We study the fluctuations of the outer domain of Hastings-Levitov clusters in the small particle limit. These are shown to be given by a continuous Gaussian process F taking values in the space of holomorphic functions on {|z| > 1}, of which we provide an explicit construction.The boundary values W of F are shown to perform an Ornstein-Uhlenbeck process on the space of distributions on the unit circle T, which can be described as the solution to the stochastic fractional heat equationwhere ∆ denotes the Laplace operator acting on the spatial component, and ξ(t, ϑ) is a spacetime white noise. As a consequence we find that, when the cluster is left to grow indefinitely, the boundary process W converges to a log-correlated Fractional Gaussian Field, which can be realised as (−∆) −1/4 W , for W complex White Noise on T.
Several stochastic processes modeling molecular motors on a linear track are given by random walks (not necessarily Markovian) on quasi 1d lattices and share a common regenerative structure. Analyzing this abstract common structure, we derive information on the large fluctuations of the stochastic process by proving large deviation principles for the first-passage times and for the position. We focus our attention on the Gallavotti-Cohen-type symmetry of the position rate function (fluctuation theorem), showing its equivalence with the independence of suitable random variables. In the special case of Markov random walks, we show that this symmetry is universal only inside a suitable class of quasi 1d lattices.
Abstract. We consider random walks on quasi one dimensional lattices, as introduced in [6]. This mathematical setting covers a large class of discrete kinetic models for noncooperative molecular motors on periodic tracks. We derive general formulas for the asymptotic velocity and diffusion coefficient, and we show how to reduce their computation to suitable linear systems of the same degree of a single fundamental cell, with possible linear chain removals. We apply the above results to special families of kinetic models, also catching some errors in the biophysics literature.
Internal DLA is a discrete model of a moving interface. On the cylinder graph ZN × Z, a particle starts uniformly on ZN × {0} and performs simple random walk on the cylinder until reaching an unoccupied site in ZN × Z ≥0 , which it occupies forever. This operation defines a Markov chain on subsets of the cylinder. We first show that a typical subset is rectangular with at most logarithmic fluctuations. We use this to prove that two Internal DLA chains started from different typical subsets can be coupled with high probability by adding order N 2 log N particles. For a lower bound, we show that at least order N 2 particles are required to forget which of two independent typical subsets the process started from.
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