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Slow Convergence in Bootstrap Percolation
, 2007
"... In the bootstrap percolation model, sites in an L by L square are initially infected independently with probability p. At subsequent steps, a healthy site becomes infected if it has at least 2 infected neighbours. As (L, p) → (∞, 0), the probability that the entire square is eventually infected is ..."
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In the bootstrap percolation model, sites in an L by L square are initially infected independently with probability p. At subsequent steps, a healthy site becomes infected if it has at least 2 infected neighbours. As (L, p) → (∞, 0), the probability that the entire square is eventually infected is known to undergo a phase transition in the parameter p log L, occurring asymptotically at λ = π 2 /18 [15]. We prove that the discrepancy between the critical parameter and its limit λ is at least Ω((log L) −1/2). In contrast, the critical window has width only Θ((log L) −1). For the socalled modified model, we prove rigorous explicit bounds which imply for example that the relative discrepancy is at least 1 % even when L = 10 3000. Our results shed some light on the observed differences between simulations and rigorous asymptotics.
Analysis of TopSwap Shuffling for Genome Rearrangements
"... Abstract: We study Markov chains which model genome rearrangements. These models are useful for studying the equilibrium distribution of chromosomal lengths, and are used in methods for estimating genomic distances. The primary Markov chain studied in this paper is the topswap Markov chain. The top ..."
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Abstract: We study Markov chains which model genome rearrangements. These models are useful for studying the equilibrium distribution of chromosomal lengths, and are used in methods for estimating genomic distances. The primary Markov chain studied in this paper is the topswap Markov chain. The topswap chain is a cardshuffling process with n cards divided over k decks, where the cards are ordered within each deck. A transition consists of choosing a random pair of cards, and if the cards lie in different decks, we cut each deck at the chosen card and exchange the tops of the two decks. We prove precise bounds on the relaxation time (inverse spectral gap) of the topswap chain. In particular, we prove the relaxation time is Θ(n + k). This resolves an open question of Durrett.
Submitted to the Annals of Applied Probability TRACER DIFFUSION AT LOW TEMPERATURE IN KINETICALLY CONSTRAINED MODELS
"... Abstract We describe the motion of a tracer in an environment given by a kinetically constrained spin model (KCSM) at equilibrium. We check convergence of its trajectory properly rescaled to a Brownian motion and positivity of the diffusion coefficient D as soon as the spectral gap of the environme ..."
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Abstract We describe the motion of a tracer in an environment given by a kinetically constrained spin model (KCSM) at equilibrium. We check convergence of its trajectory properly rescaled to a Brownian motion and positivity of the diffusion coefficient D as soon as the spectral gap of the environment is positive (which coincides with the ergodicity region under general conditions). Then we study the asymptotic behaviour of D when the density 1 − q of the environment goes to 1 in two classes of KCSM. For noncooperative models, the diffusion coefficient D scales like a power of q, with an exponent that we compute explicitly. In the case of the FredricksonAndersen onespin facilitated model, this proves a prediction made in Jung, Garrahan and Chandler (2004). For the East model, instead we prove that the diffusion coefficient is comparable to the spectral gap, which goes to zero faster than any power of q. This result proves that the trend found in numerical simulation results (Jung, Garrahan and
FREDRICKSONANDERSEN ONE SPIN FACILITATED MODEL OUT OF EQUILIBRIUM
"... ABSTRACT. We consider the Fredrickson and Andersen one spin facilitated model (FA1f) on an infinite connected graph with polynomial growth. Each site with rate one refreshes its occupation variable to a filled or to an empty state with probability p ∈ [0, 1] or q = 1 − p respectively, provided that ..."
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ABSTRACT. We consider the Fredrickson and Andersen one spin facilitated model (FA1f) on an infinite connected graph with polynomial growth. Each site with rate one refreshes its occupation variable to a filled or to an empty state with probability p ∈ [0, 1] or q = 1 − p respectively, provided that at least one of its nearest neighbours is empty. We study the nonequilibrium dynamics started from an initial distribution ν different from the stationary product pBernoulli measure µ. We assume that, under ν, the distance between two nearest empty sites has exponential moments. We then prove convergence to equilibrium when the vacancy density q is above a proper threshold q ̄ < 1. The convergence is exponential or stretched exponential, depending on the growth of the graph. In particular it is exponential on Zd for d = 1 and stretched exponential for d> 1. Our result can be generalized to other non cooperative models. 1.
Astonishing Cellular Automata
, 2007
"... Cellular automata arise naturally in the study of physical systems, and exhibit a seemingly limitless range of intriguing behaviour. Such models lend themselves naturally to simulation, yet rigorous analysis can be notoriously difficult, and can yield highly unexpected results. Bootstrap percolation ..."
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Cellular automata arise naturally in the study of physical systems, and exhibit a seemingly limitless range of intriguing behaviour. Such models lend themselves naturally to simulation, yet rigorous analysis can be notoriously difficult, and can yield highly unexpected results. Bootstrap percolation is a very simple model, originally introduced in [13], which turns out to hold many surprises. Cells arranged in an L by L square grid can be either infected or healthy. Initially, we flip a biased coin for each cell, declaring it infected with probability p. At each subsequent time step, any healthy cell with 2 or more infected neighbours becomes infected, while infected cells remain infected forever. (A cell’s neighbours are the cells immediately to its North, South, East and West interior cells have 4 neighbours while boundary cells have fewer). See Figures 1 and 2. 0 1 2 3 4 5 6 Figure 1: Bootstrap percolation on a square of size 5 at time steps