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In this paper, we provide new proofs of the existence and the condensation of Bethe roots for the Bethe Ansatz equation associated with the six-vertex model with periodic boundary conditions and an arbitrary density of up arrows (per line) in the regime $Delta<1$. As an application, we provide a short, fully rigorous computation of the free energy of the six-vertex model on the torus, as well as an asymptotic expansion of the six-vertex partition functions when the density of up arrows approaches $1/2$. This latter result is at the base of a number of recent results, in particular the rigorous proof of continuity/discontinuity of the phase transition of the random-cluster model, the localization/delocalization behaviour of the six-vertex height function when $a=b=1$ and $cge1$, and the rotational invariance of the six-vertex model and the Fortuin-Kasteleyn percolation.
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This letter is concerned with the analysis of the six-vertex model with domain-wall boundaries in terms of partial differential equations (PDEs). The models partition function is shown to obey a system of PDEs resembling the celebrated Knizhnik-Zamolodchikov equation. The analysis of our PDEs naturally produces a family of novel determinant representations for the models partition function.
We prove the vertex-reinforced jump process (VRJP) is recurrent in two dimensions for any translation invariant finite range initial rates. Our proof has two main ingredients. The first is a direct connection between the VRJP and sigma models whose target space is a hyperbolic space $mathbb{H}^n$ or its supersymmetric counterpart $mathbb{H}^{2|2}$. These results are analogues of well-known relations between the Gaussian free field and the local times of simple random walk. The second ingredient is a Mermin--Wagner theorem for these sigma models. This result is of intrinsic interest for the sigma models and also implies our main theorem on the VRJP. Surprisingly, our Mermin--Wagner theorem applies even though the symmetry groups of $mathbb{H}^n$ and $mathbb{H}^{2|2}$ are non-amenable.
In this work we elaborate on a previous result relating the partition function of the six-vertex model with domain-wall boundary conditions to eigenvalues of a transfer matrix. More precisely, we express the aforementioned partition function as a determinant of a matrix with entries being eigenvalues of the anti-periodic six-vertex models transfer matrix.
In this paper, we explain a connection between a family of free-fermionic six-vertex models and a discrete time evolution operator on one-dimensional Fermionic Fock space. The family of ice models generalize those with domain wall boundary, and we focus on two sets of Boltzmann weights whose partition functions were previously shown to generalize a generating function identity of Tokuyama. We produce associated Hamiltonians that recover these Boltzmann weights, and furthermore calculate the partition functions using commutation relations and elementary combinatorics. We give an expression for these partition functions as determinants, akin to the Jacobi-Trudi identity for Schur polynomials.
Using the results on the $1/n$-expansion of the Verblunsky coefficients for a class of polynomials orthogonal on the unit circle with $n$ varying weight, we prove that the local eigenvalue statistic for unitary matrix models is independent of the form of the potential, determining the matrix model. Our proof is applicable to the case of four times differentiable potentials and of supports, consisting of one interval.
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