Related papers: Limit shapes for the asymmetric five vertex model
A detailed combinatorial analysis of planar convex lattice polygonal lines is presented. This makes it possible to answer an open question of Vershik regarding the existence of a limit shape when the number of vertices is constrained.
We study the 19-vertex model of Statistical Mechanics in a square with the domain wall boundary condition. Using the minimal set of generating flip actions we build a parametrized dynamic version of the model. For all observed dynamic…
We consider the Schroedinger equation with a subcritical focusing power nonlinearity on a noncompact metric graph, and prove that for every finite edge there exists a threshold value of the mass, beyond which there exists a positive bound…
A variety of methods are developed for characterising the free energy landscapes of continuum, Landau-type free energy models. Using morphologies of lipid vesicles and a multistable liquid crystal device as examples, I show that the methods…
We consider the free boundary problem for the Euler--Fourier system that describes the motion of compressible, inviscid and heat-conducting fluids. The effect of surface tension is neglected and there is no heat flux across the free…
A mean field theory is developed for the calculation of the surface free energy of the staggered BCSOS, (or six vertex) model as function of the surface orientation and of temperature. The model approximately describes surfaces of crystals…
The main result of this paper is the construction of infinitely many conserved quantities (corresponding to commuting transfer-matrices) for the limit shape equation for the 6-vertex model on a cylinder. This suggests that the limit shape…
We consider the numerical investigation of surface bound orientational order using unit tangential vector fields by means of a gradient-flow equation of a weak surface Frank-Oseen energy. The energy is composed of intrinsic and extrinsic…
We study equilibrium configurations for the Euler-Plateau energy with elastic modulus, which couples an energy functional of Euler-Plateau type with a total curvature term often present in models for the free energy of biomembranes. It is…
A numerical study of an optimal control formulation for a shape optimization problem governed by an elliptic variational inequality is performed. The shape optimization problem is reformulated as a boundary control problem in a fixed…
We compute an upper bound for the value of the $L^1$-relaxed area of the graph of the vortex map $u : B_l(0)\subset \mathbb R^2 \to \mathbb R^2$, $u(x):= x/\vert x\vert$, $x \neq 0$, for all values of $l>0$. Together with a previously…
Formal Laplace operators are analyzed for a large class of resistance networks with vertex weights. The graphs are completed with respect to the minimal resistance path metric. Compactness and a novel connectivity hypothesis for the…
The dimer model is a classical statistical mechanics model which is exactly solvable in two dimensions, but about which little is known in higher dimensions. In analogy with large $N$ limits in lattice gauge theory, we study a large $N$…
An ideal conductor electrode in contact with a semi-infinite two-dimensional two-component plasma in an external potential is considered. The model is mapped onto an integrable sine-Gordon theory with Dirichlet boundary conditions. The…
We extend the concept of quintessence to a flat nonminimally coupled scalar - tensor theories of gravity. By means of Noether's symmetries for the cosmological pointlike Lagrangian L, it is possible to exhibit exact solutions for a class of…
We explicitly compute limit shapes for several grand canonical Gibbs ensembles of partitions of integers. These ensembles appear in models of aggregation and are also related to invariant measures of zero range and coagulation-fragmentation…
We consider the free-boundary motion of two perfect incompressible fluids with different densities $\rho_+$ and $\rho_-$, separated by a surface of discontinuity along which the pressure experiences a jump proportional to the mean curvature…
In this paper we present a new solution of the star-triangle relation having positive Boltzmann weights. The solution defines an exactly solvable two-dimensional Ising-type (edge interaction) model of statistical mechanics where the local…
We study the free boundary Euler equations in two spatial dimensions. We prove that if the boundary is sufficiently regular, then solutions of the free boundary fluid motion converge to solutions of the Euler equations in a fixed domain…
We provide a numerical method for computing solutions to a free boundary problem arising from the equilibrium state of a floating drop. This numerical method is based on a Newton's method for the underlying nonlinear boundary value…