The 6-vertex model with fixed boundary conditions

Abstract

We study the 6-vertex model with fixed boundary conditions. In the thermodynamical limit there is a formation of the limit shape. We collect most of the known results about the analytical properties of the free energy of the model as the function of electric fields and study the asymptotical behavior near singularities. We also study the asymptotic of limit shapes and the structure of correlation functions in the bulk.

Six-vertex model on a finite lattice: integral representations for nonlocal correlation functions

“…A prototypical example is provided by the six-vertex model with domain wall boundary conditions. Current interest in the model is mostly motivated by the occurrence of phase separation [2][3][4][5], which recently triggered a number of numerical studies [6][7][8] and analytical results [9][10][11][12]. The model is also of relevance for quantum quenches in the closely related Heisenberg XXZ quantum spin chain [13][14][15][16], and for N = 4 super-Yang-Mills theory [17][18][19].…”

The two-point correlation function in the six-vertex model

“…In this paper, we show how, based on the generated configurations, to compute numerically the two-point correlation function. When ∆ = 0 both the limit shape height function and the correlation functions are known from the exact solution because in this case the six-vertex can be mapped to a dimer model on a modified (decorated) square lattice, details can be found in [34,38]. We demonstrate that in this case the usual averaging over time in Markov process gives an excellent agreement of numerical results with the exact ones.…”

“…It was adopted to the six-vertex model in [46,34]. The random variable φ(x, y) is a free Gaussian quantum field in the Euclidean space time with the metric determined by the height function h(x, y), see for example [46,16].…”

“…The random variable φ(x, y) is a free Gaussian quantum field in the Euclidean space time with the metric determined by the height function h(x, y), see for example [46,16]. For generic values of parameters in the six-vertex model, mathematically, the variational principle is still a hypothesis [34,15]. It is proven in some special cases of stochastic weights in [5] and it follows from [7] for the free fermionic case ∆ = 0.…”

An irreversible local Markov chain that preserves the six vertex model on a torus

“…An interested reader could consult e.g. Palamarchuk-Reshetikhin [14] and references therein. Its Gibbs measures on a torus are parametrized by two nonnegative integers (in addition to the weights of the six vertex model itself).…”