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Many signals evolve in time as a stochastic process, randomly switching between states over discretely sampled time points. Here we make an explicit link between the underlying stochastic process of a signal that can take on a bounded…
Structural properties of evolving random graphs are investigated. Treating linking as a dynamic aggregation process, rate equations for the distribution of node to node distances (paths) and of cycles are formulated and solved analytically.…
We consider random networks whose dynamics is described by a rate equation, with transition rates $w_{nm}$ that form a symmetric matrix. The long time evolution of the system is characterized by a diffusion coefficient $D$. In one dimension…
We study the current of particles that move independently in a common static random environment on the one-dimensional integer lattice. A two-level fluctuation picture appears. On the central limit scale the quenched mean of the current…
We study effects of turbulent mixing on the random growth of an interface in the problem of the deposition of a substance on a substrate. The growth is modelled by the well-known Kardar--Parisi--Zhang model. The turbulent advecting velocity…
It is possible to represent each of a number of Markov chains as an evolving sequence of connected subsets of a directed acyclic graph that grow in the following way: initially, all vertices of the graph are unoccupied, particles are fed in…
We study the normal approximation of functionals of Poisson measures having the form of a finite sum of multiple integrals. When the integrands are nonnegative, our results yield necessary and sufficient conditions for central limit…
We discuss the space-time determinantal random field which arises for the PNG model in one dimension and resembles the one for Dyson's Brownian motion. The information of interest for growth processes is carried by the edge statistics of…
In a system made up of independent random walks, fluctuations of order $n^{1/4}$ from the hydrodynamic limit come from particle current across characteristics. We show that a two-parameter space-time particle current process converges to a…
We study a random graph model which combines properties of the edge percolation model on Z^d and a classical random graph G(n,c/n). We show that this model, being a homogeneous random graph, has a natural relation to the so-called "rank 1…
Percolation is a model for random damage to a network. It is one of the simplest models that displays a phase transition: when the network is severely damaged, it falls apart in many small connected components, while if the damage is light,…
We introduce a new random graph model motivated by biological questions relating to speciation. This random graph is defined as the stationary distribution of a Markov chain on the space of graphs on $\{1, \ldots, n\}$. The dynamics of this…
We prove that under certain mild moment and continuity assumptions, the $d$-dimensional Gaussian free field is the only stochastic process in $d\geq 2$ that is translation invariant, exhibits a certain scaling, and satisfies the usual…
One of the major themes of random matrix theory is that many asymptotic properties of traditionally studied distributions of random matrices are universal. We probe the edges of universality by studying the spectral properties of random…
We consider a random object that is associated with both random walks and random media, specifically, the superposition of a configuration of subcritical Bernoulli percolation on an infinite connected graph and the trace of the simple…
We study the growth of random networks under a constraint that the diameter, defined as the average shortest path length between all nodes, remains approximately constant. We show that if the graph maintains the form of its degree…
We consider the random evolution described by the motion of a particle moving on a circle alternating the angular velocities $ \pm c $ and changing rotation at Poisson random times, resulting in a telegraph process over the circle. We study…
The random reversal graph offers new perspectives, allowing to study the connectivity of genomes as well as their most likely distance as a function of the reversal rate. Our main result shows that the structure of the random reversal graph…
This paper is concerned mainly with the macroscopic fractal behavior of various random sets that arise in modern and classical probability theory. Among other things, it is shown here that the macroscopic behavior of Boolean coverage…
We consider vertex percolation on pseudo-random $d-$regular graphs. The previous study by the second author established the existence of phase transition from small components to a linear (in $\frac{n}{d}$) sized component, at…