Brownian Web and Oriented Percolation: Density Bounds
Anish Sarkar, Rongfeng Sun
备注：
12 pages. This is a proceeding article for the RIMS workshop "Applications of Renormalization Group Methods in Mathematical Sciences", held at Kyoto University from September 12th to 14th, 2011. Submitted to the RIMS Kokyuroku series
在最近的一项工作中，我们证明了在扩散标度下，时空格点Z^2上超临界定向渗流构型中最右边的无限开路径的集合在分布上收敛到布朗网。在这个证明中，FKG不等式在建立密度界方面发挥了重要作用，这是Fontes等人（2004）制定的布朗网收敛准则的一部分。在本文中，我们说明了如何在这种情况下验证Newman et al（2005）提出的另一种收敛准则，该准则涉及可以在不使用FKG不等式的情况下建立的对偶密度界。这种替代方法在某种意义上更稳健。我们还将证明，从时间0开始的最右边的无限开放路径的集合的空间密度在时间上渐近衰减为c/ \sqrt{t}，对于某些c>0。
英文摘要：
In a recent work, we proved that under diffusive scaling, the collection of rightmost infinite open paths in a supercritical oriented percolation configuration on the space-time lattice Z^2 converges in distribution to the Brownian web. In that proof, the FKG inequality played an important role in establishing a density bound, which is a part of the convergence criterion for the Brownian web formulated by Fontes et al (2004). In this note, we illustrate how an alternative convergence criterion formulated by Newman et al (2005) can be verified in this case, which involves a dual density bound that can be established without using the FKG inequality. This alternative approach is in some sense more robust. We will also show that the spatial density of the collection of rightmost infinite open paths starting at time 0 decays asymptotically in time as c/\sqrt{t} for some c>0.
Previously, Sarkar and Sun have shown that for supercritical oriented percolation in dimension $1+1$, the set of rightmost infinite open paths converges to the Brownian web after proper centering and scaling. In this note, we show that a pair of sticky Brownian webs arise naturally if one considers the set of right-most infinite open paths for two coupled percolation configurations with distinct (but close) percolation parameters. This leads to a natural conjecture on the convergence of the dynamical supercritical oriented percolation model to the so-called dynamical Brownian web.
This paper presents a new, short proof of the computation of the upper tail large deviation rate function for the Brownian directed percolation model. Through a distributional equivalence between the last passage time in this model and the largest eigenvalue in a random matrix drawn from the Gaussian Unitary Ensemble, this provides a new proof a previously known result. The method leads to associated results for the stationary Brownian directed percolation model which have not been observed before.
Brownian Web in the Scaling Limit of Supercritical Oriented Percolation in Dimension 1+1
我们证明了，在对空间和时间进行定心和扩散重新标度之后，时空格点Z^2_{even}：={（x，i）in Z^2：x+i is even}上的超临界定向渗流构形中的最右边无限开路的集合按分布收敛到布朗网.这证明了吴和张的一个猜想。我们的关键观察是，每个最右边的无限开放路径可以近似为一个渗透勘探集群，不同的勘探集群独立发展之前，他们相交。
This paper proves an equality in law between the invariant measure of a reflected system of Brownian motions and a vector of point-to-line last passage percolation times in a discrete random environment. A consequence describes the distribution of the all-time supremum of Dyson Brownian motion with drift. A finite temperature version relates the point-to-line partition functions of two directed polymers, with an inverse-gamma and a Brownian environment, and generalises Dufresne's identity. Our proof introduces an interacting system of Brownian motions with an invariant measure given by a field…
We consider a continuum percolation model on $\R^d$, $d\geq 1$.For $t,λ\in (0,\infty)$ and $d\in\{1,2,3\}$, the occupied set is given by the union of independent Brownian paths running up to time $t$ whoseinitial points form a Poisson point process with intensity $λ\textgreater{}0$.When $d\geq 4$, the Brownian paths are replaced by Wiener sausageswith radius $r\textgreater{}0$.We establish that, for $d=1$ and all choices of $t$, no percolation occurs,whereas for $d\geq 2$, there is a non-trivial percolation transitionin $t$, provided $λ$ and $r$ are chosen properly.The last statement means tha…
We consider the backbone of the infinite cluster generated by supercritical oriented site percolation in dimension 1 +1. A directed random walk on this backbone can be seen as an "ancestral line" of an individual sampled in the stationary discrete-time contact process. Such ancestral lineages were investigated in [BCDG13] where a central limit theorem for a single walker was proved. Here, we consider infinitely many coalescing walkers on the same backbone starting at each space-time point. We show that, after diffusive rescaling, the collection of paths converges in distribution to the Brownia…
In this paper we study the convergence of dynamical discrete web (DyDW) to the dynamical Brownian web (DyBW) in the path space topology. We show that almost surely the DyBW has RCLL paths taking values in an appropriate metric space and as a sequence of RCLL paths, the scaled dynamical discrete web converges to the DyBW. This proves weak convergence of the DyDW process to the DyBW process.
We prove the existence of semi-infinite geodesics for Brownian last-passage percolation (BLPP). Specifically, on a single event of probability one, there exist semi-infinite geodesics started from every space-time point and traveling in every asymptotic direction. Properties of these geodesics include uniqueness for a fixed initial point and direction, non-uniqueness for fixed direction but random initial points, and coalescence of all geodesics traveling in a common, fixed direction. Along the way, we prove that for fixed northeast and southwest directions, there almost surely exist no bi-inf…
Several authors have studied convergence in distribution to the Brownian web under diffusive scaling of Markovian random walks. In a paper by R. Roy, K. Saha and A. Sarkar, convergence to the Brownian web is proved for a system of coalescing random paths -- the Random Directed Forest -- which are not Markovian. Paths in the Random Directed Forest do not cross each other before coalescence. Here we study a generalization of the non-Markovian Random Directed Forest where paths can cross each other and prove convergence to the Brownian web. This provides an example of how the techniques to prove …
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An analog of the Trotter formula for the Arratia flow is presented. Perturbations of the Brownian web by mappings associated with an ordinary differential equation with a smooth right part are considered and proved to be convergent exclusively in the weak sense. The flow obtained as a limit is the Arratia flow with drift.
We propose a metric space of coalescing pairs of paths on which we are able to prove (more or less) directly convergence of objects such as the persistence probability in the (one dimensional, nearest neighbor, symmetric) voter model or the diffusively rescaled weight distribution in a silo model (as well as the equivalent output distribution in a river basin model), interpreted in terms of (dual) diffusively rescaled coalescing random walks, to corresponding objects defined in terms of the Brownian web.
In the zero temperature Brownian semi-discrete directed polymer we study the joint distribution of two last-passage times at positions ordered in the time-like direction. This is the situation when we have the slow de-correlation phenomenon. We compute the limiting joint distribution function in a scaling limit. This limiting distribution is given by an expansion in determinants which is not a Fredholm expansion. A somewhat similar looking formula was derived non-rigorously in a related model by Dotsenko.
In this paper we prove a duality relation between coalescence times and exit points in last-passage percolation models with exponential weights. As a consequence, we get lower bounds for coalescence times with scaling exponent 3/2, and we relate its distribution with variational problems involving the Brownian motion process and the Airy process.
The Airy line ensemble is a positive-integer indexed system of random continuous curves whose finite dimensional distributions are given by the multi-line Airy process. It is a natural object in the KPZ universality class: for example, its highest curve, the Airy$_2$ process, describes after the subtraction of a parabola the limiting law of the scaled energy of a geodesic running from the origin to a variable point on an anti-diagonal line in such problems as Poissonian last passage percolation. The ensemble of curves resulting from the Airy line ensemble after the subtraction of the same para…
The geometry of coalescing random walks, the Brownian web distance and KPZ universality
Budapest University of Technology and Economics（布达佩斯科技经济大学）；HUN-REN–BME Stochastics Research Group（匈牙利研究网络-布达佩斯科技大学随机过程研究组）；University of Toronto（多伦多大学）
The two-dimensional directed spanning forest (DSF) introduced by Baccelli and Bordenave is a planar directed forest whose vertex set is given by a homogeneous Poisson point process $\mathcal{N}$ on $\mathbb{R}^2$. If the DSF has direction $-e_y$, the ancestor $h(u)$ of a vertex $u \in \mathcal{N}$ is the nearest Poisson point (in the $L_2$ distance) having strictly larger $y$-coordinate. This construction induces complex geometrical dependencies. In this paper we show that the collection of DSF paths, properly scaled, converges in distribution to the Brownian web (BW). This verifies a conjectu…
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We consider the tributary structure of Howard's drainage model studied by Gangopadhyay et. al. Conditional on the event that the tributary survives up to time $n$, we show that, as a sequence of random metric spaces, scaled tributary converges in distribution to a continuum random tree with respect to Gromov Hausdorff topology. This verifies a prediction made by Aldous for a simpler model (where paths are independent till they coalesce) but for a different conditional set up. The limiting continuum tree is slightly different from what was surmised earlier. Our proof uses the fact that there ex…
In last passage percolation models lying in the KPZ universality class, the energy of long energy-maximizing paths may be studied as a function of the paths' pair of endpoint locations. Scaled coordinates may be introduced, so that these maximizing paths, or polymers, now cross unit distances with unit-order fluctuations, and have scaled energy, or weight, of unit order. In this article, we consider Brownian last passage percolation in these scaled coordinates. In the narrow wedge case, when one endpoint of such polymers is fixed, say at $(0,0) \in \mathbb{R}^2$, and the other is varied horizo…
In last passage percolation models lying in the KPZ universality class, long maximizing paths have a typical deviation from the linear interpolation of their endpoints governed by the two-thirds power of the interpolating distance. This two-thirds power dictates a choice of scaled coordinates, in which these maximizers, now called polymers, cross unit distances with unit-order fluctuations. In this article, we consider Brownian last passage percolation in these scaled coordinates, and prove that the probability of the presence of $k$ disjoint polymers crossing a unit-order region while beginni…
The rate of the weak convergence in the fractional step method for the Arratia flow is established in terms of the Wasserstein distance between the images of the Lebesque measure under the action of the flow. We introduce finite-dimensional densities describing sequences of collisions in the Arratia flow and derive an explicit expression for them. With the initial interval discretized, the convergence of the corresponding approximations of the point measure associated with the Arratia flow is discussed in terms of such densities.
We consider large-scale point fields which naturally appear in the context of the Kardar-Parisi-Zhang (KPZ) phenomenon. Such point fields are geometrical objects formed by points of mass concentration, and by shocks separating the sources of these points. We introduce similarly defined point fields for processes of coalescing fractional Brownian motions (cfBm). The case of the Hurst index 2/3 is of particular interest for us since, in this case, the power law of the density decay is the same as that in the KPZ phenomenon. In this paper, we present strong numerical evidence that statistical pro…
A universality property for last-passage percolation paths close to the axis
在$Z_+^2$中考虑最后一次通过定向渗流模型，其中i.i.d.其公共分布具有有限$(2+p)$阶矩的权重。我们研究了从原点到点$\big(n,n^{\lfloor a \rfloor}\big)$的通过时间的涨落。我们发现，对于合适的$a$（取决于$p$），这个数量，适当缩放，收敛分布为$n\to\infty$的Tracy-Widom分布，无论潜在的权重分布。该论点使用耦合到布朗定向渗流问题和Komlós，Major和Tusnády的强近似。
Consider a sequence of n bi-infinite and stationary Brownian queues in tandem. Assume that the arrival process entering in the first queue is a zero mean ergodic process. We prove that the departure process from the n-th queue converges in distribution to a Brownian motion as n goes to infinity. In particular this implies that the Brownian motion is an attractive invariant measure for the Brownian queueing operator. Our proof exploits the relationship between the Brownian queues in tandem and the last-passage Brownian percolation model, developing a coupling technique in the second setting. Th…
Convergence of directed forests, spanning on random subsets of lattices or on point processes, towards the Brownian web has made the subject of an abundant literature, a large part of which relies on a criterion proposed by Fontes, Isopi, Newman and Ravishankar (2004). One of their convergence condition, called (B2), states that the probability of the event that there exists three distinct paths for a time interval of length $t(>0)$, all starting within a segment of length $\varepsilon$, is of small order of $\varepsilon$. This condition is often verified by applying an FKG type correlation in…
Hack [Studies of longitudinal stream profiles in Virginia and Maryland (1957). Report], while studying the drainage system in the Shenandoah valley and the adjacent mountains of Virginia, observed a power law relation $l\sim a^{0.6}$ between the length $l$ of a stream from its source to a divide and the area $a$ of the basin that collects the precipitation contributing to the stream as tributaries. We study the tributary structure of Howard's drainage network model of headward growth and branching studied by Gangopadhyay, Roy and Sarkar [Ann. Appl. Probab. 14 (2004) 1242-1266]. We show that th…
In last passage percolation models lying in the KPZ universality class, the energy of long energy-maximizing paths may be studied as a function of the paths' pair of endpoint locations. Scaled coordinates may be introduced, so that these maximizing paths, or polymers, now cross unit distances with unit-order fluctuations, and have scaled energy, or weight, of unit order. In this article, we consider Brownian last passage percolation in these scaled coordinates. In the narrow wedge case, one endpoint of such polymers is fixed, say at $(0,0) \in \mathbb{R}^2$, and the other is varied horizontall…
We consider directed last passage percolation on $\mathbb{Z}^2$ with exponential passage times on the vertices. A topic of great interest is the coupling structure of the weights of geodesics as the endpoints are varied spatially and temporally. A particular specialization is when one considers geodesics to points varying in the time direction starting from a given initial data. This paper considers the flat initial condition which corresponds to line-to-point last passage times. Settling a conjecture by Ferrari and Spohn (SIGMA, 2016), we show that for the passage times from the line $x+y=0$ …
We consider the system of one-sided reflected Brownian motions which is in variational duality with Brownian last passage percolation. We show that it has integrable transition probabilities, expressed in terms of Hermite polynomials and hitting times of exponential random walks, and that it converges in the 1:2:3 scaling limit to the KPZ fixed point, the scaling invariant Markov process defined in [MQR17] and believed to govern the long time large scale fluctuations for all models in the KPZ universality class. Brownian last passage percolation was shown recently in [DOV18] to converge to the…
