我们表明，对于一个固定版本的Hammersley的过程，与泊松“源”的正x轴上，和泊松“汇”的正y轴上，一个孤立的第二类粒子，位于原点在时间为零，渐近移动，概率为1，沿Hammersley的过程的守恒方程的特征。这使我们能够表明，哈默斯利的过程中没有汇或源，所定义的奥尔德斯和Diaconis [Probab. Theory Related Fields 10（1995）199-213]在分布上局部收敛于泊松过程，这是Aldous和Diaconis（1995）通过使用遍历分解定理和Hammersley过程作为一维点过程的构造首次证明的结果，其在整个实数线上作为（连续）时间的函数发展。作为推论，我们得到了EL（t，t）/t收敛于2的结果，即t \to \infty，其中L（t，t）是从（0，0）到（t，t）的最长东北路径的长度.这些事实的证明既不需要遍历分解定理，也不需要次可加遍历定理。我们还证明了具有源和汇的平稳过程的伯克定理的一个版本，并简要讨论了这些结果与随机排列的最长递增子序列理论的关系。
英文摘要：
We show that, for a stationary version of Hammersley's process, with Poisson ``sources'' on the positive x-axis, and Poisson ``sinks'' on the positive y-axis, an isolated second-class particle, located at the origin at time zero, moves asymptotically, with probability 1, along the characteristic of a conservation equation for Hammersley's process. This allows us to show that Hammersley's process without sinks or sources, as defined by Aldous and Diaconis [Probab. Theory Related Fields 10 (1995) 199-213] converges locally in distribution to a Poisson process, a result first proved in Aldous and Diaconis (1995) by using the ergodic decomposition theorem and a construction of Hammersley's process as a one-dimensional point process, developing as a function of (continuous) time on the whole real line. As a corollary we get the result that EL(t,t)/t converges to 2, as t\to\infty, where L(t,t) is the length of a longest North-East path from (0,0) to (t,t). The proofs of these facts need neither the ergodic decomposition theorem nor the subadditive ergodic theorem. We also prove a version of Burke's theorem for the stationary process with sources and sinks and briefly discuss the relation of these results with the theory of longest increasing subsequences of random permutations.
We introduce two stationary versions of two discrete variants of Hammersley's process in a finite box, this allows us to recover in a unified and simple way the laws of large numbers proved by T. Sepp{ä}l{ä}inen for two generalized Ulam's problems. As a by-product we obtain an elementary solution for the original Ulam problem. We also prove that for the first process defined on Z, Bernoulli product measures are the only extremal and translation-invariant stationary measures.
We construct a stationary random tree, embedded in the upper half plane, with prescribed offspring distribution and whose vertices are the atoms of a unit Poisson point process. This process which we call Hammersley's tree process extends the usual Hammersley's line process. Just as Hammersley's process is related to the problem of the longest increasing subsequence, this model also has a combinatorial interpretation: it counts the number of heaps (i.e. increasing trees) required to store a random permutation. This problem was initially considered by Byers et. al (2011) and Istrate and Bonchis…
We study the invariant distributions of Hammersley's serial harness process in all dimensions and height fluctuations in one dimension. Subject to mild moment assumptions there is essentially one unique invariant distribution, and all other invariant distributions are obtained by adding harmonic functions of the averaging kernel. We identify one Gaussian case where the invariant distribution is i.i.d. Height fluctuations in one dimension obey the stochastic heat equation with additive noise (Edwards-Wilkinson universality). We prove this for correlated initial data subject to polynomial decay …
We consider the Hammersley interacting particle system starting from a shock initial profile with densities $λ,ρ\in\mathbb{R}$ ($λ> ρ$). The microscopic shock is taken as the position of a second-class particle initially at the origin, and the main results are: a central limit theorem for the shock; the variance of the shock equals $2[λρ(λ- ρ)]^{-1}t + O(t^{2/3})$. By using the same method of proof, we also prove similar results for first-class particles.
We discuss the order of the variance on a lattice analogue of the Hammersley process with boundaries, for which the environment on each site has independent, Bernoulli distributed values. The last passage time is the maximum number of Bernoulli points that can be collected on a piecewise linear path, where each segment has strictly positive but finite slope. We show that along characteristic directions the order of the variance of the last passage time is of order $N^{2/3}$ in the model with boundary. These characteristic directions are restricted in a cone starting at the origin, and along an…
The longest increasing subsequence problem for permutations has been studied extensively in the last fifty years. The interpretation of the longest increasing subsequence as the longest 21-avoiding subsequence in the context of permutation patterns leads to many interesting research directions. We introduce and study the statistical properties of Hammersleytype interacting particle processes related to these generalizations and explore the finer structures of their distributions. We also propose three different interacting particle systems in the plane analogous to the Hammersley process in on…
We study languages and formal power series associated to (variants of) Hammersley's process. We show that the ordinary Hammersley process yields a regular language and the Hammersley tree process yields deterministic context-free (but non-regular) languages. For the extension to intervals of the Hammersley process we show that there are two relevant formal languages. One of them leads to the same class of languages as the ordinary Hammersley tree process. The other one yields non-context-free languages. The results are motivated by the problem of studying the analog of the famous Ulam-Hammersl…
We consider Robinson-Schensted-Knuth algorithm applied to a random input and study the growth of the bottom rows of the corresponding Young diagrams. We prove multidimensional Poisson limit theorem for the resulting Plancherel growth process. In this way we extend the result of Aldous and Diaconis to more than just one row. This result can be interpreted as convergence of the multi-line Hammersley process to its stationary distribution which is given by a collection of independent Poisson point processes.
Using the fact that the Airy process describes the limiting fluctuations of the Hammersley last-passage percolation model, we prove that it behaves locally like a Brownian motion. Our method is quite straightforward, and it is based on a certain monotonicity and good control over the equilibrium measures of the Hammersley model (local comparison).
We construct a stationary ergodic process $X_1, X_2, \ldots $ such that each $X_t$ has the uniform distribution on the unit square and the length $L_n$ of the shortest path through the points $X_1, X_2, \ldots,X_n$ is not asymptotic to a constant times the square root of $n$. In other words, we show that the Beardwood, Halton and Hammersley theorem does not extend from the case of independent uniformly distributed random variables to the case of stationary ergodic sequences with uniform marginal distributions.
In this paper we will prove a shape theorem for the last passage percolation model on a two dimensional $F$-compound 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 semi-infinite geodesics in some fixed direction $α$, following an approach developed by Newman and co-authors, and applied to the classical Hammersley process by Wüthrich. These results will be crucial in the development of an upcoming paper on the relatio…
We investigate partitioning of integer sequences into heapable subsequences (previously defined and established by Mitzenmacher et al). We show that an extension of patience sorting computes the decomposition into a minimal number of heapable subsequences (MHS). We connect this parameter to an interactive particle system, a multiset extension of Hammersley's process, and investigate its expected value on a random permutation. In contrast with the (well studied) case of the longest increasing subsequence, we bring experimental evidence that the correct asymptotic scaling is $\frac{1+\sqrt{5}}{2…
We consider a variant of the continuous and discrete Ulam-Hammersley problems: we study the maximal length of an increasing path through a Poisson point process (or a Bernoulli point process) with the restriction that there must be minimal gaps between abscissae and ordinates of successive points of the path.For both cases (continuous and discrete) our approach rely on couplings with well-studied models: respectively the classical Ulam-Hammersley problem and last-passage percolation with geometric weights. Thanks to these couplings we obtain explicit limiting shapes in both settings.We also es…
Perturbation of the equilibrium for a totally asymmetric stick process in one dimension
研究了完全非对称的一维相互作用系统平衡态的小扰动的演化。我们所采用的模型是从一个标记粒子看的Hammersley过程，它可以被看作是一个在Z点上的正值棒高度相互作用的过程。已知在欧拉标度（空间和时间尺度n）下，经验杆轮廓服从Burgers方程。我们用两种方法来改进这个结果：如果过程开始时足够接近平衡，那么随着时间n^v（1 \le v <3），直到流体动力学尺度上的误差消失，动力学只是平移了初始的杆构型。在一个特殊的标度族下，微扰的时间演化是可见的：在时间n^ν，1<ν<3/2上，从平衡态开始的n^（1-ν）阶微扰遵循无粘Burgers方程。棒模型的结果来自于Hammersley过程中标记粒子的渐近结果。
