Quantitative estimates of a drainage network model Rahul Roy Indian - - PowerPoint PPT Presentation

Quantitative estimates of a drainage network model Rahul Roy Indian Statistical Institute, New Delhi joint work with Kumarjit Saha and Anish Sarkar.

Hurst exponent

A river basin network is an anisotropic system defined by a longitudinal length L∥ = L and a typical perpendicular length L⊥ ≤ L. The history of the Hurst exponent in fractional Brownian motion comes from when Harold Hurst studying the Nile river postulated the scaling L⊥ = LH, 0 ≤ H ≤ 1. The basin is said to be self affine if H < 1 and self-similar if H = 1 1.

1For Nile, Hurst obtained H = 0.90.

Hack’s law

The Hurst exponent can be viewed as a measure of the meandering

• f the river and the tributaries it gathers in the process.

Hack’s law studies the length, l, of the longest stream vis-` a-vis the drainage area, a, of the basin, i.e. the area of the land which collects the precipitation contributing to the network. John Hack observed that l ∼ ah where h is the Hack exponent 2.

2For the Shenandoah valley in Virginia, Hack calculated h = 0.571.

Fella river basin

Figure : Fella river network in Northern Italy.

From Maritan et al, Phy. Rev. E 1996

River network

‘A river network is a spanning tree defined in a lattice of arbitrary size and shape3.’ We study the Howard’s model of ‘headward growth and branching in a random fashion’ 4. There are various river network models proposed by geologists. We first discuss a few of them. Rodriguez-Iturbe and Rinaldo [1997] (Fractal River Basins) presents a survey of the development of this field.

3Maritan et al, Phy. Rev. E 1996 4Rodriguez-Iturbe and Rinaldo (1997)

Scheidegger Model

Scheidegger (1967) introduced a drainage model consisting of a two dimensional grid where each square, representing a unit area, is to be drained, to one of its two neighbours in a preferred direction. Here, the flow can happen along the chosen direction. The preferred direction is assumed to be direction of the slope. Thus, the drainage network forms an oriented network.

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Direction

Each vertex chooses one of the two downward edges with equal probability. ✉

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The Graph

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Observations

From a vertex there is exactly one edge going down. So no loops are possible. The graph is either

▶ a connected infinite tree ▶ each component a connected infinite tree

Questions :

▶ How many trees are there? ▶ Are they bi-infinite in both the direction?

Howard’s Model

Howard [1971] removed the restriction of drainage to a neighbouring square and modelled a network to include “headward growth and branching in a random fashion”. This was a model for a region where the gradient is very high and not all points are sources. The flow happens between the sources.

Howard’s Model - Source (black) Points

Declare points open with probability 0 < p < 1, independently of each other. Open points act as the sources. ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉

Every open point connects to the closest open point in the next level. If there is a choice of points, one among the choices, is chosen with equal probability. Note, unlike the Scheidegger Model, the edges are not connecting nearest neighbours.

✒✑ ✓✏ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ❄ ❄ ❄ ❄ ❄ ❅ ❅ ❅ ❅ ❅ ❘ ❄

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has a choice

Each source has exactly one child, but some may not have any ancestor. ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✒✑ ✓✏ ✉

has no ancestors

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Directed Spanning Forest (DSF)

Baccelli and Bordenave (2007) introduced a model which they called directed spanning forest. DSF appears as an essential tool for the asymptotic analysis of the Radial Spanning Tree. We start with a homogeneous Poission point process of unit intensity

• n R2. Each vertex in the point process connects to the nearest point
• f the process having a strictly smaller y co-ordinate.

Geometry of the network

All these models have two common geometric properties: (i) For d = 1 and d = 2, the graph consists of a single tree almost surely and, for d ≥ 3, the graph has infinitely many disjoint trees. (ii) There is no bi-infinite path almost surely for any d ≥ 1.

Howard’s Model: d = 1

We study Howard’s model for d = 1 only.

Notation

For an open vertex (u, t) let h(u, t) denote the open vertex on the line {y = t + 1} such that < (u, t), h(u, t) > is an edge of the random graph. Thinking of h(u, t) as the progeny of (u, t), hk(u, t) := h(hk−1(u, t) is the k th generation progeny of (u, t) on the line {y = t + k}. For an open vertex (v, s) set Ck(v, s) := {(u, s − k) ∈ V : hk(u, s − k) = (v, s)}, the k-th generation ancestors of (v, v), and, C(v, s) := ∪k≥0Ck(v, s). the set of all ancestors of (v, s).

Notation

In the terminology of drainage networks, |C(v, s)| represents the amount of water that is drained through the points (v, s). We define, H(v, s) := inf{k ≥ 1 : Ck(v, s) = ∅}, the height to the mouth of the river. Since there are no bi-infinite paths, H(v, s) < ∞ almost surely.

Notation

On the event, {H(x, t) > n}, define, for 0 ≤ k ≤ n, Rk(x, t) = max{y : (y, t − k) ∈ Ck(x, t)} and Lk(x, t) = min{y : (y, t − k) ∈ Ck(x, t)}. Clearly Rk(x, t) ≥ Lk(x, t). Set Dk = (Dk(x, t) =) Rk(x, t) − Lk(x, t). Define, a process in C[0, 1] as follows: for s ∈ [0, 1], W(x,t)

n

(s) = D[ns] + (ns − [ns])(D[ns]+1 − D[ns]) √n .

Main results: 1

Theorem 1. Exact tail decay for cluster height lim

n→∞

√nP(H(0, 0) > n) = 1 σ√π where σ(p) is a positive constant, depending on p only. Theorem 2. Scaling limit of conditional cluster width process 1 √ 2σ W(0,0)

n

|{H(0, 0) > n} ⇒ B+ where B+ is the standard Brownian meander process.

Main results: 2

Corollary 1. For u > 0, we have √nP(|Cn(0, 0)| > uσ √ 2n) → exp(−u2/2p2) σ√π where |A| denotes the cardinality of the set A. Corollary 2. For u > 0, we have P(

n

∑

k=0

|Ck(0, 0)| ≤ uσ √ 2n3|H(0, 0) > n) → P(pI ≤ u) where I = ∫ 1

0 B+(t)dt.

Brownian meander

Let {Bt : t ≥ 0} be a standard 1-dimensional Brownian motion starting from 0 and let τ := sup{t ∈ [0, 1] : Bt = 0}, the Brownian meander5 is the process B+

t

:= 1 1 − τ |Bτ+t(1−τ)|, t ∈ [0, 1].

5Durett, Iglehart, Miller (1977)

The transition density is given by: For 0 < s < t ≤ 1 and x, y > 0 ps,t(x, y)dy := P(B+

t

∈ dy|B+

s = x)

= (φt−s(y − x) + φt−s(y + x)) Φ1−t(y) − 1/2 Φ1−s(y) − 1/2, where φt and Φt are, respectively, the density and distribution functions of a N(0, t) random variable. In particular B+

1 has a Rayleigh distribution, i.e.

density f(y) = ye−y2/2, y > 0.

The Brownian Web

Intuitively, the Brownian web is a collection of 1-dimensional coalescing Brownian motions starting from every point (x, t) in the space-time plane R2 Technical difficulty: there are an uncountable number of coalescing Brownian motions! It arises naturally as the diffusive scaling limit of a system of 1-dimensional coalescing random walks starting from every vertex of the space-time lattice Zd. Arratia, 1978, 1981, T´

• th & Werner, 1998, Fontes, Isopi, Newman &

Ravishankar 2004

B1 B2 B3 B4

Formally, the Brownian web is a random variable taking values in the space H of compact sets of paths equipped with the Hausdorff metric

• dH. The space (H, dH) is also a complete separable metric space.

Let BH be the Borel σ-algebra associated with dH. The Brownian web W is an (H, BH)-valued random variable.

Theorem

There exists an (H, BH)-valued random variable W (the Brownian web) whose distribution is uniquely determined by the following: (a) For each deterministic z ∈ R2, almost surely there is a unique path πz ∈ W(z). (b) For any finite deterministic set of points z1, . . . , zk ∈ R2, the collection (πz1, . . . , πzk) is distributed as coalescing Brownian motions. (c) For any deterministic countable dense subset D ⊂ R2, almost surely, W is the closure of {πz : z ∈ D} in (Π, d). Fontes, Isopi, Newman & Ravishankar 2004

This theorem shows that the Brownian web is in some sense separable: even though there are uncountably many coalescing Brownian motions in the Brownian web, the whole collection is a.s. determined uniquely by a countable skeletal subset of paths. Under this set-up we can apply the standard theory of weak convergence to prove the convergence of various 1-dimensional coalescing systems to the Brownian web.

Howard’s model – scaled

For each (x, t) ∈ V, i.e. (x, t) an open point, let π(x,t) := {∪k≥0 < hk((x, t), hk+1(x, t) >}, i.e. the open path starting at (x, t) in the random graph, and, for n ≥ 1 and σ > 0, χ = {π(x,t) : (x, t) ∈ V} χn(σ) = {( u(1) σ√n, u(2) n ) : (u(1), u(2)) ∈ χ } χn(σ) is the set of all diffusively scaled paths from the random graph. Colletti, Dias and Fontes [2009] showed that in (H, BH) ¯ χn(σ) ⇒ W for an appropriate σ.

Duality

▶ Dual graph on 1 2Z × Z also consists of a single tree almost surely. ▶ There is no bi-infinite path almost surely.

Denote by ˆ V the set of dual vertices. Define, for the dual process ˆ π(ˆ

x, ˆ t) – the path formed by the dual edges,

ˆ χ(σ) = {ˆ π(ˆ

x, ˆ t) : (ˆ

x,ˆ t) ∈ ˆ V} and ˆ χn(σ) = {( u(1) σ√n, u(2) n ) : (u(1), u(2)) ∈ ˆ χ } . Theorem 3. (¯ χn(σ), ¯ ˆ χn(σ)) ⇒ (W, ˆ W) for an appropriate σ.

Proof of theorem 1

To show lim

n→∞

√nP(H(0, 0) > n) = 1 σ√π We start with a (H, BH) valued random variable K and for t > 0, t0, a, b ∈ R with a < b define ηK(t0, t, a, b) := |{π(t0 + t) : π ∈ K, σπ ≤ t0 and π(t0 + t) ∈ [a, b]}|, where σπ is the y-co-ordinate of the starting point of π.

(1,1) (0,1) η(0, 1, 0, 1) = 3

E(ηχn(0, 1, 0, 1)) = E ([√nσ] ∑

k=0

1[

H(k,n)>n

]) =([√nσ] + 1)E ( 1[

H(0,0)>n

]) from translation invariance =([√nσ] + 1)P(H(0, 0) > n). Can prove:

▶ ηχn(0, 1, 0, 1) ⇒ ηW(0, 1, 0, 1) (need Reimer’s inequality). ▶ The family {ηχn(0, 1, 0, 1) : n ∈ N} is uniformly integrable.

Conclusion: as n → ∞, E(ηχn(0, 1, 0, 1)) → E(ηW(0, 1, 0, 1)).

By comparing with the limit of the nearest neighbour random walk we may show E(ηW(0, 1, 0, 1)) = 1 √π to give lim

n→∞

√nP(H(0, 0) > n) = 1 σ√π

Nearest neighbour random walk

Denote the scaled nearest neighbour paths by ∆n. Same arguments show that, as n → ∞ E(η∆n(0, 1, 0, 1)) → E(ηW(0, 1, 0, 1)). Now, we have E(η∆n(0, 1, 0, 1)) = E ([√n/2] ∑

k=0

1[

H(2k,n)>n

]) =([√n/2] + 1)E ( 1[

H(0,0)>n

]) = ([√n/2] + 1)P(H(0, 0) > n) =([√n/2] + 1)P(2 simple symmetric random walks starting at 0 and 2 do not meet until time n) → 1 √π .

Brownian meander

We show two things:

▶ Convergence of the finite dimensional distributions. ▶ Tightness of the family.

We need to study a quantity similar to ηW(0, 1, 0, 1). For any (H, BH) valued random variable K and for 0 < t1 < t2 < . . . < tk ≤ 1 and for a1, a2, . . . , ak ∈ R define ηK(0, 1, 0, 1, t1, a1 . . . , tk, ak) = |{(x, 1) : x ∈ [0, 1], there exists π ∈ K0 with σπ ≤ 0π(1) = x, for each i = 1, . . . , k there exists π1

i , π2 i ∈ Kti

with |π1

i (ti) − π2 i (ti)| ≥ ai, π1 i (1) = π2 i (1) = x}|.

(0,1) (1,1)

a1 a2 a3 t1 t2 t3 t3

References [GRS04] S. Gangopadhyay, R. Roy, and A. Sarkar (2004) Random

• riented Trees: a Model of drainage networks. Ann. Appl.
• Probab. 14, 1242 -1266.

[FINR04] L.R.G. Fontes, M. Isopi, C.M. Newman and K. Ravishankar (2004) The Brownian web: characterization and convergence.

• Ann. Probab. 32, 2857 - 2883.

[CFD09] C.F . Coletti, L.R.G. Fontes, and E.S. Dias (2009) Scaling limit for a drainage network model. J. Appl. Probab. 46, 1184 - 1197. [Bol76] E. Bolthausen (1976) On a functional central limit theorem for random walks conditioned to stay positive. Ann. Probab. 4, 480 - 485. [R00] D. Reimer (2000) Proof of the van den Berg-Kesten conjecture.

• Combin. Probab. Comput. 9, 27 - 32.