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References
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Exact Solution of the Six-Vertex Model with Domain Wall Boundary Conditions
Pavel M. Bleher
Abstract The six-vertex model, or the square ice model, with domain wall boundary conditions (DWBC) has been introduced and solved for finite N by Korepin and Izergin. The solution is based on the Yang-Baxter equations and it represents the free energy in terms of an N × N Hankel determinant. Paul Zinn-Justin observed that the Izergin-Korepin formula can be re-expressed in terms of the partition function of a random matrix model with a nonpolynomial interaction. We use this observation to obtain the large N asymptotics of the six-vertex model with DWBC in the disordered phase and ferroelectric phases, and also on the critical line between these two phases. The solution is based on the Riemann-Hilbert approach.
1 Six-Vertex Model
The six-vertex model, or the model of two-dimensional ice, is stated on a square lattice with arrows on edges. The arrows obey the rule that at every vertex there are two arrows pointing in and two arrows pointing out. Such 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 Fig. 1.
We will consider the domain wall boundary conditions (DWBC), in which the arrows on the upper and lower boundaries point in 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 Fig. 2.
The name of the square ice comes from the two-dimensional arrangement of water molecules, H2O, with oxygen atoms at the vertices of the lattice and one
Pavel M. Bleher
Department of Mathematical Sciences, Indiana University-Purdue University Indianapolis, 402 N. Blackford St., Indianapolis, IN 46202, USA, e-mail: bleher@math.iupui.edu
The author is supported in part by the National Science Foundation (NSF) Grant DMS0652005.
V. Sidoraviciusˇ (ed.), New Trends in Mathematical Physics, |
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Pavel M. Bleher |
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Fig. 1 The six arrow configurations allowed at a vertex
Fig. 2 An example of 4 × 4 configuration
hydrogen atom between each pair of adjacent oxygen atoms. We place an arrow in the direction from a hydrogen atom toward an oxygen atom if there is a bond between them. Thus, as we already noticed before, there are two in-bound and two out-bound arrows at each vertex.
For each possible vertex state we assign a weight wi , i = 1, . . . , 6, and define, as usual, the partition function, as a sum over all possible arrow configurations of the product of the vertex weights,
Exact Solution of the Six-Vertex Model |
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Fig. 3 The corresponding ice model
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ZN = |
w(σ ), |
w(σ ) = |
wσ (x) = wiNi (σ ), (1) |
arrow configurations σ |
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x VN |
i=1 |
where VN is the N × N set of vertices, σ (x) {1, . . . , 6} is the vertex configuration of σ at vertex x, according to Fig. 1, and Ni (σ ) is the number of vertices of type i in the configuration σ . The sum is taken over all possible configurations obeying the given boundary condition. The Gibbs measure is defined then as
μN (σ ) = |
w(σ ) |
(2) |
ZN . |
Our main goal is to obtain the large N asymptotics of the partition function ZN . In general, the six-vertex model has six parameters: the weights wi . However, by
using some conservation laws we can reduce these to only two parameters. Namely,
first we reduce to the case |
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w1 = w2 ≡ a, |
w3 = w4 ≡ b, |
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w5 = w6 ≡ c, |
(3) |
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and then, by using the identity, |
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ZN (a, a, b, b, c, c) = cN |
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ZN |
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(4) |
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to the two parameters, a and |
b . For details on how we make this reduction, see, |
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c |
c |
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e.g., the works [1] of Allison and Reshetikhin, [11] of Ferrari and Spohn, and [7] of Bleher and Liechty.
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Pavel M. Bleher |
Fig. 4 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, where = 0, and the three dots correspond to 1-, 2-, and 3-enumeration of alternating sign matrices
2 Phase Diagram of the Six-Vertex Model
Introduce the parameter |
a2 |
+ b2 − c2 |
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(5) |
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2ab |
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The phase diagram of the six-vertex model consists of the following three regions: the ferroelectric phase region, > 1; the anti-ferroelectric phase region, < −1; and, the disordered phase region, −1 < < 1, see, e.g., [21]. In these three regions we parameterize the weights in the standard way: in the ferroelectric phase region,
a = sinh(t − γ ), |
b = sinh(t + γ ), |
c = sinh(2|γ |), |
0 < |γ | < t, |
(6) |
in the anti-ferroelectric phase region, |
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a = sinh(γ − t ), |
b = sinh(γ + t ), |
c = sinh(2γ ), |
|t | < γ , |
(7) |
and in the disordered phase region, |
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a = sin(γ − t ), |
b = sin(γ + t ), |
c = sin(2γ ), |
|t | < γ . |
(8) |
The phase diagram of the model is shown on Fig. 4.
Here we will discuss the disordered and ferroelectric phase regions, and we will use parameterizations (8) and (6).
The phase diagram and the Bethe-Ansatz solution of the six-vertex model for periodic and anti-periodic boundary conditions are thoroughly discussed in the works of Lieb [17–20], Lieb, Wu [21], Sutherland [25], Baxter [3], Batchelor, Baxter,
Exact Solution of the Six-Vertex Model |
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O’Rourke, Yung [2]. See also the work of Wu, Lin [27], in which the Pfaffian solution for the six-vertex model with periodic boundary conditions is obtained on the free fermion line, = 0.
3 Izergin-Korepin Determinantal Formula
The six-vertex model with DWBC was introduced by Korepin in [14], who derived an important recursion relation for the partition function of the model. This lead to a beautiful determinantal formula of Izergin and Korepin [12], for the partition function of the six-vertex model with DWBC. A detailed proof of this formula and its generalizations are given in the paper of Izergin, Coker, and Korepin [13]. When the weights are parameterized according to (8), the formula of Izergin is
Z |
N = |
[sin(γ + t ) sin(γ − t )]N 2 |
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τ |
N |
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( |
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n=0 |
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where τN is the Hankel determinant, |
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τN = det |
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N , |
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dt i+k−2 |
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≤ |
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and |
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sin(2γ ) |
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φ (t ) = |
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sin(γ |
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t ) |
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An elegant derivation of the Izergin-Korepin determinantal formula from the YangBaxter equations is given in the papers of Korepin and Zinn-Justin [15] and Kuperberg [16].
One of the applications of the determinantal formula is that it implies that the partition function τN solves the Toda equation,
τ |
τ |
− |
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≥ |
1, ( ) |
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(12) |
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N +1 |
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N N |
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cf. [24]. This was used by Korepin and Zinn-Justin [15] 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.
4The Six-Vertex Model with DWBC and a Random Matrix Model
Another application of the Izergin-Korepin determinantal formula is that τN can be expressed in terms of a partition function of a random matrix model. The relation to
64 |
Pavel M. Bleher |
the random matrix model was obtained and used by Zinn-Justin [29]. This relation will be very important for us. It can be derived as follows. For the evaluation of the Hankel determinant, it is convenient to use the integral representation of φ (t ), namely, to write it in the form of the Laplace transform,
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φ (t ) = |
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∞ et λm(λ)dλ, |
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−∞ |
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where |
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sinh λ2 (π − 2γ ) |
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m(λ) = |
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sinh λ2 π |
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di φ |
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and by substituting this into the Hankel determinant, (10), we obtain that |
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N |
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det(λi+k−2) |
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Consider any permutation σ SN of variables λi . From the last equation we have that
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τN = |
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where Δ(λ) is the Vandermonde determinant, |
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Equation (18) expresses τN in terms of a matrix model integral. Namely, if m(x) = e−V (x), then
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where the integration is over the space of model integral can be solved, furthermore,
N × N Hermitian matrices. The matrix in terms of orthogonal polynomials.
Exact Solution of the Six-Vertex Model |
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Introduce monic polynomials P (x) |
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xn |
+ · · · |
orthogonal on the line with |
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respect to the weight w(x) = e |
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m(x), so that |
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∞ Pn(x)Pm(x)et x m(x)dx = hnδnm. |
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τN = |
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(22) |
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hn. |
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n=0 |
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The orthogonal polynomials satisfy the three term recurrence relation, |
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xPn(x) = Pn+1(x) + QnPn(x) + RnPn−1(x), |
(23) |
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where Rn can be found as |
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hn |
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Rn = |
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(24) |
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see, e.g., [26]. This gives that |
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hn−1 |
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hn = h0 |
n |
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Rj , |
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(25) |
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j =1 |
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where |
∞ |
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sin(2γ ) |
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h0 = |
t x |
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(26) |
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e |
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m(x)dx |
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sin(γ |
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t ) sin(γ |
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t ) |
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N −1 |
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τ |
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RN −n. |
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(27) |
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n=1
5 Asymptotic Formula for the Recurrence Coefficients
We prove the following asymptotics of the recurrence coefficients Rn.
Theorem 1 (See [4]). As n → ∞,
Rn = |
n2 |
cj n−κj + cn−2 + O(n−2−ε ) , ε > 0, (28) |
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R + cos(nω) |
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γ 2 |
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j : κj ≤2 |
where the sum is finite and it goes over j = 1, 2, . . . such that κj ≤ 2,
66
R =
and
where
and
where
Pavel M. Bleher
π |
2 |
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t |
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2j |
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, ζ ≡ |
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ω |
= π(1 + ζ ); |
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κj = 1 + |
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π |
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2 cos |
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γ |
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2 |
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2γ |
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cj = |
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2γ eϕ(yj ) |
(−1)j |
sin |
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(30) |
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cos |
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2γ |
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π |
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yj = |
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(31) |
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2γ |
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ϕ(y) = − |
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ln |
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2π cos |
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∞ arg(μ + iy)f (μ)dμ + y ln y − y , |
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(32) |
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π |
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π |
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f (μ) = |
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coth μ |
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coth μ |
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− 1 − sgn μ. |
(33) |
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2γ |
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π γ 2 |
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π 2 |
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c = |
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(34) |
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6(π |
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2 |
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48 cos |
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The error term in (28) is uniform on any compact subset of the set |
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(t, γ ) : |
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(35) |
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Remark. The method of the proof allows an extension of formula (28) to an asymptotic series in negative powers of n.
Denote
FN = |
1 |
ln |
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τN |
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N 2 |
( |
N −1 n )2 |
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n=0 |
! |
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From Theorem 1 we derive the following result.
Theorem 2. As N → ∞, |
FN = F + O(N −1), |
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1 R |
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π |
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F = |
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ln |
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2 |
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2 |
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(36)
(37)
(38)
This coincides with the formula of work [29], obtained in the saddle-point approximation. Earlier it was derived in work [15], from some Ansatz for the free
Exact Solution of the Six-Vertex Model |
67 |
energy asymptotics. For the partition function ZN in (9) we obtain from Theorem 2 the formula,
1 |
ln ZN |
= |
f |
+ |
O(N −1) |
f |
= |
ln |
π [cos(2t ) − cos(2γ )] |
. (39) |
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N 2 |
4γ cos 2πγt |
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Let us compare this formula and asymptotics (28) with known exact results.
6 Previous Exact Results
The free fermion line, γ = π4 , |t | < π4 . In this case the exact result is |
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ZN = 1, |
(40) |
see, e.g., [9], which implies f = 0. This agrees with formula (39), which also gives f = 0 when γ = π4 . Moreover, the orthogonal polynomials in this case are the Meixner-Pollaczek polynomials, for which
Rn = |
4n2 |
= |
n2R |
(41) |
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γ 2 , |
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cos2 2t |
cf. [9]. Thus, formula (28) is exact on the free fermion line, with no error term. This agrees with Theorem 1, because from (30), (34), cj = c = 0 when γ = π4 .
The ASM (ice) point, γ = π3 , t = 0. In this case we obtain from (8) that
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a = b = c = |
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(42) |
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2 |
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hence |
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N 2 |
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ZN = |
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where A(N ) is the number of configurations in the six-vertex model with DWBC. There is a one-to-one correspondence between the set of configurations in the sixvertex model with DWBC and the set of N × N alternating sign matrices. By definition, an alternating sign matrix (ASM) is a matrix with the following properties:
•All entries of the matrix are −1, 0, 1;
•If we look at the sequence of (−1)’s and 1’s, they are alternating along any row and any column;
•The sum of entries is equal to 1 along any row and any column.
The above correspondence is established as follows: given a configuration of arrows on edges, we assign (−1) to any vertex of type (1) on Fig. 1, 1 to any vertex of type (2), and 0 to any vertex of other types. Then the configuration on the vertices gives
68 |
Pavel M. Bleher |
rise to an ASM, and this correspondence is one-to-one. For instance, Fig. 5 shows the ASM corresponding to the configuration of arrows on Fig. 2.
0 |
1 |
0 |
0 |
0 |
0 |
1 |
0 |
1 |
−1 |
0 |
1 |
0 |
1 |
0 |
0 |
Fig. 5 ASM for the configuration of Fig. 2
For the number of ASMs there is an exact formula:
A(N ) |
N −1 |
(3n + 1)!n! |
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(44) |
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n 0 |
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1) |
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(2n) (2n |
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This formula was conjectured in [22], [23], and proved by Zeilberger [28] by combinatorial arguments. Another proof was given by Kuperberg [16], who used formula (9). The relation to classical orthogonal polynomials was found by Colomo and Pronko [9], who used this relation to give a new proof of the ASM conjecture. The orthogonal polynomials in this case are the continuous Hahn polynomials and from [9] we find that
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R |
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n2(9n2 − 1) |
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9n2 |
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5 |
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O(n−2). |
(45) |
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4n2 − 1 |
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Formula (28) gives |
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Rn = |
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(46) |
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π 2 |
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which agrees with (45). From (44) we find that as N → ∞, |
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3√ |
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3 |
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A(N ) = C |
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N − |
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36 |
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where C > 0 is a constant, so that |
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ZN = C |
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115 |
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N → ∞. |
(48) |
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36 |
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15552N 2 |
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Formula (39) gives f |
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ln |
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which agrees with the last formula. |
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8 |
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The x = 3 ASM point, γ = |
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3N /2 |
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ZN = |
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(49) |
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Exact Solution of the Six-Vertex Model |
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69 |
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where |
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3m(m+1) m |
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(3k−1)! |
2 |
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A(2m |
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1 |
; |
3) |
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[ |
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k=1 |
(m+k)! ] |
(50) |
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A(2m |
+ |
2 |
; |
3) |
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3m (3m+2)!m! A(2m |
+ |
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1 |
3). |
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In this case A(N ; 3) counts the number of alternating sign matrices with weight 3k , where k is the number of (−1) entries. Formula (50) for A(N ; 3) was conjectured in [22], [23] and proved in [16]. The relation to classical orthogonal polynomials was again found by Colomo and Pronko [9], who used it to give a new proof of formula (50) for the 3-enumeration of ASMs. The orthogonal polynomials in this case are expressed in terms of the continuous dual Hahn polynomials and from [9] we find that
R2m = 36m2, R2m+1 = 4(3m + 1)(3m + 2). |
(51) |
In this case the subdominant term in the asymptotics of Rn exhibits a period 2 oscillation. Namely, we obtain from the last formula that
R |
n = |
9n2 |
+ |
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−1 + (−1)n |
. |
(52) |
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2 |
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This perfectly fits to the frequency value ω = π for ζ |
= 0 in (29). Moreover, |
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formula (28) gives |
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Rn |
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π 2 |
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(−1)nc1 |
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O(n−2−ε ) , |
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π 2 |
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4 |
n2 |
− |
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which agrees with (52) and it provides with the value of c1 = |
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72 |
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From (50) we find, that as m → ∞, |
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77 |
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A(2m 3) |
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3−m(2m) |
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1 |
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7776m2 + |
O(N |
−3) , |
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(53)
(54)
where C3 > 0 is a constant, and |
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2m 1 |
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A(2m + 1; 3) = C3 |
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Formula (39) gives f = ln 43 , which agrees with the last formula. |
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We have the identity, |
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see, e.g., [5], which is equivalent to the Toda equation (12). By using identity (58), we obtain from Theorem 1 the following asymptotics.
Theorem 3. As N → ∞, |
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This gives a quasiperiodic asymptotics, as N → ∞, of the second derivative of the subdominant terms.
7 Zinn-Justin’s Conjecture
Paul Zinn-Justin conjectured in [29] that |
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Formulae (40), (48), and (57) confirm this conjecture, with the value of κ given as
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Bogoliubov, Kitaev and Zvonarev obtained in [8] the asymptotics of ZN on the line ac + bc = 1, separating the disordered and antiferroelectric phases. This corresponds to the value γ = 0. They found that in this case formula (60) holds with κ = 121 .
With the help of Theorem 1 we prove the following result.
Theorem 4. We have that |
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ZN = CN κ eN 2f |
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and C > 0 is a constant.
Exact Solution of the Six-Vertex Model |
71 |
This proves the conjecture of Zinn-Justin, and it gives the exact value of the exponent κ. Let us remark, that the presence of the power-like factor N κ in the asymptotic expansion of ZN in (63) is rather unusual from the point of view of random matrix models. In the one-cut case the usual large N asymptotics of ZN in a noncritical random matrix model is the so called “topological expansion”, which gives ZN as an asymptotic series in powers of 1/N 2. For a rigorous proof of the “topological expansion” see the work of Ercolani and McLaughlin [10] (see also [5]).
8 Large N Asymptotics of ZN in the Ferroelectric Phase
Recently Bleher and Liechty [7], [6] obtained the large N asymptotics of ZN in the ferroelectric phase, > 1, and also on the critical line between the ferroelectric and disordered phases, = 1. In the ferroelectric phase we use parameterization (6) for a, b and c. The large N asymptotics of ZN in the ferroelectric phase is given by the following theorem:
Theorem 5. In the ferroelectric phase with t > γ > 0, for any ε > 0, as N → ∞,
ZN = CGN F N 2 1 + O e−N 1−ε , |
(65) |
where C = 1 − e−4γ , G = eγ −t , and F = sinh(t + γ ).
On the critical line between the ferroelectric and disordered phases we use the parameterization b = a + 1, c = 1. The main result here is the following asymptotic formula for ZN :
Theorem 6. As n → ∞, |
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72 |
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