Related papers: Some properties of rectangle and a random point
We present correspondences induced by some classical mappings between measures on an interval and measures on the unit circle. More precisely, we link their sequences of orthogonal polynomial and their recursion coefficients. We also deduce…
Delaunay triangulations of a point set in the Euclidean plane are ubiquitous in a number of computational sciences, including computational geometry. Delaunay triangulations are not well defined as soon as 4 or more points are concyclic but…
Twisted hypercubes are generalizations of the Boolean hypercube, obtained by iteratively connecting two instances of a graph by a uniformly random perfect matching. Dudek et al. showed that when the two instances are independent, these…
We present applications of rectangular matrix models to various combinatorial problems, among which the enumeration of face-bicolored graphs with prescribed vertex degrees, and vertex-tricolored triangulations. We also mention possible…
We study critical site percolation on the triangular lattice. We find the difference of the probabilities of having a percolation interface to the right and to the left of two given points in the scaling limit. This generalizes both Cardy's…
The random matrix ensembles are applied to the quantum statistical two-dimensional systems of electrons. The quantum systems are studied using the finite dimensional real, complex and quaternion Hilbert spaces of the eigenfunctions. The…
While optics and mechanics are two distinct branches of physics, they are connected. It is well known that geometrical/ray treatment of light has direct analogies to mechanical descriptions of particle motion. However, connections between…
Random matrix theory (RMT) provides a common mathematical formulation of distinct physical questions in three different areas: quantum chaos, the 1-d integrable model with the $1/r^2$ interaction (the Calogero-Sutherland-Moser system), and…
A 3-dimensional Riemannian manifold equipped with a tensor structure of type $(1,1)$, whose third power is the identity, is considered. This structure and the metric have circulant matrices with respect to some basis, i.e., these structures…
We consider arrangements of axis-aligned rectangles in the plane. A geometric arrangement specifies the coordinates of all rectangles, while a combinatorial arrangement specifies only the respective intersection type in which each pair of…
We consider symmetric partial exclusion and inclusion processes in a general graph in contact with reservoirs, where we allow both for edge disorder and well-chosen site disorder. We extend the classical dualities to this context and then…
We study graphs that are formed by independently-positioned needles (i.e., line segments) in the unit square. To mathematically characterize the graph structure, we derive the probability that two line segments intersect and determine…
We characterize the topological configurations of points and lines that may arise when placing n points on a circle and drawing the n perpendicular bisectors of the sides of the corresponding convex cyclic n-gon. We also provide exact and…
We obtain a new solution to the star-triangle relation for an Ising-type model with two kinds of spin variables at each lattice site, taking continuous real values and arbitrary integer values, respectively. The Boltzmann weights are…
This is an English translation from the Latin original of Leonhard Euler's ``Solutio facilior problematis Diophantei circa triangulum, in quo rectae ex angulis latera opposita bisecantes rationaliter exprimantur''. In this paper, Euler…
We analyse graphs in which each vertex is assigned random coordinates in a geometric space of arbitrary dimensionality and only edges between adjacent points are present. The critical connectivity is found numerically by examining the size…
A scenario is outlined for quantum measurement, assuming that self-sustaining classicality is the consequence of an attractive gravitational self-interaction acting on massive bodies, and randomness arises already in the classical domain. A…
A random planar quadrangulation process is introduced as an approximation for certain cellular automata in terms of random growth of rays from a given set of points. This model turns out to be a particular (rectangular) case of the…
In this report, the explicit probability density functions of the random Euclidean distances associated with equilateral triangles are given, when the two endpoints of a link are randomly distributed in 1) the same triangle, 2) two adjacent…
A mapping is obtained relating analytical radial Coulomb systems in any dimension greater than one to analytical radial oscillators in any dimension. This mapping, involving supersymmetry-based quantum-defect theory, is possible for…