scholarly journals Around the Razumov–Stroganov Conjecture: Proof of a Multi-Parameter Sum Rule

10.37236/1903 ◽  
2005 ◽  
Vol 12 (1) ◽  
Author(s):  
P. Di Francesco ◽  
P. Zinn-Justin
Keyword(s):  
Boundary Conditions ◽  
Periodic Boundary ◽  
Periodic Boundary Conditions ◽  
Loop Model ◽  
Vertex Model ◽  
Sum Rule ◽  
Alternating Sign Matrices ◽  
Wall Boundary Conditions ◽  
Periodic Strip ◽  
Domain Wall Boundary Conditions

We prove that the sum of entries of the suitably normalized groundstate vector of the $O(1)$ loop model with periodic boundary conditions on a periodic strip of size $2n$ is equal to the total number of $n\times n$ alternating sign matrices. This is done by identifying the state sum of a multi-parameter inhomogeneous version of the $O(1)$ model with the partition function of the inhomogeneous six-vertex model on a $n\times n$ square grid with domain wall boundary conditions.

scholarly journals EXACT SOLUTION OF THE SIX-VERTEX MODEL WITH DOMAIN WALL BOUNDARY CONDITIONS: CRITICAL LINE BETWEEN DISORDERED AND ANTIFERROELECTRIC PHASES

2012 ◽  
Vol 01 (04) ◽  
pp. 1250012 ◽  
Author(s):  
PAVEL BLEHER ◽  
THOMAS BOTHNER
Keyword(s):  
Domain Wall ◽  
Boundary Conditions ◽  
Critical Line ◽  
Hilbert Problem ◽  
Steepest Descent Method ◽  
Vertex Model ◽  
Wall Boundary ◽  
Large N ◽  
Wall Boundary Conditions ◽  
Domain Wall Boundary Conditions

In the present paper we obtain the large N asymptotics of the partition function ZN of the six-vertex model with domain wall boundary conditions on the critical line between the disordered and antiferroelectric phases. Using the weights a = 1 - x, b = 1 + x, c = 2, |x| < 1, we prove that, as N → ∞, ZN = CFN2N1/12(1 + O(N-1)), where F is given by an explicit expression in x and the x-dependency in C is determined. This result reproduces and improves the one given in the physics literature by Bogoliubov, Kitaev and Zvonarev [Boundary polarization in the six-vertex model, Phys. Rev. E65 (2002) 026126]. Furthermore, we prove that the free energy exhibits an infinite-order phase transition between the disordered and antiferroelectric phases. Our proofs are based on the large N asymptotics for the underlying orthogonal polynomials which involve a non-analytical weight function, the Deift–Zhou non-linear steepest descent method to the corresponding Riemann–Hilbert problem, and the Toda equation for the tau-function.

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