661 research outputs found

    TASEP hydrodynamics using microscopic characteristics

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    The convergence of the totally asymmetric simple exclusion process to the solution of the Burgers equation is a classical result. In his seminal 1981 paper, Herman Rost proved the convergence of the density fields and local equilibrium when the limiting solution of the equation is a rarefaction fan. An important tool of his proof is the subadditive ergodic theorem. We prove his results by showing how second class particles transport the rarefaction-fan solution, as characteristics do for the Burgers equation, avoiding subadditivity. In the way we show laws of large numbers for tagged particles, fluxes and second class particles, and simplify existing proofs in the shock cases. The presentation is self contained.Comment: 20 pages, 13 figures. This version is accepted for publication in Probability Surveys, February 20 201

    Regularity of quasi-stationary measures for simple exclusion in dimension d >= 5

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    We consider the symmetric simple exclusion process on Z^d, for d>= 5, and study the regularity of the quasi-stationary measures of the dynamics conditionned on not occupying the origin. For each \rho\in ]0,1[, we establish uniqueness of the density of quasi-stationary measures in L^2(d\nur), where \nur is the stationary measure of density \rho. This, in turn, permits us to obtain sharp estimates for P_{\nur}(\tau>t), where \tau is the first time the origin is occupied.Comment: 18 pages. Corrections after referee report. To be published in Ann Proba

    Hitting times for independent random walks on Zd\mathbb{Z}^d

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    We consider a system of asymmetric independent random walks on Zd\mathbb{Z}^d, denoted by {ηt,tR}\{\eta_t,t\in{\mathbb{R}}\}, stationary under the product Poisson measure νρ\nu_{\rho} of marginal density ρ>0\rho>0. We fix a pattern A\mathcal{A}, an increasing local event, and denote by τ\tau the hitting time of A\mathcal{A}. By using a loss network representation of our system, at small density, we obtain a coupling between the laws of ηt\eta_t conditioned on {τ>t}\{\tau>t\} for all times tt. When d3d\ge3, this provides bounds on the rate of convergence of the law of ηt\eta_t conditioned on {τ>t}\{\tau>t\} toward its limiting probability measure as tt tends to infinity. We also treat the case where the initial measure is close to νρ\nu_{\rho} without being product.Comment: Published at http://dx.doi.org/10.1214/009117906000000106 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

    Multiclass Hammersley-Aldous-Diaconis process and multiclass-customer queues

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    In the Hammersley-Aldous-Diaconis process infinitely many particles sit in R and at most one particle is allowed at each position. A particle at x$ whose nearest neighbor to the right is at y, jumps at rate y-x to a position uniformly distributed in the interval (x,y). The basic coupling between trajectories with different initial configuration induces a process with different classes of particles. We show that the invariant measures for the two-class process can be obtained as follows. First, a stationary M/M/1 queue is constructed as a function of two homogeneous Poisson processes, the arrivals with rate \lambda and the (attempted) services with rate \rho>\lambda. Then put the first class particles at the instants of departures (effective services) and second class particles at the instants of unused services. The procedure is generalized for the n-class case by using n-1 queues in tandem with n-1 priority-types of customers. A multi-line process is introduced; it consists of a coupling (different from Liggett's basic coupling), having as invariant measure the product of Poisson processes. The definition of the multi-line process involves the dual points of the space-time Poisson process used in the graphical construction of the system. The coupled process is a transformation of the multi-line process and its invariant measure the transformation described above of the product measure.Comment: 21 pages, 6 figure

    Fleming-Viot selects the minimal quasi-stationary distribution: The Galton-Watson case

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    Consider N particles moving independently, each one according to a subcritical continuous-time Galton-Watson process unless it hits 0, at which time it jumps instantaneously to the position of one of the other particles chosen uniformly at random. The resulting dynamics is called Fleming-Viot process. We show that for each N there exists a unique invariant measure for the Fleming-Viot process, and that its stationary empirical distribution converges, as N goes to infinity, to the minimal quasi-stationary distribution of the Galton-Watson process conditioned on non-extinction.Comment: 25 page

    No phase transition for Gaussian fields with bounded spins

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    Let a<b, \Omega=[a,b]^{\Z^d} and H be the (formal) Hamiltonian defined on \Omega by H(\eta) = \frac12 \sum_{x,y\in\Z^d} J(x-y) (\eta(x)-\eta(y))^2 where J:\Z^d\to\R is any summable non-negative symmetric function (J(x)\ge 0 for all x\in\Z^d, \sum_x J(x)<\infty and J(x)=J(-x)). We prove that there is a unique Gibbs measure on \Omega associated to H. The result is a consequence of the fact that the corresponding Gibbs sampler is attractive and has a unique invariant measure.Comment: 7 page

    Escape of mass in zero-range processes with random rates

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    We consider zero-range processes in Zd{\mathbb{Z}}^d with site dependent jump rates. The rate for a particle jump from site xx to yy in Zd{\mathbb{Z}}^d is given by λxg(k)p(yx)\lambda_xg(k)p(y-x), where p()p(\cdot) is a probability in Zd{\mathbb{Z}}^d, g(k)g(k) is a bounded nondecreasing function of the number kk of particles in xx and λ={λx}\lambda =\{\lambda_x\} is a collection of i.i.d. random variables with values in (c,1](c,1], for some c>0c>0. For almost every realization of the environment λ\lambda the zero-range process has product invariant measures {νλ,v:0vc}\{{\nu_{\lambda, v}}:0\le v\le c\} parametrized by vv, the average total jump rate from any given site. The density of a measure, defined by the asymptotic average number of particles per site, is an increasing function of vv. There exists a product invariant measure νλ,c{\nu _{\lambda, c}}, with maximal density. Let μ\mu be a probability measure concentrating mass on configurations whose number of particles at site xx grows less than exponentially with x\|x\|. Denoting by Sλ(t)S_{\lambda}(t) the semigroup of the process, we prove that all weak limits of {μSλ(t),t0}\{\mu S_{\lambda}(t),t\ge 0\} as tt\to \infty are dominated, in the natural partial order, by νλ,c{\nu_{\lambda, c}}. In particular, if μ\mu dominates νλ,c{\nu _{\lambda, c}}, then μSλ(t)\mu S_{\lambda}(t) converges to νλ,c{\nu_{\lambda, c}}. The result is particularly striking when the maximal density is finite and the initial measure has a density above the maximal.Comment: Published at http://dx.doi.org/10.1214/074921707000000300 in the IMS Lecture Notes Monograph Series (http://www.imstat.org/publications/lecnotes.htm) by the Institute of Mathematical Statistics (http://www.imstat.org

    Detection of spatial pattern through independence of thinned processes

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    Let N, N' and N'' be point processes such that N' is obtained from N by homogeneous independent thinning and N''= N- N'. We give a new elementary proof that N' and N'' are independent if and only if N is a Poisson point process. We present some applications of this result to test if a homogeneous point process is a Poisson point process.Comment: 11 pages, one figur