Related papers: The problem of predecessors on spanning trees
Self-avoiding walks (SAWs) and loop-erased random walks (LERWs) are two ensembles of random paths with numerous applications in mathematics, statistical physics and quantum field theory. While SAWs are described by the $n \to 0$ limit of…
The evolution of aligned DNA sequence sites is generally modeled by a Markov process operating along the edges of a phylogenetic tree. It is well known that the probability distribution on the site patterns at the tips of the tree…
We consider the survival probability $f(t)$ of a random walk with a constant hopping rate $w$ on a host lattice of fractal dimension $d$ and spectral dimension $d_s\le 2$, with spatially correlated traps. The traps form a sublattice with…
The probability distribution of the number $s$ of distinct sites visited up to time $t$ by a random walk on the fully-connected lattice with $N$ sites is first obtained by solving the eigenvalue problem associated with the discrete master…
The set of visited sites and the number of visited sites are two basic properties of the random walk trajectory. We consider two independent random walks on a hyper-cubic lattice and study ordering probabilities associated with these…
On an $r\times (n-r)$ lattice rectangle, we first consider walks that begin at the SW corner, proceed with unit steps in either of the directions E or N, and terminate at the NE corner of the rectangle. For each integer $k$ we ask for…
We study the growth of a time-ordered rooted tree by probabilistic attachment of new vertices to leaves. We construct a likelihood function of the leaves based on the connectivity of the tree. We take such connectivity to be induced by the…
This paper is a variation on the uniform spanning tree theme. We use random spanning forests to solve the following problem: for a Markov process on a finite set of size $n$, find a probability law on the subsets of any given size $m \leq…
We present an analytical approach to study simple symmetric random walks (RWs) on a crossing geometry consisting of a plane square lattice crossed by $n_l$ number of lines that all meet each other at a single point (the origin) on the…
A two-dimensional conformal field theory with a conserved $U(1)$ current $\vec J$, when perturbed by the operator ${\vec J}^{\,2}$, exhibits a line of fixed points along which the scaling dimensions of the operators with non-zero $U(1)$…
We construct forests that span $\mathbb{Z}^d$, $d\geq2$, that are stationary and directed, and whose trees are infinite, but for which the subtrees attached to each vertex are as short as possible. For $d\geq3$, two independent copies of…
We consider the percolation problem of sites on an $L\times L$ square lattice with periodic boundary conditions which were unvisited by a random walk of $N=uL^2$ steps, i.e. are vacant. Most of the results are obtained from numerical…
Loop-erased random walk, abbreviated LERW, is one of the most well-studied critical lattice models. It is the self-avoiding random walk one gets after erasing the loops from a simple random walk in order or alternatively by considering the…
Geodesic distance, sometimes called shortest path length, has proven useful in a great variety of applications, such as information retrieval on networks including treelike networked models. Here, our goal is to analytically determine the…
The hierarchical and recursive expressive capability of rooted trees is applicable to represent statistical models in various areas, such as data compression, image processing, and machine learning. On the other hand, such hierarchical…
A random walk problem with particles on discrete double infinite linear grids is discussed. The model is based on the work of Montroll and others. A probability connected with the problem is given in the form of integrals containing…
The statistics of self-avoiding random walks have been used to model polymer physics for decades. A self-avoiding walk that grows one step at a time on a lattice will eventually trap itself, which occurs after an average of 71 steps on a…
In the first part of the article our subject of interest is a simple symmetric random walk on the integers which faces a random risk to be killed. This risk is described by random potentials, which in turn are defined by a sequence of…
The self-avoiding walk on the square site-diluted correlated percolation lattice is considered. The Ising model is employed to realize the spatial correlations of the metric space. As a well-accepted result, the (generalized) Flory's mean…
The problem of missing link prediction in complex networks has attracted much attention recently. Two difficulties in link prediction are the sparsity and huge size of the target networks. Therefore, the design of an efficient and effective…