1.1 Main Results
Before we describe the model we fix some notation which describe special regions in Z
d
. For u =
u
1
, . . . , u
d
∈ Z
d
and k ≥ 1 let m
k
u = u
1
, . . . , u
d−1
, u
d
− k. Also, for k, h ≥ 1 define the regions
H
u, k = {v ∈ Z
d
: v
d
= u
d
− k and ||v − m
k
u||
L
1
≤ k},
Λu, h = {v : v ∈ Hu, k for some 1 ≤ k ≤ h}, Λu = ∪
∞ h=1
Λu, h and
B u, h = {v : v ∈ Hu, k and ||v − m
k
u||
L
1
= k for some 1 ≤ k ≤ h} . We set H
u, 0 = Λu, 0 = ;. u
Figure 3: The region Λ u, 3. The seven vertices at the bottom constitute Hu, 3 while the six
vertices on the two linear ‘boundary’ segments containing u constitute Bu, 3
We equip Ω = {0, 1}
Z
d
with the σ-algebra F generated by finite-dimensional cylinder sets and a
product probability measure P
p
defined through its marginals as P
p
{ω : ωu = 1} = 1 − P
p
{ω : ωu = 0} = p for u ∈ Z
d
and 0 ≤ p ≤ 1. 1
On another probability space Ξ, S , µ we accommodate the collection {U
u,v
: v ∈ Λu, u ∈ Z
d
} of i.i.d. uniform 0, 1 random variables. The random graph, defined on the product space Ω × Ξ, F ×
S , P := P
p
× µ, is given by the vertex set
V := V ω, ξ = {u ∈ Z
d
: ωu = 1} for ω, ξ ∈ Ω × Ξ,
and the almost surely unique edge set E =
n
u, v : u, v ∈ V , and for some h ≥ 1, v ∈ Λu, h, Λu, h − 1 ∩ V = ; and U
u,v
≤ U
u,w
for all w ∈ Λu, h ∩ V
o .
2 The graph G = V , E is the object of our study here. The construction of the edge-set ensures that,
almost surely, there is exactly one edge going ‘down’ and, as such, each connected component of the graph is a tree.
Our first result discusses the structure of the graph and the second result discusses the structure of each connected component of the graph.
Theorem 1. For 0 p 1 we have, almost surely
2163
i for d = 2, 3, the graph G is almost surely connected and consists of a connected tree ii for d ≥ 4 the graph G is almost surely disconnected and consists of infinitely components each of
which is a tree. While the model guarantees that no river source terminates in the downward direction, this is not
the case in the upward direction. This is our next result.
Theorem 2. For d ≥ 2, the graph G contains no bi-infinite path almost surely.
Our specific choice of ‘right-angled’ cones is not important for the results. Thus if, for some 1 a
∞ we had Λ
a
u, h = ∪
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