Related papers: Asymptotic shapes with free boundaries
We investigate compactness phenomena involving free boundary minimal hypersurfaces in Riemannian manifolds of dimension less than eight. We provide natural geometric conditions that ensure strong one-sheeted graphical subsequential…
We study large-scale height fluctuations of random stepped surfaces corresponding to uniformly random lozenge tilings of polygons on the triangular lattice. For a class of polygons (which allows arbitrarily large number of sides), we show…
We discuss the asymptotic behaviour of models of lattice polygons, mainly on the square lattice. In particular, we focus on limiting area laws in the uniform perimeter ensemble where, for fixed perimeter, each polygon of a given area occurs…
We complete the description, initiated in [6], of a free boundary travelling at constant speed in a half plane, where the propagation is controlled by a line having a large diffusion on its own. The main result of this work is that the free…
We present a generic and systematic approach for constructing D-dimensional lattice models with exactly solvable d-dimensional boundary states localized to corners, edges, hinges and surfaces. These solvable models represent a class of…
We prove asymptotic lower bounds on the variance of the number of vertices and missed area of random disc-polygons in convex discs whose boundary is $C_+^2$ smooth. The established lower bounds are of the same order as the upper bounds…
The asymptotic shape theorem for the contact process in random environment gives the existence of a norm $\mu$ on $\Rd$ such that the hitting time $t(x)$ is asymptotically equivalent to $\mu(x)$ when the contact process survives. We provide…
In this note, we consider the six-vertex model with domain wall boundary conditions, defined on a $M\times M$ lattice, in the inhomogeneous case where the partition function depends on 2M inhomogeneities $\lambda_j$ and $\mu_k$. For a…
We prove that simply connected H-surfaces with bounded area and free boundary in a domain necessarily concentrate at a critical point of the mean curvature of the boundary of this domain.
In this work we study the asymptotic distribution of eigenvalues in one-dimensional open sets. The method of proof is rather elementary, based on the Dirichlet lattice points problem, which enable us to consider sets with infinite measure.…
The purpose of this note is to give a succinct summary of some basic properties of T-graphs which arise in the study of the dimer model. We focus in particular on the relation between the dimer model on the heaxgonal lattice with a given…
We study a translational invariant free fermions model in imaginary time, with nearest neighbor and next-nearest neighbor hopping terms, for a class of inhomogeneous boundary conditions. This model is known to give rise to limit shapes and…
We investigate existence and regularity properties of one-phase free boundary graphs, in connection with the question of whether there exists a complete non-planar free boundary graph in high dimensions.
An asymptotic equation of motion for the pattern interface in the domain-forming reaction-diffusion systems is derived. The free boundary problem is reduced to the universal equation of non-local contour dynamics in two dimensions in the…
We analyze bound states of Robin Laplacian in infinite planar domains with a smooth boundary, in particular, their relations to the geometry of the latter. The domains considered have locally straight boundary being, for instance, locally…
This paper presents the asymptotic analysis of random lattices in high dimensions to clarify the distance properties of the considered lattices. These properties not only indicate the asymptotic value for the distance between any pair of…
Based on a fairly precise approximation to the lattice discrepancy of a Lame disc, an asymptotic formula is established for the number of lattice points in a related three-dimensional body, linearly dilated by a large real parameter x.…
We introduce a new class of discrete approximations of planar domains that we call "hedgehog domains". In particular, this class of approximations contains two-step Aztec diamonds and similar shapes. We show that fluctuations of the height…
The thermodynamic limit is foundational to statistical mechanics, underlying our understanding of many-body phases. It assumes that, as the system size grows infinitely at fixed density of particles, unambiguous macroscopic phases emerge…
By using the asymptotic theory of Pemantle and Wilson, exact asymptotic expansions of the free energy of the monomer-dimer model on rectangular $n \times \infty$ lattices in terms of dimer density are obtained for small values of $n$, at…