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We consider the problem of stochastic flow of multiple particles traveling on a closed loop, with a constraint that particles move without passing. We use a Markov chain description that reduces the problem to a generalized random walk on a hyperplane (with boundaries). By expressing positions via a moving reference frame, the geometry of the no-passing criteria is greatly simplified, with the resultant condition expressible as the coordinate system planes which bound the first orthant. To determine state transition probabilities, we decompose transitions into independent events and construct a digraph representation in which calculating transition probability is reduced to a shortest path determination on the digraph. The resultant decomposition digraph is self-converse, and we exploit that property to establish the necessary symmetries to find the stationary density for the process.
We discuss conditions for unique ergodicity of a collective random walk on a continuous circle. Individual particles in this collective motion perform independent (and different in general) random walks conditioned by the assumption that the particle
We work under the A{i}d{e}kon-Chen conditions which ensure that the derivative martingale in a supercritical branching random walk on the line converges almost surely to a nondegenerate nonnegative random variable that we denote by $Z$. It is shown t
We consider a simple random walk on $mathbb{Z}^d$ started at the origin and stopped on its first exit time from $(-L,L)^d cap mathbb{Z}^d$. Write $L$ in the form $L = m N$ with $m = m(N)$ and $N$ an integer going to infinity in such a way that $L^2 s
We consider self-avoiding walk on a tree with random conductances. It is proven that in the weak disorder regime, the quenched critical point is equal to the annealed one, and that in the strong disorder regime, these critical points are strictly dif
We consider the limit behavior of a one-dimensional random walk with unit jumps whose transition probabilities are modified every time the walk hits zero. The invariance principle is proved in the scheme of series where the size of modifications depe