Abstract
We calculate explicitly the constant factor
C
in the large
N
asymptotics of the partition function
Z_N
of the six-vertex model with domain wall boundary conditions on the critical line between the disordered and ferroelectric phases. On the critical line the weights
a,b,c
of the model are parameterized by a parameter
\alpha >1
, as
a=\frac{\alpha-1}{2}
,
b=\frac{\alpha +1}{2}
,
c=1
. The asymptotics of
Z_N
on the critical line was obtained earlier in the paper [8] of Bleher and Liechty:
Z_N=CF^{N^2}G^{\sqrt{N}}N^{1/4}(1+O(N^{-1/2}))
, where
F
and
G
are given by explicit expressions, but the constant factor
C>0
was not known. To calculate the constant
C
, we find, by using the Riemann–Hilbert approach, an asymptotic behavior of
Z_N
in the double scaling limit, as
N
and
\alpha
tend simultaneously to
\infty
in such a way that
\frac{N}{\alpha}\to t\ge 0
. Then we apply the Toda equation for the tau-function to find a structural form for
C
, as a function of
\alpha
, and we combine the structural form of
C
and the double scaling asymptotic behavior of
Z_N
to calculate
C
.