Related papers: Random Triangle Theory with Geometry and Applicati…
The algebraic area probability distribution of closed planar random walks of length N on a square lattice is considered. The generating function for the distribution satisfies a recurrence relation in which the combinatorics is encoded. A…
This paper is devoted to the Gaussian fluctuations and deviations of the traces of tridiagonal random matrix. Under quite general assumptions, we prove that the traces are approximately normal distributed. Multi-dimensional central limit…
There is a trivial $O(\frac{n^3}{T})$ time algorithm for approximate triangle counting where $T$ is the number of triangles in the graph and $n$ the number of vertices. At the same time, one may count triangles exactly using fast matrix…
Sine-skewed circular distributions are identifiable and have easily-computable trigonometric moments and a simple random number generation algorithm, whereas they are known to have relatively low levels of asymmetry. This study proposes a…
A uniform recursive tree on $n$ vertices is a random tree where each possible $(n-1)!$ labeled recursive rooted tree is selected with equal probability. In this paper we introduce and study weighted trees, a non-uniform recursive tree model…
Two-sample hypothesis testing for random graphs arises naturally in neuroscience, social networks, and machine learning. In this paper, we consider a semiparametric problem of two-sample hypothesis testing for a class of latent position…
We use the idea of the broken stick problem (which goes back to Poincare) and calculate the corresponding probabilities for the cases in which the three broken part are: the medians in a triangle, the altitudes, radii of excircles, angle…
We initiate the study of distribution testing for probability distributions over the edges of a graph, motivated by the closely related question of ``edge-distribution-free'' graph property testing. The main results of this paper are…
We show that, within a finite window of parameter space, random matrix theory (RMT) statistics emerge in observables of a finite-volume massive free scalar field theory after a local operator quench. The spacing-ratio distribution of…
An elliptic random matrix $X$ is a square matrix whose $(i,j)$-entry $X_{ij}$ is independent of the rest of the entries except possibly $X_{ji}$. Elliptic random matrices generalize Wigner matrices and non-Hermitian random matrices with…
In this paper we derive results concerning the connected components and the diameter of random graphs with an arbitrary i.i.d. degree sequence. We study these properties primarily, but not exclusively, when the tail of the degree…
Motivated by a question of Erd\"{o}s and inquiries by Beeson and Laczkovich, we explore the possible $N$ for which a triangle $T$ can tile into $N$ congruent copies of a triangle $R$. The \emph{reptile} cases (where $T$ is similar to $R$)…
The purpose of this paper is to give a selective survey on recent progress in random metric theory and its applications to conditional risk measures. This paper includes eight sections. Section 1 is a longer introduction, which gives a…
Suppose an interval is put on a horizontal line with random roughness. With probability one it is supported at two points, one from the left, and another from the right from its center. We compute probability distribution of support points…
We study the following combinatorial counting and sampling problems: can we efficiently sample from the Erd\H{o}s-R\'{e}nyi random graph $G(n,p)$ conditioned on triangle-freeness? Can we efficiently approximate the probability that $G(n,p)$…
Random geometric graphs are random graph models defined on metric spaces. Such a model is defined by first sampling points from a metric space and then connecting each pair of sampled points with probability that depends on their distance,…
Polytopes are the basic finite data structures for convex sets: they appear as feasible regions in linear optimization, as geometric summaries in algorithms, and as random objects in stochastic geometry. A natural geometric question is…
Understanding the effects of the choice of the tree on the joint distribution of a tree-structured Markov random field (MRF) is crucial for fully exploiting the intelligibility of such probabilistic graphical models. Tools must be developed…
The celebrated elliptic law describes the distribution of eigenvalues of random matrices with correlations between off-diagonal pairs of elements, having applications to a wide range of physical and biological systems. Here, we investigate…
The saltbox-roof parametric probability distribution is a special case of the triangular distribution, where only one side is truncated. Here it is presented as a single and independent distribution, where the explicit equations are defined…