2 . !2 /#3 . !2 /
;
#1 . !2 /#4 . !2 /
ˇ 0 ˛ 0 D #22 .0/
ˇ ˇ 0 D #4 .0/2
#1 . !2 /#2 . !2 /
;
#3 . !2 /#4 . !2 /
#1 . !2 /#4 . !2 /
:
#2 . !2 /#3 . !2 /
and
ˇ ˛ D #22 .0/
(2.17)
#3 . !2 /#4 . !2 /
;
#1 . !2 /#2 . !2 /
ˇ 0 ˛ D #32 .0/
ˇ ˛ 0 D #3 .0/2
#2 . !2 /#4 . !2 /
:
#1 . !2 /#3 . !2 /
For a proof of Proposition 2.1, see the next section.
#1 . !2 /#3 . !2 /
;
#2 . !2 /#4 . !2 /
790
P. M. BLEHER AND K. LIECHTY
2.3 Equilibrium Density Function
P ROPOSITION 2.2 The equilibrium density function .x/ is given by the formulae
Z
1 x
dx 0
; ˛ x ˛0;
p
˛
.x 0 ˛/.˛ 0 x 0 /.ˇ 0 x 0 /.ˇ x 0 /
1
;
˛0 x ˇ0;
(2.18) .x/ D
2Z
dx 0
1 ˇ
; ˇ 0 x ˇ:
p
x
.x 0 ˛/.x 0 ˛ 0 /.x 0 ˇ 0 /.ˇ x 0 /
Also,
Z
ˇ
(2.19)
.x/dx D
0
1C
:
2
The resolvent !.´/ of the equilibrium measure is given as
Z 1
d´0
(2.20)
!.´/ D
;
p
.´0 ˛/.´0 ˛ 0 /.´0 ˇ 0 /.´0 ˇ/
´
where integration takes place on the sheet of
p
p
R.´0 / .´0 ˛/.´0 ˛ 0 /.´0 ˇ 0 /.´0 ˇ/
p
for which R.´0 / > 0 for ´0 > ˇ, with cuts on Œ˛; ˛ 0 and Œˇ 0 ; ˇ.
For a proof of this proposition, see the next section.
2.4 g-function
Define the g-function on C n Œ1; ˇ as
Z ˇ
(2.21)
g.´/ D
log.´ x/d0 .x/
˛
where we take the principal branch for logarithm.
Properties of g.´/.
(1) g.´/ is analytic in C n .1; ˇ.
(2) For large ´,
(2.22)
g.´/ D log ´ 1
X
gj
;
´j
Z
gj D
j D1
ˇ
˛
xj
d0 .x/:
j
(3) g 0 .´/ D !.´/.
(4) From the first relation in (2.12) we have that
(2.23)
gC .x/ C g .x/ D jxj x C l
for x 2 Œ˛; ˛ 0 [ Œˇ 0 ; ˇ;
EXACT SOLUTION OF THE SIX-VERTEX MODEL
791
where gC and g refer to the limiting values of g from the upper and lower halfplanes, respectively. By differentiating this equation we obtain that
0
0
(2.24) !C .x/ C ! .x/ D gC
.x/ C g
.x/ D sgn x for x 2 Œ˛; ˛ 0 [ Œˇ 0; ˇ:
Consider the function
f .x/ D gC .x/ C g .x/ .jxj x C l/:
(2.25)
We have from (2.23) and (2.24) that
f .x/ D f 0 .x/ D 0 for x D ˛; ˛ 0 ; ˇ 0 ; ˇ;
(2.26)
and from (2.20) that
1
(2.27) f 00 .x/ D p
.x ˛/.x ˛ 0 /.x ˇ 0 /.x ˇ/
for x 2 .1; ˛/ [ .˛ 0 ; ˇ 0 / [ .ˇ; 1/:
Since
f 00 .x/ < 0 for x 2 .1; ˛/ [ .ˇ; 1/
(2.28)
and
f 00 .x/ > 0 for x 2 .˛ 0 ; ˇ 0 /; x 6D 0;
(2.29)
D jxj x C l
we obtain that
(2.30)
gC .x/ C g .x/ > jxj x C l
< jxj x C l
for x 2 Œ˛; ˛ 0 [ Œˇ 0 ; ˇ;
for x 2 .˛ 0 ; ˇ 0 /;
for x 2 R n Œ˛; ˇ:
(5) Equation (2.21) implies that the function
‚
(2.31)
G.x/ gC .x/ g .x/
is pure imaginary for all real x, and
(2.32)
G.x/ D
for 1 < x ˛;
2
i
Rx
2
i 2
i ˛ .s/ ds
x
2
i 1C
2 2
Rˇ
2
i x .s/ ds
for ˛ x ˛ 0 ;
0
for ˇ x < 1:
for ˛ 0 x ˇ 0 ;
for ˇ 0 x ˇ;
From (2.30) and (2.32) we obtain that
(
Rx
jxj x C l ˙ Œ2
i 2
i ˛ .s/ ds
Rˇ
(2.33) 2g˙ .x/ D
jxj x C l ˙ 2
i x .s/ ds
for ˛ x ˛ 0 ;
for ˇ 0 x ˇ;
792
P. M. BLEHER AND K. LIECHTY
(6) Also, from (2.32)
ˇ
dG.x C iy/ ˇˇ
(2.34)
D 2
.x/ > 0;
ˇ
dy
yD0
x 2 .˛; ˇ/:
Observe that from (2.23) we have that
(2.35) G.x/ D 2gC .x/V .x/l D Œ2g .x/V .x/l;
x 2 Œ˛; ˛ 0 [Œˇ 0 ; ˇ;
where V .x/ jxj x.
2.5 Evaluation of the Lagrange Multiplier l
P ROPOSITION 2.3 The Lagrange multiplier l solves the equation
#10 .0/
:
(2.36)
e l=2 D
2e#1 .!/
For a proof of this proposition, see the next section.
3 Proofs of Propositions 2.1, 2.2, and 2.3
P ROOF OF P ROPOSITION 2.1: Following Zinn-Justin [31], we make the following elliptic change of variables:
Z ´
1p 0
d´0
(3.1) u.´/ D
;
.ˇ ˛/.ˇ ˛ 0 /
p
2
.´0 ˛/.´0 ˛ 0 /.´0 ˇ 0 /.´0 ˇ/
ˇ
p
where integration takes place on the sheet on R.´0 / specified in Proposition 2.2.
To understand this integral in terms of the usual elliptic integrals, we first make the
change of variables
.ˇ ´0 /.ˇ 0 ˛/
;
(3.2)
v.´0 / D 0
.ˇ ´0 /.ˇ ˛/
so that
ˇ 0 .ˇ ˛/v ˇ.ˇ 0 ˛/
:
(3.3)
´0 D
.ˇ ˛/v .ˇ 0 ˛/
Note that v.ˇ/ D 0, v.ˇ 0 / D 1, and v.˛/ D 1. When we substitute v into
equation (3.1), we have
Z v.´/
1
dv
(3.4)
u.´/ D
;
p
2k 0
v.v 1/.v 1=k 2 /
where
s
.ˇ ˛/.ˇ 0 ˛ 0 /
:
(3.5)
kD
.ˇ 0 ˛/.ˇ ˛ 0 /
p
We next take v 0 D v, obtaining
Z pv.´/
dv 0
;
p
(3.6)
u.´/ D
.1 v 02 /.1 k 2 v 02 /
0
EXACT SOLUTION OF THE SIX-VERTEX MODEL
u(β’ )iK ’
u(α’ )KiK
a
u
’
u(α )K
u(β)0
u(β’ )iK
793
u(α’ )KiK ’
’
F IGURE 3.1. The u-plane. Here u D u1 u.1/ and a D u C iK 0 .
which corresponds to
(3.7)
p
v.´/ D sn.u; k/, so that
.ˇ ´/.ˇ 0 ˛/
D sn2 .u/;
.ˇ 0 ´/.ˇ ˛/
sn.u/ D sn.u; k/:
Notice that u maps the upper ´-plane conformally and bijectively onto the rectangle Œ0; K Œ0; iK 0 , and the lower ´-plane conformally and bijectively onto the
rectangle Œ0; K ŒiK 0 ; 0, where
Z 1
dv 0
;
p
K D u.˛/ D
.1 v 02 /.1 k 2 v 02 /
0
(3.8)
Z 1=k
dv 0
p
K 0 D iu.ˇ 0 / D
.v 02 1/.1 k 2 v 02 /
1
are the usual complete integrals of the first kind. More specifically (see Figure 3.1),
(1) The upper (respectively, lower) cusp of the interval Œˇ; ˇ 0 is mapped onto
the interval Œ0; iK 0 (respectively, Œ0; iK 0 ).
(2) The upper (respectively, lower) cusp of the interval Œ˛; ˛ 0 is mapped onto
the interval ŒK; K C iK 0 (respectively, ŒK; K iK 0 ).
(3) The interval Œˇ 0 ; ˛ 0 is mapped onto the interval ŒiK 0 ; K C iK 0 or the interval ŒiK 0 ; K iK 0 , depending on the path of integration.
(4) The remaining part of the real axis, Œ1; ˛ [ Œˇ; 1, is mapped onto the
interval Œ0; K, with u.1/ D u D u1 .
We will denote the rectangle Œ0; K ŒiK 0 ; iK 0 as R, the fundamental domain of
the function ´.u/. We can now define
(3.9)
!.u/
Q
D !.´.u// for u 2 R:
794
P. M. BLEHER AND K. LIECHTY
The Euler-Lagrange equation (2.13) and the equation (2.14) then become
!.u/
Q
C !.u/
Q
D 1
(3.10)
!.u/
Q
C !.u
Q
C 2K/ D 1 i
!.u
Q C 2iK 0 / !.u/
Q
D
for u 2 ŒiK 0 ; iK 0 ;
for u 2 ŒK iK 0 ; K C iK 0 ;
for u 2 ŒiK 0 ; K iK 0 :
The function !.´/ is analytic in C nŒ˛; ˇ but can be analytically continued from
either above or below through any of the cuts Œ˛; ˛ 0 , Œ˛ 0 ; ˇ 0 , and Œˇ 0 ; ˇ. These
analytic continuations in the ´-plane give an analytic continuation of !Q in the uplane into a neighborhood of R, which can then be continued by equations (3.10)
to the entire u-plane. We therefore have that !Q is analytic and satisfies equations
(3.10) throughout the u-plane. The first two equations of (3.10) can be combined as
!.u
Q C 2K/ D !.u/
Q
2:
(3.11)
It therefore follows that !Q is a linear function of u, as its derivative is a doubly
periodic entire function. We also know from the fact that !.´/ 1´ at infinity that
(3.12)
2
.u u1 / C O.u u1 /2
!.u/
Q
D p
.ˇ 0 ˛/.ˇ ˛ 0 /
in some neighborhood of u1 , where u1 is the image of infinity under the map
u.´/. It thus follows from (3.10), (3.11), and (3.12) that
!.u/
Q
D
(3.13)
1
.u u1 /
K
and that
K0
D
;
K
2
(3.14)
p
(3.15)
.ˇ 0 ˛/.ˇ ˛ 0 / D 2K;
1
u1
D
:
K
2
(3.16)
From (3.7) we obtain that
ˇ0 ˛
D sn2 .u1 /:
ˇ˛
(3.17)
This implies that
cn2 .u1 / D 1 sn2 .u1 / D 1 (3.18)
ˇ0 ˛
ˇ ˇ0
D
;
ˇ˛
ˇ˛
dn2 .u1 / D 1 k 2 sn2 .u1 /
D1
.ˇ ˛/.ˇ 0 ˛ 0 / .ˇ 0 ˛/
ˇ ˇ0
D
:
.ˇ 0 ˛/.ˇ ˛ 0 / .ˇ ˛/
ˇ ˛0
EXACT SOLUTION OF THE SIX-VERTEX MODEL
795
From equations (3.15), (3.17), and (3.18) we obtain the distances between the
turning points in terms of u1 :
ˇ ˛ D 2K
(3.19)
dn.u1 /
;
sn.u1 / cn.u1 /
ˇ ˇ 0 D 2K
ˇ ˛ 0 D 2K
cn.u1 /
;
sn.u1 / dn.u1 /
cn.u1 / dn.u1 /
:
sn.u1 /
The functions sn, cn, and dn are expressed in terms of Jacobi theta functions as
follows (see, e.g., [29]),
sn.u/ D
(3.20)
u
/
#3 .0/ #1 . 2K
u ;
#2 .0/ #4 . 2K /
dn.u/ D
cn.u/ D
u
/
#4 .0/ #2 . 2K
u ;
#2 .0/ #4 . 2K /
u
/
#4 .0/ #3 . 2K
u :
#3 .0/ #4 . 2K /
By (3.14), the half-period ratio and the elliptic nome q of the theta functions are
D
(3.21)
i
iK 0
D
K
2
and
qDe
K 0
K
De
2
2
:
If we take into account the fact that
#3 .0/2 D
(3.22)
2K
along with equation (3.16), we can write equations for the distances between the
turning points that involve only the original parameters:
ˇ ˛ D #22 .0/
(3.23)
#3 . !2 /#4 . !2 /
;
#1 . !2 /#2 . !2 /
ˇ 0 ˛ D #32 .0/
ˇ ˛ 0 D #3 .0/2
#1 . !2 /#3 . !2 /
;
#2 . !2 /#4 . !2 /
#2 . !2 /#4 . !2 /
;
#1 . !2 /#3 . !2 /
giving (2.17). These equations determine the endpoints ˛; ˛ 0 ; ˇ 0 ; ˇ up to a shift.
To fix the shift we use equation (2.12) at the points ˛ 0 and ˇ 0 to obtain
Z ˇ0
!.´ C i0/ C !.´ i0/ d´ D .1 /ˇ 0 C .1 C /˛ 0 :
(3.24)
˛0
Writing this integral in terms of u gives
Z
(3.25)
KCiK 0
iK 0
1
.u u1 /r 0 .u/du
K
Z KiK 0
1
C
.u u1 /r 0 .u/du D .1 /ˇ 0 C .1 C /˛ 0
K
iK 0
796
P. M. BLEHER AND K. LIECHTY
where
(3.26)
ˇ
ˇ 0 .ˇ ˛/ sn2 .u/ ˇ.ˇ 0 ˛/
r.u/ D
D
.ˇ ˛/ sn2 .u/ .ˇ 0 ˛/
1
ˇ 0 sn2 .u/
sn2 .u1 /
sn2 .u/
sn2 .u1 /
;
d
r.u/:
du
r 0 .u/ D
Note that r.˙iK 0 / D ˇ 0 and r.K ˙ iK 0 / D ˛ 0 .
Integrating by parts gives
(3.27)
2
..K u1 /˛ 0 C ˇ 0 u1 / K
Z
Z
KCiK 0
iK 0
KiK 0
iK 0
r.u/
du
K
r.u/
du D .1 /ˇ 0 C .1 C /˛ 0
K
or equivalently
Z
(3.28)
KCiK 0
iK 0
Z
r.u/du C
KiK 0
iK 0
r.u/du D 0:
We can evaluate these integrals by first writing r.u/ in the form
sn2 .u/
ˇ ˇ0
(3.29)
r.u/ D ˇ C
sn2 .u1 / 1 sn22 .u/
sn .u /
1
and using the functions
‚.u/ D #4
(3.30)
u 2K
;
Z.u/ D
‚0 .u/
:
‚.u/
The addition formulae for the sn and Z functions are (see [29])
(3.31)
sn.u ˙ a/ D
sn.u/ cn.a/ dn.a/ ˙ sn.a/ cn.u/ dn.u/
;
1 k 2 sn2 .a/ sn2 .u/
Z.u ˙ a/ D Z.u/ ˙ Z.a/ k 2 sn.u/ sn.a/ sn.u ˙ a/:
Thus we have
Z.u a/ Z.u C a/ C 2Z.a/
D k 2 sn.u/ sn.a/.sn.u C a/ C sn.u a//
(3.32)
k 2 sn.u/ sn.a/Œ2 sn.u/ cn.a/ dn.a/
1 k 2 sn2 .a/ sn2 .u/
2k 2 sn.a/ cn.a/ dn.a/ sn2 .u/
:
D
1 k 2 sn2 .a/ sn2 .u/
D
EXACT SOLUTION OF THE SIX-VERTEX MODEL
797
We also have the half- and quarter-period identities
sn.u C iK 0 / D
(3.33)
1
;
k sn.u/
cn.u C iK 0 / D
dn.u C iK 0 / D i
i dn.u/
;
k sn.u/
cn.u/
:
sn.u/
In particular, notice that sn.u11 / D k sn.u1 CiK 0 /. Using the addition formulae
(3.32), we can write r.u/ as
ˇ ˇ0
r.u/ D ˇ C
2k 2 sn.a/ cn.a/ dn.a/ sn2 .u1 /
(3.34)
.Z.u a/ Z.u C a/ C 2Z.a//;
where a D u1 C iK 0 (see Figure 3.1). From (3.33) and (3.19), it follows that
(3.35)
ˇ ˇ0
D K:
2k 2 sn.u1 C iK 0 / cn.u1 C iK 0 / dn.u1 C iK 0 / sn2 .u1 /
Thus we can write (3.34) as
r.u/ D ˇ K Z.u u1 iK 0 /
(3.36)
Z.u C u1 C iK 0 / C 2Z.u1 C iK 0 / :
If we write u D x C iK 0 in the first integral of (3.28) and u D x iK 0 in the
second, we obtain
Z K
2ˇ 4KZ.u1 C iK 0 / KŒZ.x u1 / Z.x C u1 C 2iK 0 /
(3.37)
0
C Z.x u1 2iK 0 / Z.x C u1 / dx D 0:
From the periodic properties of #4 , it follows that
(3.38)
Z.u ˙ 2iK 0 / D Z.u/ i
;
K
so we can write (3.37) as
Z K
2ˇ 4KZ.u1 C iK 0 / 2
i
(3.39)
0
C 2KŒZ.x C u1 / Z.x u1 / dx D 0:
This equation is readily integrated, as Z is the logarithmic derivative of the ‚
function. Integrating gives
(3.40)
‚.x C u1 / K
0 D .2ˇ 4KZ.u1 C iK 0/ 2
i /x C 2K log
‚.x u1 / xD0
‚.K C u1 / ‚.u1 /
D 2Kˇ 4K 2 Z.u1 C iK 0 / 2K
i C 2K log
:
‚.K u1 / ‚.u1 /
798
P. M. BLEHER AND K. LIECHTY
The logarithmic term in this equation is 0 due to the evenness and periodicity (period 2K) of the ‚-function and the fact that the relevant term in the integration is
real on the entire contour of integration. Thus we have that
ˇ D 2KZ.u1 C iK 0 / C i:
(3.41)
From (1.41), we can deduce that
Z.u1 C iK / D 2K
0
(3.42)
#20 . !2 /
Ci
#2 . !2 /
and write (3.41) as
ˇ D (3.43)
#20 . !2 /
:
#2 . !2 /
This equation, together with equations (3.23) and (3.21), determine the endpoints
˛; ˛ 0 ; ˇ 0 ; ˇ. In fact, similar to (3.43) we have the following explicit formulae for
the other endpoints:
(3.44)
˛ D #10 . !2 /
;
#1 . !2 /
˛ 0 D #40 . !2 /
;
#4 . !2 /
ˇ 0 D #30 . !2 /
;
#3 . !2 /
which follow from (3.23), (3.43), and the identities (see [29]),
#10 .´/#4 .´/ #4 .0/2 #2 .´/#3 .´/
;
#1 .´/
# 0 .´/#2 .´/ #2 .0/3 #3 .´/#4 .´/
;
#20 .´/ D 1
#1 .´/
# 0 .´/#3 .´/ #3 .0/2 #2 .´/#4 .´/
:
#30 .´/ D 1
#1 .´/
#40 .´/ D
(3.45)
Similarly, in addition to the formulae (3.23) for distances between turning points,
we get (2.16).
P ROOF OF P ROPOSITION 2.2: From equations (3.13), (3.1), and (3.15) we obtain formula (2.20); compare [31]. From formula (2.20) and equations (2.8) and
(2.14), we obtain that the equilibrium density function .x/ is given by formulae
(2.18). We are left to prove formula (2.19).
By (3.1), (3.15), and (2.18), on the interval Œˇ 0 ; ˇ,
.x/ D
(3.46)
It follows that
(3.47)
Z
ˇ
ˇ0
1
uC .x/ for x 2 Œˇ 0 ; ˇ:
iK
1
.x/dx D
iK
Z
ˇ
ˇ0
1
uC .x/dx D
iK
Z
0
iK 0
ur 0 .u/du;
EXACT SOLUTION OF THE SIX-VERTEX MODEL
799
where r.u/ is defined in (3.26). If we use equation (3.36) together with formula
(3.41), we can write r.u/ as
r.u/ D i
KŒZ.u u1 iK 0 / Z.u C u1 C iK 0 /:
(3.48)
Integrating (3.47) by parts, we get
Z 0
ur 0 .u/du D
(3.49)
iK 0
‚.u1 iK 0 /‚.u1 C 2iK 0 /
ˇ iK K K log
‚.u1 /‚.u1 C iK 0 /
0
0
0
:
Using the fact that ‚ is an even function and the identity
‚.u C 2iK 0 / D e i e i e
(3.50)
i u
K
‚.u/;
we can write (3.49) as
Z 0
K 0 i
u1
0
0
0
0
ur .u/du D ˇ iK K C K i
C K
K
(3.51)
iK 0
0 0
D i.K
ˇ K u1 /:
Remark: There is some question here as to which branch of the logarithm to take,
but it is clear that we have chosen the correct branch, as it is the only one that gives
Rˇ
0 < ˇ 0 .x/dx < 1.
Thus, from (3.47) and (3.51), we have
Z ˇ
1
i.K
ˇ 0 K 0 u1 /
.x/dx D
iK
ˇ0
(3.52)
ˇ0
1
ˇ 0 K 0 u1
D1
I
D1
K
K
2
2
hence by (2.18),
Z
(3.53)
ˇ
Z
.x/dx D
0
D
ˇ0
0
ˇ0
2
Z
.x/dx C
C1
ˇ
ˇ0
.x/dx
ˇ0
1
1C
D
;
2
2
2
which proves formula (2.19).
P ROOF OF P ROPOSITION 2.3: By taking x D ˇ we obtain from (2.30) that
(3.54)
l D 2g.ˇ/ V .ˇ/ D 2g.ˇ/ .1 /ˇ:
We also have that
(3.55)
lim Œg.A/ log A D 0I
A!1
800
P. M. BLEHER AND K. LIECHTY
hence
l D 2 lim Œg.A/ g.ˇ/ log A .1 /ˇ
A!1
Z A
!.´/ d´ log A .1 /ˇ:
D 2 lim
(3.56)
A!1
ˇ
Writing this integral in terms of u (so ´ D r.u/) gives
Z
(3.57)
1
0
l D 2 lim
.u u1 /r .u/du C log A .1 /ˇ
A!1 0 K
Z B
1
0
.u u1 /r .u/du C log r.B/ .1 /ˇ;
D 2 lim
B!u1
0 K
B
where A D r.B/. Integrating by parts gives
l D 2 lim
B!u1
(3.58)
ˇB
ˇ
1
.u u1 /r.u/ˇˇ
K
uD0
Z B
1
r.u/du C log r.B/ .1 /ˇ:
K 0
From (3.26) we obtain that r.0/ D ˇ and
.ˇ ˇ 0 / sn.u1 /
2 sn0 .u1 /
.ˇ ˇ 0 / sn.u1 /
D KI
D
2 cn.u1 / dn.u1 /
lim .B u1 /r.B/ D (3.59)
B!u1
hence
(3.60)
ˇ.10 /
l D 2 1 C
2
Z B
1
r.u/du log r.B/ .1 /ˇ
2 lim
B!u1 K 0
Z B
1
r.u/du log r.B/ :
D 2 2 lim
B!u1 K 0
Using equation (3.48) for r.u/, we immediately get that
(3.61)
1
K
Z
B
0
B
i
‚.B C u1 C iK 0 /
r.u/du D
C log
:
K
‚.B u1 iK 0 /
EXACT SOLUTION OF THE SIX-VERTEX MODEL
801
Now using equation (3.26) for r.u/, we have
lim
B!u1
(3.62)
1
K
Z
B
r.u/du log r.B/
0
‚.B C u1 C iK 0 / sn2 .u1 / sn2 .B/
u1 i
C lim log
D
B!u1
K
‚.B u1 iK 0 / ˇ sn2 .u1 / ˇ 0 sn2 .B/
u1 i
‚.2u1 C iK 0 /2 sn.u1 / sn0 .u1 /
D
C log
K
‚0 .iK 0 /.ˇ ˇ 0 / sn2 .u1 /
i u1
2e K #1 . uK1 /
u1 i
C log
D
K
#10 .0/
2#1 .!/
D log
:
#10 .0/
Plugging this into (3.60) gives
#10 .0/
;
l D 2 C 2 log
2#1 .!/
(3.63)
and thus we obtain that
e l=2 D
(3.64)
#10 .0/
:
2e#1 .!/
4 Riemann-Hilbert Approach: Interpolation Problem
The Riemann-Hilbert approach to discrete orthogonal polynomials is based on
the following interpolation problem (IP), which was introduced by Borodin and
Boyarchenko [9] as the “discrete Riemann-Hilbert problem.” See also the monograph [2] of Baik, Kriecherbauer, McLaughlin, and Miller, in which it is called the
“interpolation problem.”
We will consider the lattice Ln defined in (1.26) and the weight wn .x/ defined
in (1.24).
I NTERPOLATION P ROBLEM. For a given n D 0; 1; : : :, find a 2 2 matrixvalued function Pn .´/ D .Pnij .´//1i;j 2 with the following properties:
(1) Analyticity. Pn .´/ is an analytic function of ´ for ´ 2 C n Ln .
(2) Residues at poles. At each node x 2 Ln , the elements Pn11 .´/ and Pn21 .´/
of the matrix Pn .´/ are analytic functions of ´, and the elements Pn12 .´/
and Pn22 .´/ have a simple pole with the residues
(4.1)
Res Pnj 2 .´/ D wn .x/Pnj1 .x/;
´Dx
j D 1; 2:
(3) Asymptotics at infinity. There exists a function r.x/ > 0 on Ln such that
(4.2)
lim r.x/ D 0;
x!1
802
(4.3)
P. M. BLEHER AND K. LIECHTY
and such that as ´ ! 1, Pn .´/ admits the asymptotic expansion
n
P1
P2
´
0
Pn .´/ I C
C 2 C ;
0 ´n
´
´
[
1
D x; r.x/ ;
´2Cn
x2Ln
where D.x; r.x// denotes a disk of radius r.x/ > 0 centered at x and I is
the identity matrix.
It is not difficult to see (see [2, 9]) that the IP has a unique solution, which is
Pnn .´/
C.wn Pnn /.´/
(4.4)
Pn .´/ D
.hn;n1 /1 Pn;n1 .´/ .hn;n1 /1 C.wn Pn;n1 /.´/
where the Cauchy transformation C is defined by the formula
X f .x/
;
(4.5)
C.f /.´/ D
´x
x2Ln
and Pnk .´/ D ´k C are the orthogonal polynomials defined in (1.27). Because
of the orthogonality condition, as ´ ! 1,
C.wn Pnk /.´/ D
1
1
1
X
X
X
wn .x/Pnk .x/
xj
wn .x/Pnk .x/
´x
´j C1
j D0
x2Ln
x2Ln
(4.6)
D
hnk
C
´kC1
1
X
j DkC2
aj
;
´j
which justifies asymptotic expansion (4.3). We have that
h1
n;n1 D ŒP1 21 :
hnn D ŒP1 12 ;
(4.7)
5 Reduction of IP to RHP
5.1 Preliminary Considerations
We would like to reduce the interpolation problem to a Riemann-Hilbert problem (RHP). Introduce the function
….´/ D
(5.1)
2
n
´
sin
:
n
2
Observe that
….xk / D 0;
0
… .xk / D .1/ ;
k
(5.2)
for xk D
i n
xk
exp
2
2k
2 Ln :
n
D .1/k
EXACT SOLUTION OF THE SIX-VERTEX MODEL
Introduce the upper-triangular matrices
Du˙ .´/
(5.3)
n .´/ ˙
1 w….´/
e
D
0
1
and the lower-triangular matrices
Dl˙
(5.4)
D
….´/1
wn1.´/ e ˙
0
i n´
2
i n´
2
!
!
….´/
1
0
….´/1
i n´
D
1
0
….´/
….´/w
e ˙ 2
n .´/
Define the matrix-valued functions,
(
Run D Pn .´/ (5.5)
and
(
Rln D Pn .´/ (5.6)
DuC .´/
Du .´/
803
!
0
1
:
when Im ´ 0;
when Im ´ 0;
DlC .´/ when Im ´ 0;
Dl .´/ when Im ´ 0:
From (4.4) we have that
Run .´/
0
nn .´/ ˙
wn .´/P
e
….´/
Pnn .´/
D@
h1
n;n1 Pn;n1 .´/
i n´
2
i n´
wn .´/h1
n;n1 Pn;n1 .´/ ˙ 2
e
….´/
C C.wn Pnn /.´/
C h1
n;n1 C.wn Pn;n1 /.´/
1
A
when ˙ Im ´ 0;
and
Rln .´/
0
D
C.wn Pnn /.´/ ˙ i n´
Pnn .´/
e 2
….´/ wn .´/
@ 1
1
h
C.wn Pn;n1 /.´/ ˙ i n´
hn1 Pn;n1 .´/
n;n1 wn .´/
e 2
….´/
1
….´/C.wn Pnn /.´/
A
….´/h1
C.w
P
/.´/
n n;n1
n;n1
when ˙ Im ´ 0:
Observe that the functions Run .´/ and Rln .´/ are meromorphic on the closed
quadrants of the complex plane, and they are two-valued on the real and imaginary
axes. Their possible poles are located on the lattice Ln . An important result is that,
in fact, due to some cancellations, they do not have any poles at all. We have the
following proposition:
P ROPOSITION 5.1 The matrix-valued functions Run .´/ and Rln .´/ have no poles,
and on the real line they satisfy the following jump conditions at x 2 R:
!
n iwn .x/
1
(5.7)
;
RunC .x/ D Run .x/jRu.x/; jRu .x/ D
0
1
804
P. M. BLEHER AND K. LIECHTY
and
(5.8)
RlnC .x/
D
Rln .x/jRl .x/;
jRl .x/
D
1
i
wn
n .x/
0
:
1
P ROOF : It follows from the definition of Run .´/ that all possible poles of Run .´/
are located on the lattice Ln . Let us show that the residue of all these poles is equal
to 0. Consider any xk 2 Ln . The residue of the matrix element Run;12 .´/ at xk is
equal to
(5.9)
Res Run;12 .´/ D ´Dxk
wn .xk /Pnn .xk /
.1/k C wn .xk /Pnn .xk / D 0:
.1/k
Similarly, we get that
Res Rn;22 .´/ D 0I
(5.10)
´Dxk
hence Run .´/ has no pole at xk .
Likewise, the residue of the matrix element Rln;11 .´/ at xk is equal to
(5.11)
Res Rln;11 .´/ D
´Dxk
Pnn .xk / wn .xk /Pnn .xk /.1/k
D 0:
wn .xk /
.1/k
In the same way we obtain that
Res Rn;21 .´/ D 0:
(5.12)
´Dxk
In the entry Rln;21 .´/, the pole of the function C.wn Pn /.´/ at ´ D xk is cancelled
by the zero of the function ….´/; hence Rln;21 .´/ has no pole at xk . Similarly,
Rln;22 .´/ has no pole at xk as well; hence Rln .´/ has no pole at xk .
Let us evaluate the jump matrices at x 2 R. From (5.5) we have that
!
wn .x/
nx
1 ….x/ 2i sin 2
jRu .x/ D Du .x/1 DuC .x/ D
0
1
!
(5.13)
1 n iwn .x/
D
;
0
1
which proves (5.7). Similarly,
jRl .x/
(5.14)
which proves (5.8).
D
Dl .x/1 DlC .x/
D
1
1
2i sin nx
….x/w
2
n .x/
1
0
D
;
i
wn
1
n .x/
0
1
EXACT SOLUTION OF THE SIX-VERTEX MODEL
Ω∆
+
Ω+
Ω−
α’
805
Ω+
β’
Ω∆
−
Ω−
F IGURE 5.1. The contour †.
α’
β’
0
F IGURE 5.2. The contour †R .
5.2 Reduction of IP to RHP
Let us discuss how to reduce the interpolation problem to a Riemann-Hilbert
problem. We follow the work [2] with some modifications. Denote
(5.15)
D Ln \ Œ˛ 0 ; ˇ 0 ;
r D Ln n :
Consider the oriented contour † on the complex plane depicted in Figure 5.1, in
which the horizontal lines are Im ´ D "; 0; ", where " > 0 is a small positive
constant that will be determined later, and the vertical segments pass through the
r
points ´ D ˛ 0 and ´ D ˇ 0 . Also consider the regions ˙ and ˙ bounded by the
contour †; see Figure 5.1. Observe that the regions r
˙ consist of two connected
.
components, to the left and to the right of ˙
Define
K R .´/K
1
u
n
n
Kn Rln .´/K1
n
Kn Pn .´/K1
n
n
(5.16)
Rn .´/ D
for ´ 2 r
˙;
for ´ 2 ˙;
otherwise:
1 0 where Kn D 0 ni .
Define a contour †R by adding to the contour † a vertical segment Œi"; i";
see Figure 5.2. If A C is a set on the complex plane and b 2 C, then, as usual,
we denote
(5.17)
A C b D f´ D a C b; a 2 Ag:
P ROPOSITION 5.2 The matrix-valued function Rn .´/ has the following jumps on
the contour †R :
(5.18)
RnC .´/ D Rn .´/jR .´/;
806
where
ˆ
P. M. BLEHER AND K. LIECHTY
1 wn .´/
0
1
!
when ´ 2 .1; ˛ 0 / [ .ˇ 0 ; 1/;
!
1
0
when ´ 2 Œ˛ 0 ; ˇ 0 ;
2 w .´/1 1
/
. n
n
1
0
i n´
˙ 2
(5.19)
jR .´/ D
i wn .´/e
A
@1 n
….´/
Kn Du˙ .´/K1
n D
0
1
when ´ 2 .1; ˛ 0 / [ .ˇ 0 ; 1/ ˙ i";
0
1
….´/1
0
@
A
Kn Dl˙ .´/K1
˙ i n´
n D
2
i n e
….´/
wn .´/
when ´ 2 .˛ 0 ; ˇ 0 / ˙ i";
Kn Dl˙ .´/1 Du˙ .´/K1
n D
….´/
i n´
˙ i n´
2
n i wn .´/e
i n´
i ˙ 2
n
e
!
i
1 ˙ 2
n
wn .´/ e
when ´ 2 .0; ˙i"/ C ˛ 0 or ´ 2 .0; ˙i"/ C ˇ 0 ;
Kn D0˙ .´/K1
n
when ´ 2 .0; ˙i"/;
and
(5.20)
D0˙ .´/ D
1
0
i n´
n´ e ˙ 2
2 sinh.n´/e….´/
1
!
:
Notice that the jumps on vertical contours close to the origin, D0˙ .´/, are exponentially close to the identity matrix.
6 First Transformation of the RHP
Define the matrix function Tn .´/ as follows from the equation
(6.1)
Rn .´/ D e
nl
2 3
l
Tn .´/e n.g.´/ 2 /3 ;
where the Lagrange multiplier l and the function g.´/ are as described in Section 2
and 3 D . 10 10 / is the third Pauli matrix. Then Tn .´/ satisfies the following
Riemann-Hilbert problem:
(1) Tn .´/ is analytic in C n †R .
(2) TnC .´/ D Tn .´/jT .´/ for ´ 2 †R , where
(
l
l
e n.g .´/ 2 /3 jR .´/e n.gC .´/ 2 /3 for ´ 2 R
(6.2)
jT .´/ D
l
l
for ´ 2 †R n R:
e n.g.´/ 2 /3 jR .´/e n.g.´/ 2 /3
EXACT SOLUTION OF THE SIX-VERTEX MODEL
(3) Tn .´/ I C
T1
´
C
T2
´2
807
C as ´ ! 1.
From (2.22) we have that
g.´/ D log ´ C O.´1 /
(6.3)
as ´ ! 1:
This implies that
ŒT1 12 D e nl ŒR1 12 :
(6.4)
Let’s take a closer look at the behavior of the jump matrix jT described in (6.2)
on the horizontal segments of †R . We have that
(6.5) jT .´/ D
!
8 nG.´/
e
e n.gC .´/Cg .´/V .´/l/
ˆ
ˆ
when ´ 2 .1; ˛ 0 / [ .ˇ 0 ; 1/;
ˆ
ˆ
ˆ
0
e nG.´/
ˆ
ˆ
!
ˆ
ˆ
ˆ
e nG.´/
0
ˆ
ˆ
ˆ
when ´ 2 .˛ 0 ; ˇ 0 /;
ˆ
ˆ
/2 e n.gC .´/Cg .´/V .´/l/ e nG.´/
. n
ˆ
ˆ
ˆ
1
0
ˆ
ˆ
e ˙nG.´/
ˆ
< 1 ˙
i n "n
@
when ´ D x C i" 2 .˛; ˛ 0 / [ .ˇ 0 ; ˇ/ ˙ i";
1e e A
ˆ
1
ˆ
ˆ 00
1
ˆ
ˆ
ˆ 1 ˙ en.2g.´/lV .´//
ˆ
i nx "n
ˆ
ˆ
@
ˆ
1e e A when ´ D x ˙ i" 2 .1; ˛/ [ .ˇ; 1/ ˙ i";
ˆ
ˆ
ˆ
0
1
ˆ
ˆ
!
ˆ
ˆ
ˆ
….´/1
0
ˆ
ˆ
when ´ D x ˙ i" 2 .˛ 0 ; ˇ 0 / ˙ i":
:̂ i n ˙ i nx
2 e n.2g.´/lV .´//
….´/
e
˚
According to the properties of the g-function, we have the following proposition:
P ROPOSITION 6.1 The jump function jT has the following large-n asymptotics on
the real axis:
!
e nG.´/
0
for ´ 2 .˛ 0 ; ˇ 0 /;
O.e nC.´/ / e nG.´/
!
e nG.´/
1
for ´ 2 .˛; ˛ 0 / [ .ˇ 0 ; ˇ/;
0
e nG.´/
!
(6.6) jT .´/ D
1 O.e nC.´/ /
for ´ 2 .1; ˛/ [ .ˇ; 1/;
0
1
!
n
1 e ˙nG.´/ O.e /
for ´ 2 .˛; ˛ 0 / [ .ˇ 0 ; ˇ/ ˙ i";
0
1
where C.´/ is a positive continuous function on any subset of the given interval
that is bounded away from the endpoints of each interval and satisfies
(6.7)
C.´/ > cj´ C 1j
for some c > 0:
808
P. M. BLEHER AND K. LIECHTY
α
α’
β’
0
β
F IGURE 7.1. The contour †S .
7 Second Transformation of the RHP
The second transformation is based on two observations. The first is the wellknown “opening of the lenses” in a neighborhood of the unconstrained support of
the equilibrium measure. Namely, notice that, for x 2 .˛; ˛ 0 / [ .ˇ 0 ; ˇ/, the jump
matrix jT .x/ factors as
(7.1)
jT .x/ D
e nG.´/
1
0
e nG.´/
D
1
e nG.x/
0
1
0 1
1 0
1
e nG.x/
0
1
D j .x/jM jC .x/;
which allows us to reduce the jump matrix jT to the one jM plus asymptotically
small jumps on the lens boundaries. The second observation consists of two facts.
First, the jumps on the segments Œ˛ 0 ; ˇ 0 ˙ i" behave, for large n, as ˙e ˙i n´=.2 / .
Second, note that, for x 2 Œ˛ 0 ; ˇ 0 , G.x/ is a linear function with slope i . With
these facts in mind, we make the second transformation of the RHP. Let
(7.2) Sn .´/ D
8
ˆ
Tn .´/jC .´/1
ˆ
ˆ
ˆ
ˆ
ˆ
Tn .´/j .´/
ˆ
ˆ
!
ˆ
i n´
ˆ
2
ˆ
0
ˆ
< Tn .´/ n i e
i i n´
e 2
0
n
!
ˆ
i n´
ˆ
ˆ
ˆ
e 2
0
ˆ
n
i
ˆ
Tn .´/
ˆ
ˆ
n i i n´
ˆ
e 2
0
ˆ
ˆ
:̂ T .´/
n
for ´ 2 f.˛; ˛ 0 / [ .ˇ 0 ; ˇ/g .0; i "/;
for ´ 2 f.˛; ˛ 0 / [ .ˇ 0 ; ˇ/g .0; i "/;
for ´ 2 .˛ 0 ; ˇ 0 / .0; i "/;
for ´ 2 .˛ 0 ; ˇ 0 / .0; i "/;
otherwise:
This function satisfies a similar RHP to T, but jumps now occur on a new contour, †S , which is obtained from †R by adding the two segments .˛ i"; ˛ C i"/
and .ˇ i"; ˇ C i"/; see Figure 7.1.
EXACT SOLUTION OF THE SIX-VERTEX MODEL
809
On the horizontal segments for which the jump function jS differs from jT , we
have that, as n ! 1,
(7.3) jS .´/ D
!
8
0
1
ˆ
ˆ
ˆ
ˆ
ˆ
1 0
ˆ
ˆ
!
ˆ
ˆ
ˆ
1 C O.e "n= / O.e nŒG.´/
= ˆ
ˆ
ˆ
ˆ
ˆ
e nG.´/
1
ˆ
ˆ
!
ˆ
ˆ
ˆ
"n=
nŒG.´/
=
ˆ
/
O.e
1
C
O.e
ˆ
ˆ
ˆ
<
e nG.´/
1
!
n= /
ˆ
1
C
O.e
0
ˆ
ˆ
ˆ
ˆ
n i n.2g.´/lV .´//
ˆ
1 C O.e n= /
ˆ
e
ˆ
!
ˆ
ˆ
ˆ
1 C O.e n= /
0
ˆ
ˆ
ˆ
ˆ
i n.2g.´/lV .´//
ˆ
1 C O.e n= /
ˆ n
ˆ
e
ˆ
!
ˆ
ˆ
ˆ
e n i.1C/
0
ˆ
:̂ n.gC .´/Cg .´/lV .´//
e n i.1C/
e
for ´ 2 .˛; ˛ 0 / [ .ˇ 0 ; ˇ/;
for ´ i" 2 .˛; ˛ 0 / [ .ˇ 0 ; ˇ/;
for ´ C i" 2 .˛; ˛ 0 / [ .ˇ 0 ; ˇ/;
for ´ 2 fŒ˛ 0 ; ˇ 0 C i"g;
for ´ 2 fŒ˛ 0 ; ˇ 0 i"g;
for ´ 2 Œ˛ 0 ; ˇ 0 :
By formula (2.32) for the G-function and the upper constraint (2.10) on the density , we obtain that, for sufficiently small > 0,
0 < Re G.x ˙ i/ D 2
.x/ C O. 2 /
<
C O. 2 / for x 2 .˛; ˛ 0 / [ .ˇ 0 ; ˇ/:
This, combined with property (2.30) of the g-function, implies that all jumps on
horizontal segments are exponentially close to the identity matrix, provided that
they are bounded away from the segment Œ˛; ˇ. For what follows we denote
Z ˇ
.x/dx D C n
.1 C /;
(7.4)
n D C n2
0
so that
e n i.1C / D e i n :
(7.5)
8 Model RHP
The model RHP appears when we drop in the jump matrix jS .´/ the terms that
vanish as n ! 1:
(1) M.´/ is analytic in C n Œ˛; ˇ.
(2) MC .´/ D M .´/jM .´/ for ´ 2 Œ˛; ˇ, where
!
0 1
for ´ 2 Œ˛; ˛ 0 [ Œˇ 0 ; ˇ;
(8.1)
jM .´/ D
1 0
e i n 3
for ´ 2 Œ˛ 0 ; ˇ 0 :
810
P. M. BLEHER AND K. LIECHTY
(3) As ´ ! 1;
M1
M2
C 2 C :
´
´
This model problem was first solved, in the general multicut case, in [12], and
is solved in two steps. In the first step, we solve the following auxiliary RHP:
(1) Q.´/ is analytic in C n Œ˛; ˛ 0 [ Œˇ 0 ; ˇ.
0 1 / for ´ 2 Œ˛; ˛ 0 [ Œˇ 0 ; ˇ.
(2) QC .´/ D Q .´/. 1
0
1
(3) Q.´/ D I C O.´ / as ´ ! 1.
This RHP has the unique solution (see [12])
!
1
1
M.´/ I C
(8.2)
Q.´/ D
(8.3)
.´/C .´/
2
.´/ 1 .´/
2i
where
.´/ D
(8.4)
.´/ .´/
2i
.´/C 1 .´/
2
.´ ˛/.´ ˇ 0 /
.´ ˛ 0 /.´ ˇ/
1=4
with cuts on Œ˛; ˛ 0 [ Œˇ 0 ; ˇ, taking the branch such that .´/ 1 as ´ ! 1.
To solve the model RHP described in (8.1) and (8.2), we again use elliptic functions. Define the function
#3 .s C d C c/
(8.5)
f .s/ D
#3 .s C d /
where #3 is as defined in (1.37) with elliptic nome
2
i
i
2
;
D
De
qDe
2
and d and c are arbitrary complex numbers. Notice that f has the periodic properties
f .s C / D f .s/;
(8.6)
f .s C / D e 2ic f .s/;
and that f is an even function. Now let
u.´/ D
u.´/
Q
D
2K
2
(8.7)
Z
´
ˇ
d´0
p
R.´0 /
where u is as defined in (3.1). Then uQ is two-valued on Œ˛; ˇ and satisfies
uQ C .x/ uQ .x/ D (8.8)
for x 2 Œ˛ 0 ; ˇ 0 :
Also,
; uQ ˙ .˛ 0 / D ˙
; uQ ˙ .ˇ 0 / D ˙ ; uQ ˙ .ˇ/ D 0I
2
2
2
2
p
p
compare Figure 3.1. Because R.x/C D R.x/ for x 2 Œ˛; ˛ 0 [ Œˇ 0 ; ˇ, it
immediately follows that
(8.9)
(8.10)
uQ ˙ .˛/ D
uQ C .x/ C uQ .x/ D 0
for x 2 Œˇ 0 ; ˇ;
EXACT SOLUTION OF THE SIX-VERTEX MODEL
811
and that
(8.11)
uQ C .x/ C uQ .x/ D uQ C .˛ 0 / uQ C .ˇ 0 / C uQ .˛ 0 / uQ .ˇ 0 /
D
for x 2 Œ˛; ˛ 0 :
We now define
f1 .´/ D
(8.12)
Q
C d C 2n /
#3 .u.´/
;
#3 .u.´/
Q
C d/
#3 .u.´/
Q
C d C 2n /
;
f2 .´/ D
#3 .u.´/
Q
C d/
for ´ 2 C n Œ˛; ˇ;
where d is an arbitrary complex number. It then follows from (8.6) and (8.8) that
f1C .x/ D e i n f1 .x/;
(8.13)
f2C .x/ D e
i n
for x 2 Œ˛ 0 ; ˇ 0 ;
f2 .x/;
and from (8.6), (8.10), and (8.11) that
(8.14)
f1C .x/ D f2 .x/ and
f2C .x/ D f1 .x/ for x 2 Œ˛; ˛ 0 [ Œˇ 0 ˇ:
Define the matrix-valued function
0
(8.15)
F.´/ D
n
#3 .u.´/Cd
Q
1C 2 /
#
.
u.´/Cd
Q
3
1/
@
n
#3 .u.´/Cd
Q
2C 2 /
#3 .u.´/Cd
Q
2/
1
n
Q
#3 .u.´/Cd
1C 2 /
#3 .u.´/Cd
Q
1/
A
n
Q
#3 .u.´/Cd
2C 2 /
#3 .u.´/Cd
Q
2/
where d1 and d2 are yet undetermined complex constants. Then, from (8.13) and
(8.14) we have that
i n
e
0
FC .x/ D F .x/
for x 2 Œ˛ 0 ; ˇ 0 ;
0
e i n
(8.16)
0 1
FC .x/ D F .x/
for x 2 Œ˛; ˛ 0 [ Œˇ 0 ; ˇ:
1 0
We can now combine (8.3) and (8.15) to obtain
M.´/ D
(8.17)
0
n
Q
.´/C 1 .´/ #3 .u.´/Cd
1C 2 /
1 @
2
#3 .u.´/Cd
Q
1/
F.1/
n
Q
.´/ 1 .´/ #3 .u.´/Cd
2C 2 /
2i
#3 .u.´/Cd
Q
2/
where
(8.18)
1
n
Q
.´/ 1 .´/ #3 .u.´/Cd
1C 2 /
2i
#3 .u.´/Cd
Q
1/
A
n
Q
.´/C 1 .´/ #3 .u.´/Cd
2C 2 /
2
#3 .u.´/Cd
Q
2/
0
F.1/ D
#3 .u
Q 1 Cd1 C 2n /
@ #3 .uQ 1 Cd1 /
0
1
0
#3 .u
Q 1 Cd2 C 2n /
#3 .u
Q 1 Cd2 /
A
Q
This matrix satisfies conditions (8.1) and (8.2) of the model
and uQ 1 u.1/.
RHP, but may not be analytic on C n Œ˛; ˇ, as it may have some poles at the zeroes
of #3 .˙u.´/
Q
C d1;2 /. However, we can choose the constants d1 and d2 such that
812
P. M. BLEHER AND K. LIECHTY
these zeroes coincide with the zeroes of .´/ ˙ 1 .´/ and are thus cancelled in
the product.
First consider the zeroes of .´/˙ 1 .´/. These are the zeroes of 2 .´/˙1 and
thus of 4 .´/ 1; thus there is only one zero, which uniquely solves the equation
p.´/ (8.19)
.´ ˛/.´ ˇ 0 /
D 1;
.´ ˛ 0 /.´ ˇ/
which is
(8.20)
x0 D
ˇ˛ 0 ˛ˇ 0
2 .˛ 0 ; ˇ 0 /:
.˛ 0 ˛/ C .ˇ ˇ 0 /
It is easy to check that .x0 / D 1; thus x0 is the unique zero of .´/ 1 .´/,
whereas there are no zeroes of .´/ C 1 .´/ on the specified sheet. We use here
the change of variables v defined in (3.2). Notice that, by (3.19),
(8.21)
v.x0 / D
ˇ0 ˛
dn2 .u1 /
D
;
ˇ0 ˛0
k 2 cn2 .u1 /
implying that
sn2 .u.x0 // D
(8.22)
dn2 .u1 /
:
k 2 cn2 .u1 /
Since x0 2 .˛ 0 ; ˇ 0 /, we must have u.x0 / 2 .iK 0 ; K C iK 0 / (if we choose to
take uC ). Since sn2 is a one-to-one function on this interval there is a unique point
u0 2 .iK 0 ; K C iK 0 / such that sn2 .u0 / D dn2 .u1 /=k 2 cn2 .u1 /. The simple
period identity
sn.u C K C iK 0 / D
(8.23)
dn.u/
k cn.u/
along with (8.22) gives that we must have
u0 D u.x0 / D K u1 C iK 0 I
(8.24)
thus
.K u1 C iK 0 / D
C uQ 1 :
2K
2
2
We now consider zeroes of the function #3 .u.´/
Q
d / #3 .u.´/
Q
C d /. The
zeroes of this function are the solutions to the equation
(8.26)
u.´/
Q
d D .2m C 1/ C .2k C 1/
2
2
for any m; k 2 Z. Because uQ maps the first sheet of X to the rectangular domain
Œ0; 2 Œ 2 ; 2 , it is clear that this equation can have at most one solution, and
without any loss of generality we may take m D k D 0. Then, if we want the
solution of this equation to be x0 , we need to let
(8.27)
d D u.x
Q 0 / .1 C / D uQ 1 :
2
(8.25)
u.x
Q 0/ D
EXACT SOLUTION OF THE SIX-VERTEX MODEL
813
This choice of d also ensures that #3 .u.´/
Q
C d / #3 .u.´/
Q
d / has no zeroes
on the first sheet of X . We can then let
d1 D d;
(8.28)
d2 D d;
so that (8.17) and (8.18) become
0
M.´/ D
Q
C 2n /
.´/C 1 .´/ #3 .u.´/Cd
1 @
2
#3 .u.´/Cd
Q
/
F.1/
Q
C 2n /
.´/ 1 .´/ #3 .u.´/d
2i
#3 .u.´/d
Q
/
0
(8.29)
D
1
1
Q
.´/C 1 .´/ #3 .u.´/C.nC
2 /!/
!
2
#4 .u.´/C
Q
1 @
2/
F.1/
1
Q
.´/ 1 .´/ #3 .u.´/C.n
2 /!/
!
2i
#4 .u.´/
Q
2/
where
(8.30)
1
Q
C 2n /
.´/ 1 .´/ #3 .u.´/Cd
2i
#3 .u.´/Cd
Q
/
A
Q
C 2n /
.´/C 1 .´/ #3 .u.´/d
2
#3 .u.´/d
Q
/
1
0
F.1/ D
#3 . 2n /
@ #3 .0/
0
0
#3 . 2n /
#3 .0/
AD
1
Q
.´/ 1 .´/ #3 .u.´/.nC
2 /!/
!
2i
#4 .u.´/
Q
2/
A
1
Q
.´/C 1 .´/ #3 .u.´/.n
2 /!/
!
2
#4 .u.´/C
Q
2/
#4 .n!/
#3 .0/
0
#4 .n!/
#3 .0/
0
!
;
solving the model RHP. The asymptotics at infinity are
M.´/ D I C
(8.31)
M1
C O.´2 /
´
where the matrix M1 has the form
(8.32) M1 D
0
@
1
#3 . u
Q 1 d / .ˇ ˇ 0 /C.˛ 0 ˛/
Q 1 d C 2n /#3 .u
4i
#3 .u
Q 1 d C 2n /#3 .u
Q 1 d /
Q 1 Cd / .ˇ ˇ 0 /C.˛ 0 ˛/
#3 .u
Q 1 Cd C 2n /#3 .u
4i
#3 . u
Q 1 Cd C 2n /#3 .u
Q 1 Cd /
A
:
The matrix M1 can be written in a cleaner fashion and in terms of the original
parameters as follows:
P ROPOSITION 8.1 We have that
(8.33)
ŒM1 12 D
iA.!/#4 ..n C 1/!/
;
#4 .n!/
ŒM1 21 D
A.!/#4 .n!/
;
i#4 ..n 1/!/
where
(8.34)
!D
.1 C /
;
2
A.!/ D
#10 .0/
:
2#1 .!/
For a proof of this proposition, see appendix D of [6]. Notice that since M solves
the model RHP, we have that
(8.35)
det M.´/ D 1;
´ 2 C:
814
P. M. BLEHER AND K. LIECHTY
9 Parametrix at Outer Turning Points
We now consider small disks D.˛; "/ and D.ˇ; "/ centered at the outer turning
points. Denote D D D.˛; "/ [ D.ˇ; "/. We will seek a local parametrix Un .´/
defined on D such that
˚
(9.1)
Un .´/ is analytic on D n †S ;
(9.2)
UnC .´/ D Un .´/jS .´/ for ´ 2 D \ †S ;
Un .´/ D M.´/ I C O.n1 / uniformly for ´ 2 @D:
(9.3)
We first construct the parametrix near ˇ. The jumps jS are given by
0 1
1 0
!
for ´ 2 .ˇ "; ˇ/;
1
(9.4) jS .´/ D
e nG.´/
1
e nG.´/
!
0
1
!
0
1
for ´ 2 .ˇ; ˇ C i"/;
for ´ 2 .ˇ; ˇ i"/;
e nG.´/ e n.gC .´/Cg .´/V .´/l/
0
e nG.´/
!
for ´ 2 .ˇ; ˇ C "/:
If we let
(9.5)
Un .´/ D Qn .´/e n.g.´/
V .´/
l
2 2 /3
;
then the jump conditions on Qn become
†
QnC .´/ D Qn .´/jQ .´/
(9.6)
where
(9.7)
jQ .´/ D
0 1
1 0
1 0
1 1
!
1 0
1 1
!
1 1
0 1
!
for ´ 2 .ˇ "; ˇ/;
!
for ´ 2 .ˇ; ˇ C i"/;
for ´ 2 .ˇ; ˇ i"/;
for ´ 2 .ˇ; ˇ C "/;
where orientation is from left to right on horizontal contours, and down to up on
vertical contours, according to Figure 7.1.
EXACT SOLUTION OF THE SIX-VERTEX MODEL
815
Qn can be constructed using Airy functions. The Airy function solves the differential equation y 00 D ´y and has the following asymptotics at infinity:
5 3=2
1
32 ´3=2
3
1 ´
Ai.´/ D p 1=4 e
C O.´ / ;
48
2 ´
(9.8)
7 3=2
1 1=4 2 ´3=2
0
3
1C ´
C O.´ / ;
Ai .´/ D p ´ e 3
48
2 as ´ ! 1 with arg ´ 2 .
C "; "/ for any " > 0. If we let
(9.9)
y0 .´/ D Ai.´/;
y1 .´/ D ! Ai.!´/;
y2 .´/ D ! 2 Ai.! 2 ´/;
where ! D e 2 i=3 , then the functions y0 , y1 , and y2 satisfy the relation
†
y0 .´/ C y1 .´/ C y2 .´/ D 0:
(9.10)
If we take
(9.11)
ˆˇ .´/ D
y0 .´/ y2 .´/
y00 .´/ y20 .´/
!
y1 .´/ y2 .´/
y10 .´/ y20 .´/
!
y2 .´/ y1 .´/
y20 .´/ y10 .´/
!
y0 .´/ y1 .´/
y00 .´/ y10 .´/
!
for arg ´ 2 .0; 2 /;
for arg ´ 2 . 2 ; /;
for arg ´ 2 .
; 2 /;
for arg ´ 2 . 2 ; 0/;
then ˆˇ satisfies jump conditions similar to (9.7), but for jumps on rays emanating
from the origin rather than from ˇ. We thus need to map the disk D.ˇ; "/ onto
some convex neighborhood of the origin in order to take advantage of the function
ˆˇ . Our mapping should match the asymptotics of the Airy function in order to
have the matching property (9.3).
To this end notice that, by (2.18), for t 2 Œˇ 0 ; ˇ, as t ! ˇ,
.t/ D C.ˇ t/1=2 C O..ˇ t/3=2 /;
(9.12)
C > 0:
It follows that, as ´ ! ˇ for ´ 2 Œˇ 0 ; ˇ,
Z ˇ
(9.13)
.t/dt D C0 .ˇ ´/3=2 C O..ˇ ´/5=2 /;
´
Thus,
2
C0 D C:
3
2=3
Z
3
ˇ
.t/dt
2 ´
is analytic at ˇ and so extends to a conformal map from D.ˇ; "/ (for small enough
") onto a convex neighborhood of the origin. Furthermore,
(9.14)
ˇ .´/ D (9.15)
ˇ .ˇ/
D 0;
0
ˇ .ˇ/
> 0I
816
P. M. BLEHER AND K. LIECHTY
therefore ˇ is real negative on .ˇ "; ˇ/ and real positive on .ˇ; ˇ C "/. Also,
we can slightly deform the vertical pieces of the contour †S close to ˇ so that
ˇ fD.ˇ; "/
(9.16)
We now set
Qn .´/ D Eˇn .´/ˆˇ n2=3
(9.17)
so that
(9.18)
\ †S g D ."; "/ [ .i"; i"/:
Un .´/ D Eˇn .´/ˆˇ n2=3
ˇ .´/
ˇ .´/
e n.g.´/
V .´/
l
2 2 /3
where
Eˇn .´/ D M.´/Lˇn .´/1 ;
(9.19)
Lˇn .´/
n1=6
1
D p
2 1=4
.´/
ˇ
n1=6
0
!
0
1=4
.´/
ˇ
1 i
;
1 i
1=4
, which is positive on .ˇ; ˇ C "/ and has a cut
ˇ
ˇ
on .ˇ "; ˇ/. We claim that En .´/ is analytic in D.ˇ; "/; thus Un .´/ has the
ˇ
jump conditions of jS . This is clear, as both M and Ln have the same constant
jump, . 10 10 /, on the interval .ˇ "; ˇ and are analytic elsewhere. The only other
ˇ
possible singularity for either M or Ln is the isolated singularity at ˇ, and this
ˇ
is at most a fourth-root singularity and thus removable. It follows that En .´/ D
ˇ
M.´/Ln .´/1 is analytic on D.ˇ; "/; thus Un has the prescribed jumps in D.ˇ; "/.
where we take the branch of
We are left only to prove the matching condition (9.3). Using (9.8), one can
check that, for ´ in each of the sectors of analyticity, ˆˇ .n2=3 ˇ .´// satisfies the
following asymptotics as n ! 1:
ˆˇ .n2=3
(9.20)
ˇ .´//
1
1
D p n 6 3
2 e
2
3n
ˇ .´/
ˇ .´/
3=2 3
1
4 3
1 i
1 i
C
ˇ .´/
3=2
48n
5
7
5i
C O.n2 /
7i
;
where we always take the principal branch of ˇ .´/3=2 . As such,
two-valued for ´ 2 .ˇ "; ˇ/, so that
Z ˇ
2
3=2
.t/dt:
D i
(9.21)
ˇ .x/
3
x
˙
Notice that, by (2.30) and (2.33), for x 2 .ˇ "; ˇ/,
Z ˇ
(9.22)
2g˙ .x/ V .x/ D l ˙ 2
i
.t/dt:
x
ˇ .´/
3=2
is
EXACT SOLUTION OF THE SIX-VERTEX MODEL
817
This implies that
Z
Œ2gC .ˇ/ V .ˇ/ Œ2gC .x/ V .x/ D 2
i
.t/dt;
x
ˇ
Z
(9.23)
Œ2g .ˇ/ V .ˇ/ Œ2g .x/ V .x/ D 2
i
ˇ
.t/dt:
x
Combining these equations with (9.21) gives
2
1
3=2
(9.24)
D .2g˙ .ˇ/ V .ˇ// .2g˙ .x/ V .x// :
ˇ .x/
3
2
˙
This equation can be extended into the upper and lower planes, respectively, giving
(9.25)
2
3
ˇ .´/
3=2
D
1
.2g˙ .ˇ/ V .ˇ// .2g.´/ V .´// for ˙ Im ´ > 0:
2
Since, by (9.22), 2g˙ .ˇ/ V .ˇ/ D l, we get that
2
3
(9.26)
ˇ .´/
3=2
D g.´/ C
l
V .´/
C
2
2
for ´ throughout D.ˇ; "/. Plugging (9.20) and (9.26) into (9.18), we get
(9.27)
1
1
Un .´/ D M.´/Lˇn .´/1 p n 6 3
2 3=2
1 i
ˇ .´/
C
1 i
48n
V .´/
l
2 2 /3
e n.g.´/
D M.´/ I C
ˇ .´/
14 3
5
5i
2
C O.n /
7 7i
V .´/
l
e n.g.´/ 2 2 /3
3=2 1 6i
ˇ .´/
2
C O.n / :
6i 1
48n
Thus we have that Un satisfies conditions (9.1), (9.2), and (9.3).
A similar construction gives the parametrix at ˛ (see [6]). Namely, if we let
2=3
Z
3
´
.t/dt
;
(9.28)
˛ .´/ D 2 ˛
then ˛ is analytic throughout D.˛; "/, real-valued on the real line, and has negative derivative at ˛. Also, let
1
0
(9.29)
ˆ˛ .´/ D ˆˇ .´/
:
0 1
Then we can take
(9.30)
Un .´/ D E˛n .´/ˆ˛ .n2=3
˛ .´//e
l
n.g.´/ V .´/
2 2 /3
818
P. M. BLEHER AND K. LIECHTY
for ´ 2 D.˛; "/, where
E˛n .´/ D M.´/L˛n .´/1 ;
(9.31)
1
L˛n .´/ D p
2 n1=6
!
1 i
:
1=4
1 i
˛ .´/
1
6i
6i
1
1=4
.´/
˛
n1=6
0
0
Similar to (9.27), we get, as n ! 1,
(9.32)
Un .´/ D M.´/ I C
˛ .´/
3=2
48n
C O.n
2
/ :
10 Parametrix at the Inner Turning Points
We now consider small disks D.˛ 0 ; "/ and D.ˇ 0 ; "/ centered at the inner turning
points. Denote DQ D D.˛ 0 ; "/ [ D.ˇ 0 ; "/. We will seek a local parametrix Un .´/
defined on DQ such that
(10.1)
Un .´/ is analytic on DQ n †S :
(10.2)
UnC .´/ D Un .´/jS .´/
for ´ 2 DQ \ †S :
(10.3)
Un .´/ D M.´/.I C O.n1 //
Q
uniformly for ´ 2 @D:
We first construct the parametrix near ˛ 0 . Let
(10.4)
Q n .´/e Un .´/ D Q
†
i n´
2 3
e n.g.´/
V .´/
l
2 2 /3
for ˙ Im ´ 0:
Q n are
Then the jumps for Q
(10.5)
jQQ .´/ D
0 1
1 0
!
1
0
1 1
!
1 1
0
1
!
1 1
0 1
for ´ 2 .˛ 0 "; ˛ 0 /;
!
for ´ 2 .˛ 0 ; ˛ 0 C "/;
for ´ 2 .˛ 0 ; ˛ 0 C i"/;
for ´ 2 .˛ 0 ; ˛ 0 i"/;
where orientation is taken from left to right on horizontal contours, and down to
up on vertical contours according to Figure 7.1 (for a calculation of the jumps see
†
EXACT SOLUTION OF THE SIX-VERTEX MODEL
819
appendix C in [6]). We now take
(10.6)
!
y2 .´/ y0 .´/
y20 .´/ y00 .´/
!
y2 .´/ y1 .´/
y20 .´/ y10 .´/
!
y1 .´/ y2 .´/
y10 .´/ y20 .´/
!
y1 .´/ y0 .´/
y10 .´/ y00 .´/
ˆ˛ 0 .´/ D
for arg ´ 2 0; 2 ;
for arg ´ 2
2;
;
for arg ´ 2 ; 2 ;
for arg ´ 2 2 ; 0 :
Q n , but for jumps emanating from the
Then ˆ˛ 0 .´/ solves a RHP similar to that of Q
origin rather than from ˛ 0 .
Notice that, by (2.18), for t 2 Œ˛; ˛ 0 , as t ! ˛ 0 ,
(10.7)
.t/ D
1
C.˛ 0 t/1=2 C O..˛ 0 t/3=2 /;
2
C > 0:
It follows that, as ´ ! ˛ 0 for ´ 2 Œ˛; ˛ 0 ,
Z ˛0 1
(10.8)
.t/ dt D C0 .˛ 0 ´/3=2 C O..˛ 0 ´/5=2 /;
2
´
Thus,
3
˛ 0 .´/ D 2
(10.9)
˛0 Z
´
2
C0 D C:
3
2=3
1
.t/ dt
2
˛0,
is analytic at
and so extends to a conformal map from D.˛ 0 ; "/ onto a convex
neighborhood of the origin. Furthermore,
˛ 0 .˛
(10.10)
0
/ D 0;
0
0
˛ 0 .˛ / > 0I
"; ˛ 0 / and real
positive on .˛ 0 ; ˛ 0 C "/.
consequently, ˛ 0 is real negative on .˛ 0 Again, we can slightly deform the vertical pieces of the contour †S close to ˛ 0 so
that
˛ 0 fD.˛
(10.11)
0
; "/ \ †S g D ."; "/ [ .i"; i"/:
We now take
Q n .´/ D E˛ 0 .´/ˆ˛ 0 .n2=3
Q
n
(10.12)
˛ 0 .´//
where
0
E˛n .´/ D M.´/e ˙
(10.13)
1
0
L˛n .´/ D p
2 i n
2 3
n1=6
Q n .´/1
L
for ˙ Im ´ 0;
!
1=4
.´/
0
1
i
˛0
;
1 i
0
n1=6 1=4
0 .´/
˛
820
P. M. BLEHER AND K. LIECHTY
that is positive on .˛ 0 ; ˛ 0 C "/ and has a cut on
1=4
and we take the branch of ˛ 0
.˛ 0 "; ˛ 0 /. Un then becomes
(10.14)
Un .´/ D M.´/e ˙
e
i n
2 3
i n´
2 3
e
0
L˛n .´/1 ˆ˛ 0 .n2=3
l
n.g.´/ V .´/
2 2 /3
˛ 0 .´//
for ˙ Im ´ 0:
The function ˆ˛ 0 .n2=3 ˛ 0 .´// has the jumps jS , and we claim that the prefactor
0
E˛n is analytic in D.˛ 0 ; "/, and thus does not change these jumps. This can be
seen, as
(10.15)
MC .´/e
i n
2 3
D M .´/e thus the jump for the function M.´/e ˙
(10.16) e
i n
2 3
jM e
i n
2 3
D
(10.17)
i n
2 3
e
i n
2 3
e
i n
2 3
e
i n
2 3
jM e
i n
2 3
D
e
i n
2 3
jM e
i n
2 3
I
is
!
0 1 i n 3
e 2
1 0
e i n 3 e
†
or, equivalently,
i n
2 3
i n
2 3
!
0 1
1 0
!
1 0
0 1
for ´ 2 .˛ 0 "; ˛ 0 /;
for ´ 2 .˛ 0 ; ˛ 0 C "/;
for ´ 2 .˛ 0 "; ˛ 0 /;
for ´ 2 .˛ 0 ; ˛ 0 C "/;
0
which is exactly the same as the jump conditions for L˛n . Thus
0
E˛n .´/ D M.´/e ˙
i n
2 3
0
L˛n .´/1
0
has no jumps in D.˛ 0 ; "/. The only other possible singularity for E˛n is at ˛ 0 , and
0
this singularity is at most a fourth-root singularity and thus removable. Thus, E˛n
Q n has the prescribed jumps.
is analytic in D.˛ 0 ; "/, and Q
We are left to check that Un satisfies the matching condition (10.3). The large-n
asymptotics of ˆ˛ 0 .n2=3 ˛ 0 .´// are given in the different regions of analyticity as
follows:
(10.18)
ˆ˛ 0 .n2=3 ˛ 0 .´//
1
1
1
D p n 6 3 ˛ 0 .´/ 4 3
2 3=2 5i
i
1
˛ 0 .´/
˙
7i
i 1
48n
2
e 3n
3=2 3
˛ 0 .´/
for ˙ Im ´ > 0;
5
2
C O.n /
7
EXACT SOLUTION OF THE SIX-VERTEX MODEL
821
where we always take the principal branch of ˛ 0 .´/3=2 . As such,
two-valued for x 2 .˛ 0 "; ˛/, so that
Z ˛0 1
2
3=2
D i
.t/ dt
˛ 0 .x/
3
2
x
˙
(10.19)
Z ˛0
i 0
.t/dt:
D .˛ x/ ˙ i
2
x
˛ 0 .´/
3=2
is
From (2.30) and (2.33), we have that
Z
2gC .x/ V .x/ D l C 2
i
(10.20)
2g .x/ V .x/ D l 2
i
ˇ
.t/dt;
x
Z ˇ
.t/dt;
x
for x 2 .˛ 0 "; ˛ 0 /. These equations imply that
.2g˙ .x/ V .x// .2g˙ .˛ 0 / V .˛ 0 // D ˙2
i
(10.21)
Z
˛0
.t/dt:
x
We can therefore write (10.19) as
2
i
3=2
D .˛ 0 x/
˛ 0 .x/
3
2
˙
(10.22)
1
C .2g˙ .x/ V .x// .2g˙ .˛ 0 / V .˛ 0 // :
2
We can extend these equations into the upper and lower half-plane, respectively,
obtaining
(10.23)
2
3
˛ 0 .´/
3=2
D
1
i 0
.˛ ´/ C .2g.´/ V .´// .2g˙ .˛ 0 / V .˛ 0 //
2
2
for ˙ Im ´ > 0:
Using (10.20) at x D ˛ 0 , we can write
2
3
(10.24)
˛ 0 .´/
3=2
i 0
V .´/ l
.˛ ´/ C g.´/ 2
2
2
Z ˇ
.t/dt for ˙ Im ´ > 0
i
D
˛0
or, equivalently,
(10.25)
2
3
˛ 0 .´/
3=2
D g.´/ V .´/ l i´ i.n /
˙
2
2
2
2n
for ˙ Im ´ > 0:
822
P. M. BLEHER AND K. LIECHTY
Plugging (10.18) and (10.24) into (10.14) gives
Un .´/ D M.´/e ˙
1
1
1
0
L˛n .´/1 p n 6 3 ˛0 .´/ 4 3
2 3=2 5i 5
1
˛ 0 .´/
2
C O.n /
˙
7i 7
1
48n
i n
2
i
i
l
3
e n.g.´/ 2 e
(10.26)
V .´/
2 /3
e
i n
2
i n´
3 ˙ i2 3 ˙ i n´
2 3 2 3
e
e
e
n.g.´/ V 2.´/ 2l /3
1
1
1
0
L˛n .´/1 p n 6 3 ˛0 .´/ 4 3
2 3=2 i n
5 5i
1
i
˛ 0 .´/
2
C O.n / e 2 3
C
7 7i
1 i
48n
3=2
i n
n
1
6i
˛ 0 .´/
˙ i
2
3
3
e 2
C O.n /
D M.´/ I C
e 2
6i
1
48n
3=2 1
6i e ˙i n
˛ 0 .´/
2
C
O.n
D M.´/ I C
/
6i e i n
1
48n
for ˙ Im ´ > 0:
D M.´/e ˙
i n
2
3
We can make a similar construction near ˇ 0 (see [6]). Let
2=3
Z 3
´ 1
(10.27)
.t/dt
:
ˇ 0 .´/ D 2 ˇ 0 2
This function is analytic in D.ˇ 0 ; "/ and has negative derivative at ˇ 0 ; thus Im ´
and Im ˇ 0 .´/ have opposite signs for ´ 2 D.ˇ 0 ; "/. Also, let
1
0
0
(10.28)
ˆˇ 0 .´/ D ˆ˛ .´/
:
0 1
Then we can take for ´ 2 D.ˇ 0 ; "/,
(10.29)
Un .´/ D M.´/e ˙
e
i n
2 3
i n´
2 3
e
0
Lˇn .´/1 ˆˇ 0 .n2=3
l
n.g.´/ V .´/
2 2 /3
where
(10.30)
0
Lˇn .´/
1
D p
2 n1=6
1=4
.´/
ˇ0
0
ˇ 0 .´//
for ˙ Im ´ > 0;
!
1
i
:
1=4
1 i
0 .´/
0
n1=6
Similar to (10.26), we obtain
3=2 1
ˇ 0 .´/
Un .´/ D M.´/ I C
i n
6ie
(10.31)
48n
for ˙ Im ´ > 0:
ˇ
6ie ˙i n
1
C O.n
2
/
EXACT SOLUTION OF THE SIX-VERTEX MODEL
823
F IGURE 11.1. The contour †X .
11 The Third and Final Transformation of the RHP
We now consider the contour †X , consisting of the circles @D.˛; "/, @D.˛ 0 ; "/,
@D.ˇ 0 ; "/, and @D.ˇ; "/, all oriented counterclockwise, together with the parts of
†S n .Œ˛; ˛ 0 [ Œˇ 0 ; ˇ/ that lie outside of the disks D.˛; "/, D.˛ 0 ; "/, D.ˇ 0 ; "/, and
D.ˇ; "/; see Figure 11.1.
We let
(11.1)
8
Sn .´/M.´/1
ˆ
ˆ
ˆ
< for ´ outside disks D.˛; "/; D.˛ 0 ; "/; D.ˇ 0 ; "/; D.ˇ; "/;
Xn .´/ D
1
ˆ
ˆ Sn .´/Un .´/
:̂
for ´ inside disks D.˛; "/; D.˛ 0 ; "/; D.ˇ 0 ; "/; D.ˇ; "/:
Then Xn .´/ solves the following RHP:
(1) Xn .´/ is analytic on C n †X .
(2) Xn .´/ has the jump properties
XnC .x/ D Xn .´/jX .´/
(11.2)
where
(11.3)
(
jX .´/ D
M.´/Un .´/1
M.´/jS M.´/1
for ´ on the circles;
otherwise:
(3) As ´ ! 1,
X1
X2
C 2 C :
´
´
Additionally, we have that jX .´/ is uniformly close to the identity in the following
sense:
(
I C O.n1 /
uniformly on the circles,
(11.5)
jX .´/ D
C.´/n
/ on the rest of †X ;
I C O.e
Xn .´/ I C
(11.4)
where C.´/ is a positive, continuous function satisfying (6.7). If we set
jX0 .´/ D jX .´/ I;
(11.6)
then (11.5) becomes
(11.7)
jX0 .´/ D
(
O.n1 /
O.e C.´/n /
uniformly on the circles,
on the rest of †X :
The solution to the RHP for Xn is based on the following lemma:
824
P. M. BLEHER AND K. LIECHTY
L EMMA 11.1 Suppose v.´/ is a function on †X solving the equation
Z
v.u/jX0 .u/
1
(11.8)
v.´/ D I du for ´ 2 †X
2
i
´ u
†X
where ´ means the value of the integral on the minus side of †X . Then
Z
v.u/jX0 .u/
1
du for ´ 2 C n †X
(11.9)
Xn .´/ D I 2
i †X ´ u
solves the RHP for Xn .
The proof of this lemma is immediate from the jump property of the Cauchy
transform. By assumption
Xn .´/ D v.´/;
(11.10)
and the additive jump of the Cauchy transform gives
(11.11)
XnC .´/ Xn .´/ D v.´/jX0 .´/ D Xn .´/jX0 .´/I
thus XnC .´/ D Xn .´/jX .´/. Asymptotics at infinity are given by (11.9).
The solution to equation (11.8) is given by a series of perturbation theory. Namely, the solution is
v.´/ D I C
(11.12)
1
X
vk .´/
kD1
where
(11.13)
vk .´/ D 1
2
i
Z
vk1 .u/jX0 .u/
du;
´u
v0 .´/ D I:
†X
This function clearly solves (11.8) provided the series converges, which it does for
sufficiently large n. Indeed, by (11.5),
k
1
C
for some constant C > 0I
(11.14)
jvk .´/j n
1 C j´j
thus the series (11.12) is dominated by a convergent geometric series and thus
converges absolutely. This in turn gives
(11.15)
Xn .´/ D I C
1
X
Xn;k .´/
kD1
where
(11.16)
1
Xn;k .´/ D 2
i
Z
†X
vk1 .u/jX0 .u/
du:
´u
EXACT SOLUTION OF THE SIX-VERTEX MODEL
We will need to compute
(11.17)
1
Xn;1 .´/ D 2
i
Z
825
jX0 .u/
du:
´u
†X
12 Evaluation of X1
We are interested in the matrix X1 , which gives the ´1 -term of Xn .´/ at infinity;
see (11.4). By (11.9),
Z
1
v.u/jX0 .u/duI
(12.1)
X1 D 2
i
†X
hence by (11.12) and (11.14),
(12.2)
X1 D 1
2
i
Z
jX0 .u/du C O.n2 /:
†X
We would like to evaluate the integral
1
2
i
(12.3)
Z
jX0 .u/du;
†X
with an error of the order of n2 . By (11.7), it is enough to evaluate this integral
over the circles @D.˛; "/, @D.˛ 0 ; "/, @D.ˇ 0 ; "/, and @D.ˇ; "/. It can be shown (see
[6]) that the matrix-valued function jX0 .´/ is analytic in the punctured disks; hence
(12.4)
X1 D Res C Res0 C Res C Res jX0 .´/ C O.n2 /:
´D˛
´D˛
´Dˇ 0
´Dˇ
In particular, we are interested in the [12] entry of this matrix. Calculation of these
residues gives that
c.n/
C O.n2 /;
n
where c.n/ is an explicit quasi-periodic function of n; see [6].
(12.5)
ŒX1 12 D
13 Large-n Asymptotic Formula for hn
We evaluate the large-n asymptotic behavior of hnn and then we use formula
(1.28). By (4.7), hnn D ŒP1 12 , and by (5.16),
n
i
(13.1)
ŒP1 12 D ŒR1 12 I
hence
(13.2)
hnn
n
i
:
D ŒR1 12 826
P. M. BLEHER AND K. LIECHTY
Furthermore, from (6.4) we obtain that
n
i
hnn D e nl ŒT1 12 ;
(13.3)
and from (7.2) that
hnn D e ŒS1 12
nl
(13.4)
n
i
:
It follows from (11.1) that
S 1 D M1 C X 1 :
(13.5)
By (8.33),
#10 .0/
.1 C /
iA#4 ..n C 1/!/
;
!D
; AD
:
#4 .n!/
2
2#1 .!/
Combining this with (12.5) gives
iA#
.n
C
1/!
c.n/
n
i
4
nl
2
C
C O.n / :
(13.7)
hnn D e
#4 .n!/
n
By (2.36),
(13.6)
ŒM1 12 D
e l=2 D
(13.8)
hence
hnn
(13.9)
#10 .0/
A
D I
2e#1 .!/
e
2n n
i
A
#4 ..n C 1/!/ c.n/
2
C
C O.n / D
iA
e
#4 .n!/
n
2nC1
n
A
#4 .n C 1/!
c1 .n/
1C
C O.n2 /
D
2n
e #4 .n!/
n
where
c.n/#4 .n!/
:
iA#4 ..n C 1/!/
From (1.28) and the Stirling formula we obtain that
2n
1
hn
n2n hnn
e
hnn
2
1
C O.n / I
(13.11)
D
D
.nŠ/2
.nŠ/2 .2 /2n
2
2
n
6n
hence by (13.9),
2n
e
1 n
A2nC1 #4 .n C 1/!
hn
D
.nŠ/2
2
2
n
e 2n #4 .n!/
1
c1 .n/
2
(13.12)
C O.n /
1C
n
6n
c2 .n/
2nC1 #4 .n C 1/!
2
1C
C O.n / ;
DG
#4 .n!/
n
(13.10)
c1 .n/ D
EXACT SOLUTION OF THE SIX-VERTEX MODEL
827
where
#10 .0/
A
D
;
2
4 #1 .!/
Observe that c1 .n/ has the form
(13.13)
1
c2 .n/ D c1 .n/ :
6
GD
c1 .n/ D f .n!; !/;
(13.14)
where f .x; !/ is a real analytic function, periodic with respect to both x and !, of
periods and 2
, respectively. Remarkably, it can be shown using classical theta
function identities (see, e.g., [29]), that f .x; !/ does not depend on either x or !.
In fact (see [6]),
1
(13.15)
c1 .n/ I
6
thus
c2 .n/ 0:
(13.16)
We can summarize these results in the following proposition:
P ROPOSITION 13.1 As n ! 1,
(13.17)
.n
C
1/!
#
hn
4
.1 C O.n2 //;
D G 2nC1
.nŠ/2
#4 .n!/
where
GD
(13.18)
#10 .0/
:
4 #1 .!/
14 Large-n Asymptotics of Zn
By substituting (13.17) into (1.23) we obtain that
(14.1)
n1
Y hk
n
2
D 2n
Qn1
2
.kŠ/2
kD0 .kŠ/
kD0
n1
Y
n2
2kC1 #4 ..k C 1/!/
2
G
.1 C O.k //
D 2 h0
#4 .k!/
kD1
2
D C #4 .n!/.2G/n .1 C O.n1 //;
where C > 0 does not depend on n. Thus, by (1.14),
2
(14.2)
Œsinh. t/ sinh. C t/n n
2
D C #4 .n!/F n .1 C O.n1 //;
Zn D
Qn1
2
. kD0 kŠ/
where
(14.3)
F D 2G sinh. t/ sinh. C t/ D
Theorem 1.1 is proved.
sinh. t/ sinh. C t/#10 .0/
:
2 #1 .!/
828
P. M. BLEHER AND K. LIECHTY
Acknowledgment. The first author is supported in part by National Science
Foundation (NSF) Grant DMS-0652005.
Both authors would like to thank the Centre de Recherches Mathématiques at
l’Université de Montréal, where much of this work was performed, for its hospitality during their visit in fall 2008.
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PAVEL B LEHER
Indiana University-Purdue University
Indianapolis
Department of Mathematical Sciences
402 N. Blackford Street
Indianapolis, IN 46202
E-mail: [email protected]
K ARL L IECHTY
Indiana University-Purdue University
Indianapolis
Department of Mathematical Sciences
402 N. Blackford Street
Indianapolis, IN 46202
E-mail: kliechty@
math.iupui.edu
Received May 2009.