A Note on Anisotropic Percolation

Truncated Connectivities in a Highly Supercritical Anisotropic Percolation Model

“…In spite of its apparent simplicity, this conjecture (without restrictions) remains open. A more detailed discussion about this conjecture and an affirmative answer when the parameters p h and p v are sufficiently small can be found in [3].…”

Monotonicity and phase diagram for multirange percolation on oriented trees

“…Note that S(A, e) ⊆S(A, e) ⊆ prog(e). Also note that, if e 1 , e 2 ∈ (A) are distinct, then R(A, e 1 ) and R(A, e 2 ) are disjoint, by (8). Finally, note that for every e ∈ (A), we have Π(A) ∩ prog(e) = Π(S(A, e)), so that (11) can be restated as…”

“…Indeed, assume e, f ∈ (A), e ≠ f , and e ′ = ⟨u ′ , v ′ ⟩, f ′ = ⟨w ′ , x ′ ⟩ are long edges with v ′ ∈ prog(e), x ′ ∈ prog(f ). Then, since (8) gives prog(e) ∩ prog(f ) = ∅, we obtain v ′ ≠ x ′ , so e ′ ≠ f ′ .…”

“…Proof To prove (8), assume that there are two distinct hubs e = ⟨u, v⟩, e ′ = ⟨u ′ , v ′ ⟩ ∈ (A) ∶ prog(e) ∩ prog(e ′ ) ≠ ∅.…”

“…In the same spirit, for unoriented percolation on , if the parameter is smaller for horizontal edges than for vertical ones, the above probability should be larger than the probability of the origin being connected to ( m + n , m − n ). This has only been proved under the assumption that the ratio between horizontal and vertical parameters is small enough .…”