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Onedimensional stochastic growth and Gaussian . . .
, 2005
"... In this review paper we consider the polynuclear growth (PNG) model in one spatial dimension and its relation to random matrix ensembles. For curved and flat growth the scaling functions of the surface fluctuations coincide with limit distribution functions coming from certain Gaussian ensembles of ..."
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Cited by 21 (9 self)
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In this review paper we consider the polynuclear growth (PNG) model in one spatial dimension and its relation to random matrix ensembles. For curved and flat growth the scaling functions of the surface fluctuations coincide with limit distribution functions coming from certain Gaussian ensembles of random matrices. This connection can be explained via point processes associated to the PNG model and the random matrices ensemble by an extension to the multilayer PNG and multimatrix models, respectively. We also discuss other models which are equivalent to the PNG model: directed polymers, the longest increasing subsequence problem, Young tableaux, a directed percolation model, kinkantikink gas, and Hammersley process.
BUSEMANN FUNCTIONS AND EQUILIBRIUM MEASURES IN LAST PASSAGE PERCOLATION
, 901
"... Abstract. The interplay between twodimensional percolation growth models and onedimensional particle processes has always been a fruitful source of interesting mathematical phenomena. In this paper we develop a connection between the construction of Busemann functions in the Hammersley lastpassag ..."
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Cited by 11 (4 self)
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Abstract. The interplay between twodimensional percolation growth models and onedimensional particle processes has always been a fruitful source of interesting mathematical phenomena. In this paper we develop a connection between the construction of Busemann functions in the Hammersley lastpassage percolation model with i.i.d. random weights, and the existence, ergodicity and uniqueness of equilibrium measures for the related (multiclass) interacting particle process. As we shall see, in the classical Hammersley model where each point has weight one, this approach brings a new and rather geometrical solution of the longest increasing subsequence problem, as well as a detailed description of the scaling behavior of the Busemann function along different directions. 1.
A shape theorem and semiinfinite geodesics for the Hammersley model with random weights
 ALEA Lat. Am. J. Probab. Math. Stat
"... Abstract. In this paper we will prove a shape theorem for the lastpassage percolation model on a two dimensional Fcompound Poisson process, called the Hammersley model with random weights. We will also provide diffusive upper bounds for shape fluctuations. Finally we will indicate how these result ..."
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Cited by 9 (5 self)
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Abstract. In this paper we will prove a shape theorem for the lastpassage percolation model on a two dimensional Fcompound Poisson process, called the Hammersley model with random weights. We will also provide diffusive upper bounds for shape fluctuations. Finally we will indicate how these results can be used to prove existence and coalescence of semiinfinite geodesics in some fixed direction α, following an approach developed by Newman and coauthors Howard and Newman (2001); Licea and Newman (1996); Newman (1995), and applied to the classical Hammersley process by Wüthrich in Wüthrich (2002). These results will be crucial in the development of an upcoming paper on the relation between Busemann functions and equilibrium measures in lastpassage percolation models Cator and Pimentel (2009). 1.
Multiclass HammersleyAldousDiaconis process and multiclasscustomer queues
, 2007
"... In the HammersleyAldousDiaconis 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 ..."
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Cited by 6 (2 self)
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In the HammersleyAldousDiaconis 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 twoclass 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 λ and the (attempted) services with rate ρ> λ. 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 nclass case by using n − 1 queues in tandem with n − 1 prioritytypes of customers. A multiline 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 multiline process involves the dual points of the spacetime Poisson process used in the graphical construction of the system. The coupled process is a transformation of the multiline process and its invariant measure the transformation described above of the product measure.
Behavior of a secondclass particle in Hammersley process (preprint
"... In the case of a rarefaction fan in a nonstationary Hammersley process, we explicitly calculate the asymptotic behavior of the process as we move out along a ray, and the asymptotic distribution of the angle within the rarefaction fan of a second class particle and a dual second class particle. Fur ..."
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Cited by 2 (1 self)
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In the case of a rarefaction fan in a nonstationary Hammersley process, we explicitly calculate the asymptotic behavior of the process as we move out along a ray, and the asymptotic distribution of the angle within the rarefaction fan of a second class particle and a dual second class particle. Furthermore, we consider a stationary Hammersley process and use the previous results to show that trajectories of a second class particle and a dual second class particles touch with probability one, and we give some information on the area enclosed by the two trajectories, up until the first intersection point. This is linked to the area of influence of an added Poisson point in the plane. 1
Behavior of a second class particle and its dual in Hammersley’s process.
, 2005
"... In the case of a rarefaction fan in a nonstationary Hammersley process, we explicitly calculate the asymptotic behavior of the process as we move out along a ray, and the asymptotic distribution of the angle within the rarefaction fan of a second class particle and a dual second class particle. Fur ..."
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In the case of a rarefaction fan in a nonstationary Hammersley process, we explicitly calculate the asymptotic behavior of the process as we move out along a ray, and the asymptotic distribution of the angle within the rarefaction fan of a second class particle and a dual second class particle. Furthermore, we show that a second class particle and a dual second class particle touch with probability one, and we give some information on the area enclosed by the two trajectories, up until the first intersection point. The concept of a dual second class particle is natural within a Hammersley process, but we have not encountered it, or problems related to it, in the literature on discrete interacting particle processes. 1