We introduce a system of one-dimensional coalescing nonsimple random walks with long range jumps allowing crossing paths and exibiting dependence before coalescence. We show that under diffusive scaling this system converges in distribution to the Brownian Web.2000 Mathematics Subject Classification. primary 60K35.
We study the evolution of the interface for the one-dimensional voter model. We show that if the random walk kernel associated with the voter model has finite γth moment for some γ > 3, then the evolution of the interface boundaries converge weakly to a Brownian motion under diffusive scaling. This extends recent work of Newman, Ravishankar and Sun. Our result is optimal in the sense that finite γth moment is necessary for this convergence for all γ ∈ (0, 3). We also obtain relatively sharp estimates for the tail distribution of the size of the equilibrium interface, extending earlier results of Cox and Durrett, and Belhaouari, Mountford and Valle
We provide a process on the space of coalescing cadlag stable paths and show convergence in the appropriate topology for coalescing stable random walks on the integer lattice.MSC 2010: 82B44, 82D30, 60K37
We show that for the voter model on {0, 1}ℤ corresponding to a random walk with kernel p(·) and starting from unanimity to the right and opposing unanimity to the left, a tight interface between zeros and ones exists if p(·) has finite second moment but does not if p(·) fails to have finite moment of order α for some α < 2.
Consider the basic algorithm to perform the transformation n → n + 1 changing digits of the d-adic expansion of n one by one. We obtain a family of Markov chains on the non-negative integers through sucessive and independent applications of the algorithm modified by a parametrized stochastic rule that randomly prevents one of the steps in the algorithm to finish. The objects of study in this paper are the spectra of the transition operators of these Markov chains. The spectra of these Markov chains turn out to be fibered Julia sets of fibered polynomials. This enable us to analyze their topological and analytical properties with respect to the underlying parameters of the Markov chains.
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