Related papers: Representations of stack triangulations in the pla…
Light scattering in random media is usually considered within the framework of the three-dimensional Anderson universality class, with modifications for the vector nature of electromagnetic waves. We propose that the linear dispersiveness…
An intuitive property of a random graph is that its subgraphs should also appear randomly distributed. We consider graphs whose subgraph densities exactly match their expected values. We call graphs with this property for all subgraphs with…
We prove that uniform random triangulations whose genus is proportional to their size $n$ have diameter of order $\log n$ with high probability. We also show that in such triangulations, the distances between most pairs of points differ by…
In this paper, we consider a set of similar triangles with parallel sides, along with a set of points in the plane. It turns out that the set $\mathbb{R}_2= \{\pm <x >=\pm (x^2,x,1); x\in\mathbb{R} \}$ describes this set of triangles quite…
The problem of continuum percolation in dispersions of rods is reformulated in terms of weighted random geometric graphs. Nodes (or sites or vertices) in the graph represent spatial locations occupied by the centers of the rods. The…
We describe a previously unexplored effect of the continuous spontaneous localization model whereby a correlation develops in the distributions of two nearby non-interacting particles following a period of diffusion. We propose the use of…
Random operators constitute fundamental building blocks of models of complex systems yet are far from fully understood. Here, we explain an asymmetry emerging upon repeating identical isotropic (uniformly random) operations. Specifically,…
We compute the number of triangulations of a convex $k$-gon each of whose sides is subdivided by $r-1$ points. We find explicit formulas and generating functions, and we determine the asymptotic behaviour of these numbers as $k$ and/or $r$…
This paper discusses the mathematical representation of an empirically observed phenomenon, referred to as Incremental Similarity. We discuss this feature from the viewpoint of stochastic processes and present a variety of non-trivial…
The act of a person juggling can be viewed as a Markov process if we assume that the juggler throws to random heights. I make this association for the simplest reasonable model of random juggling and compute the steady state probabilities…
A graph drawing in the plane is called an almost embedding if the images of any two non-adjacent simplices (i.e. vertices or edges) are disjoint. Almost embeddings (more precisely, their higher-dimensional analogues) naturally appear in…
The distributions of the random distances associated with hexagons, rhombuses and triangles have been derived and verified in the existing work. All of these geometric shapes are related to each other and have various applications in…
We construct a matrix model equivalent (exactly, not asymptotically), to the random plane partition model, with almost arbitrary boundary conditions. Equivalently, it is also a random matrix model for a TASEP-like process with arbitrary…
We study the construction of substitution tilings of the plane based on certain simplicial configurations of tangents of the deltoid with evenly distributed orientations. The random tiling ensembles are obtained as a result of tile…
It has been known for years how random height variations of a repeated nano-scale structure can give rise to smooth angular color variations instead of the well-known diffraction pattern experienced if no randomization is present. However,…
The paper concerns lattice triangulations, that is, triangulations of the integer points in a polygon in $\mathbb{R}^2$ whose vertices are also integer points. Lattice triangulations have been studied extensively both as geometric objects…
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…
The randomization effect of the two-way (particle-flow) interaction has been studied and quantified using the notion of distributed chaos and the results of numerical simulations and laboratory measurements. It is shown, in particular, that…
In this note I introduce a new approach to (or rather a new language for) representation theory of groups. Namely, I propose to consider a (complex) representation of a group $G$ as a sheaf on some geometric object (a stack). This point of…
We apply a framework for the description of random tilings without height representation, which was proposed recently, to the special case of quasicrystalline random tilings. Several important examples are discussed, thereby demonstrating…