Scaling Limit for a Drainage Network Model

Coletti, Cristian F.; Fontes, Luiz Renato; Dias, Elaine Santos

doi:10.1239/jap/1261670696

Cited by 21 publications

(56 citation statements)

References 13 publications

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“…We should mention here that the Brownian web appears as a universal scaling limit for various network models (see Fontes et al [9], Ferrari, Fontes and Wu [8], Coletti, Fontes and Dias [5]). It is reasonable to expect that with suitable modifications our method will give similar results in other network models.…”

mentioning

confidence: 97%

Hack’s law in a drainage network model: A Brownian web approach

Roy¹,

Saha²,

Sarkar

2016

Ann. Appl. Probab.

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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 ∼ 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 (2004Probab. 14 ( ) 1242Probab. 14 ( -1266. We show that the exponent of Hack's law is 2/3 for Howard's model. Our study is based on a scaling of the process whereby the limit of the watershed area of a stream is area of a Brownian excursion process. To obtain this, we define a dual of the model and show that under diffusive scaling, both the original network and its dual converge jointly to the standard Brownian web and its dual.

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mentioning

confidence: 97%

Hack’s law in a drainage network model: A Brownian web approach

Roy¹,

Saha²,

Sarkar

2016

Ann. Appl. Probab.

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“…However, FKG property is difficult to verify for many models where paths have complex interactions, e.g., Howard's model [14], discrete directed spanning forest [20], directed spanning forest [2,8] etc. Among these, for Howard's model, only a partial version of the FKG property is proved till now [5]. The main contribution of this paper is that, we propose an alternate method for verification of (B2) based on a bound for expected first collision time and show that this method is applicable for many non-crossing path models with certain Markovian properties and homogeneity assumptions (see Proposition 3.2 for detail).…”

Section: Main Result: An Alternative Verification Of (B2)mentioning

confidence: 99%

“…Let X be the collection of all paths obtained by following the edges and for γ, σ > 0, let X n = X n (γ, σ) be the collection of scaled paths with normalization constants γ, σ. Coletti et al [5,6] proved that, for γ = 1 and σ = σ 0 (p) with σ 0 (p) as given by σ 0 (p) = (1−p)(2−2p+p 2 ) p 2 (2−p) 2 1/2 , X n converges in distribution to the Brownian web. Again, to verify Condition (B2) they proved a partial FKG inequality for the Howard's model.…”

Section: Howard's Modelmentioning

confidence: 99%

Collision Times of Random Walks and Applications to the Brownian Web

Coupier

Saha

Sarkar

et al. 2020

Genealogies of Interacting Particle Systems

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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 ε, is of small order of ε. This condition is often verified by applying an FKG type correlation inequality together with a coalescing time tail estimate for two paths. For many models where paths have complex interactions, it is hard to establish FKG type inequalities. In this article, we show that for a non-crossing path model, with certain assumptions, a suitable upper bound on expected first collision time among three paths can be obtained directly using Lyapunov functions. This, in turn, provides an alternate verification of Condition (B2). We further show that in case of independent simple symmetric one dimensional random walks or in case of independent Brownian motions, the expected value can be computed explicitly. We apply this alternate method of verification of (B2) to several models in the basin of attraction of the Brownian web studied earlier in the literature ([22], [14], [11]).

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“…With the ingredients developed above, we can now follow ideas developed in [11] for instance. Recall that τ R− = inf{k ∈ N, D k ≤ 0} and define θ = inf{t ≥ 0, B t = −d}.…”

Section: Step 4: Introduction Of An Auxiliary Markov Chainmentioning

confidence: 99%

Directed, cylindric and radial Brownian webs

Coupier¹,

Marckert²,

Tran³

2019

Electron. J. Probab.

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The Brownian web (BW) is a collection of coalescing Brownian paths (W (x,t) , (x, t) ∈ R 2 ) 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 is locally similar to the planar BW but has several important macroscopic differences. For example, in the CBW, the coalescence time between two paths admits exponential moments and the CBW as its dual contain each a.s. a unique bi-infinite path. This pair of bi-infinite paths is distributed as a pair of reflected Brownian motions on the cylinder. Projecting the CBW on the radial plane, we obtain a radial Brownian web (RBW), i.e. a family of coalescing paths where under a natural parametrization, the angular coordinate of a trajectory is a Brownian motion. Recasting some of the discrete radial forests of the literature on the cylinder, we propose rescalings of these forests that converge to the CBW, and deduce the global convergence of the corresponding rescaled radial forests to the RBW. In particular, a modification of the radial model proposed in Coletti and Valencia is shown to converge to the CBW.

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Scaling Limit for a Drainage Network Model

Cited by 21 publications

References 13 publications

Hack’s law in a drainage network model: A Brownian web approach

Hack’s law in a drainage network model: A Brownian web approach

Collision Times of Random Walks and Applications to the Brownian Web

Directed, cylindric and radial Brownian webs

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