arXiv subjects

Karl Liechty

Publications and source records attributed to Karl Liechty.

22 records · Page 2Linked to original sources

arXiv2010

Uniform Asymptotics for Discrete Orthogonal Polynomials with Respect to Varying Exponential Weights on a Regular Infinite Lattice

We consider the large-$N$ asymptotics of a system of discrete orthogonal polynomials on an infinite regular lattice of mesh $\frac{1}{N}$, with weight $e^{-NV(x)}$, where $V(x)$ is a real analytic function with sufficient growth at infinity. The proof is based on formulation of an interpolation problem for discrete orthogonal polynomials, which can be converted to a Riemann-Hilbert problem, and steepest descent analysis of this Riemann-Hilbert problem.

math-ph↗

arXiv2009

Exact solution of the six-vertex model with domain wall boundary conditions. Antiferroelectric phase

We obtain the large $n$ asymptotics of the partition function $Z_n$ of the six-vertex model with domain wall boundary conditions in the antiferroelectric phase region, with the weights $a=\sinh(\ga-t), b=\sinh(\ga+t), c=\sinh(2\ga), |t|<\ga$. We prove the conjecture of Zinn-Justin, that as $n\to\infty$, $Z_n=Cþ_4(n\om) F^{n^2}[1+O(n^{-1})]$, where $\om$ and $F$ are given by explicit expressions in $\ga$ and $t$, and $þ_4(z)$ is the Jacobi theta function. The proof is based on the Riemann-Hilbert approach to the large $n$ asymptotic expansion of the underlying discrete orthogonal polynomials and on the Deift-Zhou nonlinear steepest descent method.

math-ph↗

arXiv2008

Exact solution of the six-vertex model with domain wall boundary condition. Critical line between ferroelectric and disordered phases

This is a continuation of the papers [4] of Bleher and Fokin and [5] of Bleher and Liechty, in which the large $n$ asymptotics is obtained for the partition function $Z_n$ of the six-vertex model with domain wall boundary conditions in the disordered and ferroelectric phases, respectively. In the present paper we obtain the large $n$ asymptotics of $Z_n$ on the critical line between these two phases.

math-ph↗

arXiv2008

Exact solution of the six-vertex model with domain wall boundary conditions. Ferroelectric phase

This is a continuation of the paper [4] of Bleher and Fokin, in which the large $n$ asymptotics is obtained for the partition function $Z_n$ of the six-vertex model with domain wall boundary conditions in the disordered phase. In the present paper we obtain the large $n$ asymptotics of $Z_n$ in the ferroelectric phase. We prove that for any $\ep>0$, as $n\to\infty$, $Z_n=CG^nF^{n^2}[1+O(e^{-n^{1-\ep}})]$, and we find the exact value of the constants $C,G$ and $F$. The proof is based on the large $n$ asymptotics for the underlying discrete orthogonal polynomials and on the Toda equation for the tau-function.

math-ph↗