In this paper we construct an object which we call the full Brownian web (FBW) and prove that the collection of all space-time trajectories of a class of one-dimensional stochastic flows converges weakly, under diffusive rescaling, to the FBW. The (forward) paths of the FBW include the coalescing Brownian motions of the ordinary Brownian web along with bifurcating paths. Convergence of rescaled stochastic flows to the FBW follows from general characterization and convergence theorems that we present here combined with earlier results of Piterbarg.
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.
The Brownian web is a random object that occurs as the scaling limit of an infinite system of coalescing random walks. Perturbing this system of random walks by, independently at each point in space-time, resampling the random walk increments, leads to some natural dynamics. In this paper we consider the corresponding dynamics for the Brownian web. In particular, pairs of coupled Brownian webs are studied, where the second web is obtained from the first by perturbing according to these dynamics. A stochastic flow of kernels, which we call the erosion flow, is obtained via a filtering construct…
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 …
We introduce a system of one-dimensional coalescing nonsimple random walks with long range jumps allowing crossing paths and exibiting dependence before coalescence. We show that under diffusive scaling this system converges in distribution to the Brownian Web.
The Brownian web is a collection of one-dimensional coalescing Brownian motions starting from everywhere in space and time, and the Brownian net is a generalization that also allows branching. They appear in the diffusive scaling limits of many one-dimensional interacting particle systems with branching and coalescence. This article gives an introduction to the Brownian web and net, and how they arise in the scaling limits of various one-dimensional models, focusing mainly on coalescing random walks and random walks in i.i.d. space-time random environments. We will also briefly survey models a…
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.
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.
The Brownian Web (BW) is the random network formally consisting of the paths of coalescing one-dimensional Brownian motions starting from every space-time point in ${\mathbb R}\times{\mathbb R}$. We extend the earlier work of Arratia and of Tóth and Werner by providing characterization and convergence results for the BW distribution, including convergence of the system of all coalescing random walkssktop/brownian web/finale/arXiv submits/bweb.tex to the BW under diffusive space-time scaling. We also provide characterization and convergence results for the Double Brownian Web, which combines th…
We prove that, after centering and diffusively rescaling space and time, the collection of rightmost infinite open paths in a supercritical oriented percolation configuration on the space-time lattice Z^2_{even}:={(x,i) in Z^2: x+i is even} converges in distribution to the Brownian web. This proves a conjecture of Wu and Zhang. Our key observation is that each rightmost infinite open path can be approximated by a percolation exploration cluster, and different exploration clusters evolve independently before they intersect.
The Brownian web is a random variable consisting of a Brownian motion starting from each space-time point on the plane. These are independent until they hit each other, at which point they coalesce. Tsirelson mentions this model in his paper "Scaling limit, Noise, Stability", along with planar percolation, in suggesting the existence of a two-dimensional black noise. A two-dimensional noise is, roughly speaking, a random object on the plane whose distribution is translation invariant and whose behavior on disjoint subsets is independent. Black means sensitive to the resampling of sets of arbit…
The Brownian web (BW) is the random network formally consisting of the paths of coalescing one-dimensional Brownian motions starting from every space-time point in R\timesR. We extend the earlier work of Arratia and of Toth and Werner by providing a new characterization which is then used to obtain convergence results for the BW distribution, including convergence of the system of all coalescing random walks to the BW under diffusive space-time scaling.
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…
Consider the $d$ dimensional lattice $\mathbb{Z}^d$ where each vertex is open or closed with probability $p$ or $1-p$ respectively. An open vertex $\mathbb{u} := (\mathbb{u}(1), \mathbb{u}(2),...,\mathbb{u}(d))$ is connected by an edge to another open vertex which has the minimum $L_1$ distance among all the open vertices with $\mathbb{x}(d)>\mathbb{u}(d)$. It is shown that this random graph is a tree almost surely for $d=2$ and 3 and it is an infinite collection of disjoint trees for $d\geq 4$. In addition for $d=2$, we show that when properly scaled, family of its paths converges in distribu…
The Brownian web (BW) is a collection of coalescing Brownian paths indexed by the plane. It appears in particular as continuous limit of various discrete models of directed forests of coalescing random walks and navigation schemes. Radial counterparts have been considered but global invariance principles are hard to establish. In this paper, we consider cylindrical forests which in some sense interpolate between the directed and radial forests: we keep the topology of the plane while still taking into account the angular component. We define in this way the cylindric Brownian web (CBW), which …
We study a system of coalescing random walks on the integer lattice $\mathbb{Z}^{d}$ in which the walk is oriented in the $d$-th direction and follows certain specified rules. We first study the geometry of the paths and show that, almost surely, the paths from a graph consisting of just one tree for dimentions $d=2,3$ and infinitely many disjoint trees for dimensions $d\geq 4$. Also, there is no bi-infinite path in the graph almost surely for $d\geq 2$. Subsequently, we prove that for $d=2$ the diffusive scaling of this system converges in distribution to the Brownian web.
The Brownian web can be roughly described as a family of coalescing one-dimensional Brownian motions starting at all times in $\R$ and at all points of $\R$. It was introduced by Arratia; a variant was then studied by Toth and Werner; another variant was analyzed recently by Fontes, Isopi, Newman and Ravishankar. The two-dimensional \emph{Poisson tree} is a family of continuous time one-dimensional random walks with uniform jumps in a bounded interval. The walks start at the space-time points of a homogeneous Poisson process in $\R^2$ and are in fact constructed as a function of the point proc…
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 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 t…
Coarsening on a one-dimensional lattice is described by the voter model or equivalently by coalescing (or annihilating) random walks representing the evolving boundaries between regions of constant color and by backward (in time) coalescing random walks corresponding to color genealogies. Asympotics for large time and space on the lattice are described via a continuum space-time voter model whose boundary motion is expressed by the {\it Brownian web} (BW) of coalescing forward Brownian motions. In this paper, we study how small noise in the voter model, corresponding to the nucleation of rando…
The Brownian Web (BW) is a family of coalescing Brownian motions starting from every point in space and time $\R\times\R$. It was first introduced by Arratia, and later analyzed in detail by Tóth and Werner. More recently, Fontes, Isopi, Newman and Ravishankar gave a characterization of the BW, and general convergence criteria allowing either crossing or noncrossing paths, which they verified for coalescing simple random walks. Later Ferrari, Fontes, and Wu verified these criteria for a two dimensional Poisson Tree. In both cases, the paths are noncrossing. In this thesis, we formulate new con…
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 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…
Certain one-dimensional nearest-neighbor random walks in i.i.d. random space-time environments are known to have diffusive scaling limits. In the continuum limit, the random environment is represented by a `stochastic flow of kernels', which is a collection of random kernels that can be loosely interpreted as the transition probabilities of a Markov process in a random environment. The theory of stochastic flows of kernels was introduced by Le Jan and Raimond, who showed that each such flow is characterized by its n-point motions. We focus on a class of stochastic flows of kernels with Brownia…
We present a new proof of the extended arc-sine law related to Walsh's Brownian motion, known also as Brownian spider. The main argument mimics the scaling property used previously, in particular by D. Williams in the 1-dimensional Brownian case, which can be generalized to the multivariate case. A discussion concerning the time spent positive by a skew Bessel process is also presented.
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…
The d-inverse is a generalized notion of inverse of a stochastic process having a certain tendency of increasing expectations. Scaling limit of the d-inverse of Brownian motion with functional drift is studied. Except for degenerate case, the class of possible scaling limits is proved to consist of the d-inverses of Brownian motion without drift, one with explosion in finite time, and one with power drift.
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…
We construct a measure valued Markov process which we call infinite canonical super-Brownian motion, and which corresponds to the canonical measure of super-Brownian motion conditioned on non-extinction. Infinite canonical super-Brownian motion is a natural candidate for the scaling limit of various random branching objects on $\Z^d$ when these objects are (a) critical; (b) mean-field and (c) infinite. We prove that ICSBM is the scaling limit of the spread-out oriented percolation incipient infinite cluster above 4 dimensions and of incipient infinite branching random walk in any dimension. We…
The cactus of a pointed graph is a discrete tree associated with this graph. Similarly, with every pointed geodesic metric space $E$, one can associate an $\R$-tree called the continuous cactus of $E$. We prove under general assumptions that the cactus of random planar maps distributed according to Boltzmann weights and conditioned to have a fixed large number of vertices converges in distribution to a limiting space called the Brownian cactus, in the Gromov-Hausdorff sense. Moreover, the Brownian cactus can be interpreted as the continuous cactus of the so-called Brownian map.
Fisher-KPP equation is proved to be the scaling limit of a system of Brownian particles with local interaction. Particles proliferate and die depending on the local concentration of other particles. Opposite to discrete models, controlling concentration of particles is a major difficulty in Brownian particle interaction; local interactions instead of mean field or moderate ones makes it more difficult to implement the law of large numbers properties. The approach taken here to overcome these difficulties is largely inspired by A. Hammond and F. Rezakhanlou [10] implemented there in the mean fr…
Arratia, and later Tóth and Werner, constructed random processes that formally correspond to coalescing one-dimensional Brownian motions starting from every space-time point. We extend their work by constructing and characterizing what we call the {\em Brownian Web} as a random variable taking values in an appropriate (metric) space whose points are (compact) sets of paths. This leads to general convergence criteria and, in particular, to convergence in distribution of coalescing random walks in the scaling limit to the Brownian Web.
We introduce and study the random non-compact metric space called the Brownian plane, which is obtained as the scaling limit of the uniform infinite planar quadrangulation. Alternatively, the Brownian plane is identified as the Gromov-Hausdorff tangent cone in distribution of the Brownian map at its root vertex, and it also arises as the scaling limit of uniformly distributed (finite) planar quadrangulations with n faces when the scaling factor tends to 0 less fast than n^{-1/4}. We discuss various properties of the Brownian plane. In particular, we prove that the Brownian plane is homeomorphi…
We consider a variant of the radial spanning tree introduced by Baccelli and Bordenave. Like the original model, our model is a tree rooted at the origin, built on the realization of a planar Poisson point process. Unlike it, the paths of our model have independent jumps. We show that locally our diffusively rescaled tree, seen as the collection of the paths connecting its sites to the root, converges in distribution to the Brownian Bridge Web, which is roughly speaking a collection of coalescing Brownian bridges starting from all the points of a planar strip perpendicular to the time axis, an…
We consider the additive martingale $W_t(λ)$ and the derivative martingale $\partial W_t(λ)$ for one-dimensional supercritical super-Brownian motions with general branching mechanism. In the critical case $λ=λ_0$, we prove that $\sqrt{t}W_t(λ_0)$ converges in probability to a positive limit, which is a constant multiple of the almost sure limit $\partial W_\infty(λ_0)$ of the derivative martingale $\partial W_t(λ_0)$. We also prove that, on the survival event, $\limsup_{t\to\infty}\sqrt{t}W_t(λ_0)=\infty$ almost surely.
We present different continuous models of random geometry that have been introduced and studied in the recent years. In particular, we consider the Brownian map, which is the universal scaling limit of large planar maps in the Gromov-Hausdorff sense, and the Brownian disk, which appears as the scaling limit of planar maps with a boundary. We discuss the construction of these models, and we emphasize the role played by Brownian motion indexed by the Brownian tree.
We prove that uniform random quadrangulations of the sphere with $n$ faces, endowed with the usual graph distance and renormalized by $n^{-1/4}$, converge as $n\to\infty$ in distribution for the Gromov-Hausdorff topology to a limiting metric space. We validate a conjecture by Le Gall, by showing that the limit is (up to a scale constant) the so-called {\em Brownian map}, which was introduced by Marckert & Mokkadem and Le Gall as the most natural candidate for the scaling limit of many models of random plane maps. The proof relies strongly on the concept of {\em geodesic stars} in the map, whic…
We study the convergence in rough path topology of a certain class of discrete processes, the hidden Markov walks, to a Brownian motion with an area anomaly. This area anomaly, which is a new object, keeps track of the time-correlation of the discrete models and brings into light the question of embeddings of discrete processes into continuous time. We also identify an underlying combinatorial structure in the hidden Markov walks, which turns out to be a generalization of the occupation time from the classical ergodic theorem in the spirit of rough paths.
Stochastic networks based on random point sets as nodes have attracted considerable interest in many applications, particularly in communication networks, including wireless sensor networks, peer-to-peer networks and so on. The study of such networks generally requires the nodes to be independently and uniformly distributed as a Poisson point process. In this work, we venture beyond this standard paradigm and investigate the stochastic geometry of networks obtained from \textit{directed spanning forests} (DSF) based on randomly perturbed lattices, which have desirable statistical properties as…
We provide a decomposition of the trace of the Brownian motion into a simple path and an independent Brownian soup of loops that intersect the simple path. More precisely, we prove that any subsequential scaling limit of the loop erased random walk is a simple path (a new result in three dimensions), which can be taken as the simple path of the decomposition. In three dimensions, we also prove that the Hausdorff dimension of any such subsequential scaling limit lies in $(1,\frac53]$. We conjecture that our decomposition characterizes uniquely the law of the simple path. If so, our results woul…
We study non-compact scaling limits of uniform random planar quadrangulations with a boundary when their size tends to infinity. Depending on the asymptotic behavior of the boundary size and the choice of the scaling factor, we observe different limiting metric spaces. Among well-known objects like the Brownian plane or the infinite continuum random tree, we construct two new one-parameter families of metric spaces that appear as scaling limits: the Brownian half-plane with skewness parameter $θ$ and the infinite-volume Brownian disk of perimeter $σ$. We also obtain various coupling and limit …
The Brownian excursion measure is a conformally invariant infinite measure on curves. It figured prominently in one of the first major applications of SLE, namely the explicit calculations of the planar Brownian intersection exponents from which the Hausdorff dimension of the frontier of the Brownian path could be computed (Lawler, Schramm, and Werner, 2001). In this paper we define the simple random walk excursion measure and show that for any bounded, simply connected Jordan domain D, the simple random walk excursion measure on D converges in the scaling limit to the Brownian excursion measu…
The Brownian web (BW), which developed from the work of Arratia and then Tóth and Werner, is a random collection of paths (with specified starting points) in one plus one dimensional space-time that arises as the scaling limit of the discrete web (DW) of coalescing simple random walks. Two recently introduced extensions of the BW, the Brownian net (BN) constructed by Sun and Swart, and the dynamical Brownian web (DyBW) proposed by Howitt and Warren, are (or should be) scaling limits of corresponding discrete extensions of the DW -- the discrete net (DN) and the dynamical discrete web (DyDW). T…
We consider the model of the Brownian plane, which is a pointed non-compact random metric space with the topology of the complex plane. The Brownian plane can be obtained as the scaling limit in distribution of the uniform infinite planar triangulation or the uniform infinite planar quadrangulation and is conjectured to be the universal scaling limit of many others random planar lattices. We establish sharp bounds on the probability of having a short cycle separating the ball of radius $r$ centered at the distinguished point from infinity. Then we prove a strong version of the spatial Markov p…
We prove that, after suitable rescaling, the simple random walk on the trace of a large critical branching random walk converges to the Brownian motion on the integrated super-Brownian excursion.
Consider a family of random ordered graph trees $(T_n)_{n\geq 1}$, where $T_n$ has $n$ vertices. It has previously been established that if the associated search-depth processes converge to the normalised Brownian excursion when rescaled appropriately as $n\rightarrow\infty$, then the simple random walks on the graph trees have the Brownian motion on the Brownian continuum random tree as their scaling limit. Here, this result is extended to demonstrate the existence of a diffusion scaling limit whenever the volume measure on the limiting real tree is nonatomic, supported on the leaves of the l…
National University of Singapore（新加坡国立大学）；The Czech Academy of Sciences, Institute of Information Theory and Automation（捷克科学院信息与自动化研究所）；East China Normal University（华东师范大学）
It is well known (Donsker's Invariance Principle) that the random walk converges to Brownian motion by scaling. In this paper, we will prove that the scaled local time of the $(1,L)-$random walk converges to that of the Brownian motion. The results was proved by Rogers (1984) in the case $L=1$. Our proof is based on the intrinsic multiple branching structure within the $(1,L)-$random walk revealed by Hong and Wang (2013).
We obtain the Brownian net of Sun and Swart (2008) as the scaling limit of the paths traced out by a system of continuous (one-dimensional) space and time branching and coalescing random walks. This demonstrates a certain universality of the net, which we have not seen explored elsewhere. The walks themselves arise in a natural way as the ancestral lineages relating individuals in a sample from a biological population evolving according to the spatial Lambda-Fleming-Viot process. Our scaling reveals the effect, in dimension one, of spatial structure on the spread of a selectively advantageous …
