Related papers: On the oddness of percolation
We consider the coincidence problem for the square lattice that is translated by an arbitrary vector. General results are obtained about the set of coincidence isometries and the coincidence site lattices of a shifted square lattice by…
Random optical fields with two widely different correlation lengths generate far field speckle spots that are themselves highly speckled. We call such patterns speckled speckle, and study their critical points (singularities and stationary…
The spectral, dynamical and topological properties of physical systems described by non-Hermitian (including $\mathcal{PT}$-symmetric) Hamiltonians are deeply modified by the appearance of exceptional points and spectral singularities. Here…
A lattice-based model for continuum percolation is applied to the case of randomly located, partially aligned sticks with unequal lengths in 2D which are allowed to cross each other. Results are obtained for the critical number of sticks…
We study long-range percolation on the hierarchical lattice of order $N$, where any edge of length $k$ is present with probability $p_k=1-\exp(-\beta^{-k} \alpha)$, independently of all other edges. For fixed $\beta$, we show that the…
We consider connectivity properties of certain i.i.d. random environments on $\Z^d$, where at each location some steps may not be available. Site percolation and oriented percolation can be viewed as special cases of the models we consider.…
In this work, we reconsider the study of 2D materials involving double lattice structures associated with periodic polygons. In tessellated periodic representation, it appears two periodic polygons of $k$ sides of unequal side lengths at…
The structural properties of packed soft-core particles provide a platform to understand the cross-pollinated physical concepts in solid-state- and soft-matter physics. Confined on spherical surface, the traditional differential geometry…
A geometric study of twin and grain boundaries in crystals and quasicrystals is achieved via coincidence site lattices (CSLs) and coincidence site modules (CSMs), respectively. Recently, coincidences of shifted lattices and multilattices…
We study spatially localized optical vortices created by self-trapping of partially incoherent light with a phase dislocation in a biased photorefractive crystal. In a contrast to the decay of coherent self-trapped vortex beams due to the…
We build a multifractal object and use it as a support to study percolation. We identify some differences between percolation in a multifractal and in a regular lattice. We use many samples of finite size lattices and draw the histogram of…
We describe the linear and nonlinear stability and instability of certain symmetric configurations of point vortices on the sphere forming relative equilibria. These configurations consist of one or two rings, and a ring with one or two…
The Localization Landscape Theory (LLT) offers a classical analogy for understanding Anderson localization through an effective confining potential, whose percolation threshold has been proposed to mark the mobility edge. While this…
It is shown that there are primitive substitution tilings with dense tile orientations invariant under n-fold rotation for n=2,3,4,5,6,8. The proof for dense tile orientations uses a general result about irrationality of angles in certain…
The spectral and transport properties of a non-Hermitian tight-binding lattice with unidirectional hopping are theoretically investigated in three different geometrical settings. It is shown that, while for the infinitely-extended (open)…
We give a sharp spectral condition for the existence of odd cycles in a graph of given order. We also prove a related stability result.
We define two classes of colorings that have odd or even chirality on hexagonal lattices. This parity is an invariant in the dynamics of all loops, and explains why standard Monte-Carlo algorithms are nonergodic. We argue that adding the…
We generalize the standard site percolation model on the $d$-dimensional lattice to a model on random tessellations of $\mathbb R^d$. We prove the uniqueness of the infinite cluster by adapting the Burton-Keane argument…
The periodic tiling conjecture asserts that any finite subset of a lattice $\mathbb{Z}^d$ which tiles that lattice by translations, in fact tiles periodically. In this work we disprove this conjecture for sufficiently large $d$, which also…
In this article, we study a bond percolation model on a horizontally stretched square lattice, constructed by stretching the distances between the columns of $\mathbb{Z}_+^2$ according to a collection of independent and identically…