Applying the technique of characteristic functions developped for one-dimensional regular surfaces (curves) with compact support, we obtain the distribution of hitting probabilities for a wide class of finite membranes on square lattice. Then we generalize it to multi-dimensional finite membranes on hypercubic lattice. Basing on these distributions, we explicitly construct the Brownian self-tra...

We report on the first experimental observation of photonic Bloch oscillations and Zener tunneling in two-dimensional periodic systems. We study the propagation of an optical beam in a square lattice superimposed on a refractive index ramp. We observe oscillations of the beam inside the first Brilloin zone and tunneling of light from the first to the higher-order bands of the lattice band gap s...

We present integrable lattice equations on a two dimensional square lattice with coupled vertex and bond variables. In some of the models the vertex dynamics is independent of the evolution of the bond variables, and one can write the equations as non-autonomous “Yang-Baxter maps”. We also present a model in which the vertex and bond variables are fully coupled. Integrability is tested with alg...

The transmission properties of different metallic photonic lattices ~square and rectangular! have been experimentally studied. A numerical algorithm based on time domain finite differences has been used for simulating these photonic structures. The introduction of defects in the two-dimensional metallic lattice modifies its transmission spectrum. If metal rods are eliminated from ~or added to! ...

We study nematic equilibria on a square with tangent Dirichlet conditions on the edges, in three different modelling frameworks: (i) the off-lattice Hard Gaussian Overlap and Gay–Berne models; (ii) the lattice-based Lebwohl–Lasher model; and the (iii) two-dimensional Landau-de Gennes model. We compare the modelling predictions, identify regimes of agreement and in the Landaude Gennes case, find...

We generate experimentally different types of two-dimensional Bloch waves of a square photonic lattice by employing the phase imprinting technique.We probe the local dispersion of the Bloch modes in the photonic lattice by analyzing the linear diffraction of beams associated with the high-symmetry points of the Brillouin zone, and also distinguish the regimes of normal, anomalous, and anisotrop...

We present integrable lattice equations on a two dimensional square lattice with coupled vertex and bond variables. In some of the models the vertex dynamics is independent of the evolution of the bond variables, and one can write the equations as non-autonomous “Yang-Baxter maps”. We also present a model in which the vertex and bond variables are fully coupled. Integrability is tested with alg...

This is the third part of the five-part saga on the umbral transfermatrix method, based on Gian-Carlo Rota’s seminal notion of the umbra. In this article we describe the Maple package ZOO that for any specific k, automatically constructs an umbral scheme for enumerating “k-board” lattice animals (polyominoes) on the two-dimensional square lattice. Such umbral schemes enable counting these impor...

We study rooted spiral trees in 2, 3 and 4 dimensions on a hyper cubic lattice using exact enumeration and Monte-Carlo techniques. On the square lattice, we also obtain exact lower bound of 1.93565 on the growth constant λ. Series expansions give θ = −1.3667± 0.0010 and ν = 0.6574± 0.0010 on a square lattice. With Monte-Carlo simulations we get the estimates as θ = −1.364±0.010, and ν = 0.656±0...