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Hugo Duminil-Copin Marcelo R. Hilário Gady Kozma Vladas Sidoravicius 《Israel Journal of Mathematics》2018,225(1):479-501
We study bond percolation on the square lattice with one-dimensional inhomogeneities. Inhomogeneities are introduced in the following way: A vertical column on the square lattice is the set of vertical edges that project to the same vertex on Z. Select vertical columns at random independently with a given positive probability. Keep (respectively remove) vertical edges in the selected columns, with probability p (respectively 1?p). All horizontal edges and vertical edges lying in unselected columns are kept (respectively removed) with probability q (respectively 1 ? q). We show that, if p > pc(Z2) (the critical point for homogeneous Bernoulli bond percolation), then q can be taken strictly smaller than pc(Z2) in such a way that the probability that the origin percolates is still positive. 相似文献
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According to a celebrated result of Kesten (Acta Math 131:207–248, 1973), random difference equations have a power-law distribution tail in the asymptotic sense. Empirical evidence shows that classical
estimators of tail exponent of random difference equations, such as Hill estimator, are extremely biased for larger values
of tail exponents. It is argued in this work that the bias occurs because the power-tail region is too far in the tail from
a practical perspective. This is supported by analysis of a few examples where a stationary distribution of random difference
equation is known explicitly, and by proving a weaker form of the so-called second order regular variation of distribution
tails of random difference equations, which measures deviations from the asymptotic power tail. The latter, in particular,
suggests a specific second order term for a distribution tail. Estimation of tail exponents can be adapted by taking this
second order term into account. One such method available in the literature is examined, and a new, simple, regression type
estimator is proposed. Simulation study shows that the proposed estimator works very well. ARCH models of interest in Finance
and multiplicative cascades used in Physics are considered as motivating examples throughout the work. Extension to multidimensional
random difference equations with nonnegative entries is also considered. 相似文献
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We study a continuous time growth process on (d?1) associated to the following interacting particle system: initially there is only one simple symmetric continuous time random walk of total jump rate one located at the origin; then, whenever a random walk visits a site still unvisited by any other random walk, it creates a new independent random walk starting from that site. Let us call Pd the law of such a process and S0d(t) the set of sites, visited by all walks by time t. We prove that there exists a bounded, non-empty, convex set , such that for every ε>0, Pd-a.s. eventually in t, the set Sd0(t) is within an ε neighborhood of the set [Cdt], where for we define . Moreover, for d large enough, the set Cd is not a ball under the Euclidean norm. We also show that the empirical density of particles within Sd0(t) converges weakly to a product Poisson measure of parameter one. To cite this article: A.F. Ram??rez, V. Sidoravicius, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 821–826. 相似文献
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Harry Kesten Bernardo N. B. de Lima Vladas Sidoravicius Maria Eulália Vares 《纯数学与应用数学通讯》2014,67(6):871-905
An ordered pair of semi‐infinite binary sequences (η,ξ) is said to be compatible if there is a way of removing a certain number (possibly infinite) of ones from η and zeroes from ξ that would map both sequences to the same semi‐infinite sequence. This notion was introduced by Peter Winkler, who also posed the following question: η and ξ being independent i.i.d. Bernoulli sequences with parameters p′ and p, respectively, does there exist (p′, p) so that the set of compatible pairs has positive measure? It is known that this does not happen for p and p′ very close to . In the positive direction, we construct, for any ? > 0, a deterministic binary sequence η? whose set of zeroes has Hausdorff dimension larger than 1 ? ? and such that ?p {ξ : (η?,ξ) is compatible } > 0 for p small enough, where ?p stands for the product Bernoulli measure with parameter p. © 2014 Wiley Periodicals, Inc. 相似文献
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Luiz Renato G. Fontes Eduardo Jordo Neves Vladas Sidoravicius 《Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques》2000,36(6):787
We study a one-dimensional infinite system of particles driven by a constant positive force F which acts only on the leftmost particle which is regarded as the tracer particle (t.p.). All other particles are field neutral, do not interact among themselves, and independently of each other with probability 0<p≤1 are either perfectly inelastic and “stick” to the t.p. after the first collision, or with probability 1−p are perfectly elastic, mechanically identical and have the same mass m. At initial time all particles are at rest, and the initial measure is such that the interparticle distances ξi's are i.i.d. r.v.'s. with absolutely continuous density. We show that for any value of the field F>0, the velocity of the t.p. converges to a limit value, which we compute. 相似文献
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It is known that Hermite processes have a finite-time interval representation. For fractional Brownian motion, the representation has been well known and plays a fundamental role in developing stochastic calculus for the process. For the Rosenblatt process, the finite-time interval representation was originally established by using cumulants. The representation was extended to general Hermite processes through the convergence of suitable partial sum processes. We provide here an alternative and different proof for the finite-time interval representation of Hermite processes. The approach is based on regularization of Hermite processes and the fractional Gaussian noises underlying them, and does not use cumulants nor convergence of partial sums. 相似文献
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We investigate random interlacements on ?d, d ≥ 3. This model, recently introduced in [8], corresponds to a Poisson cloud on the space of doubly infinite trajectories modulo time shift tending to infinity at positive and negative infinite times. A nonnegative parameter u measures how many trajectories enter the picture. Our main interest lies in the percolative properties of the vacant set left by random interlacements at level u. We show that for all d ≥ 3 the vacant set at level u percolates when u is small. This solves an open problem of [8], where this fact has only been established when d ≥ 7. It also completes the proof of the nondegeneracy in all dimensions d ≥ 3 of the critical parameter u* of [8]. © 2008 Wiley Periodicals, Inc. 相似文献
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Operator fractional Brownian motions (OFBMs) are zero mean, operator self-similar (o.s.s.) Gaussian processes with stationary
increments. They generalize univariate fractional Brownian motions to the multivariate context. It is well-known that the
so-called symmetry group of an o.s.s. process is conjugate to subgroups of the orthogonal group. Moreover, by a celebrated
result of Hudson and Mason, the set of all exponents of an operator self-similar process can be related to the tangent space
of its symmetry group. 相似文献