A Remark on Monotonicity in Bernoulli Bond Percolation

Abstract

Consider an anisotropic independent bond percolation model on the d-dimensional hypercubic lattice, d 2, with parameter p. We show that the two point connectivity function P p ({(0, . . . , 0) ↔ (n, 0, . . . , 0)}) is a monotone function in n when the parameter p is close enough to 0. Analogously, we show that truncated connectivity function P p ({(0, . . . , 0) ↔ (n, 0, . . . , 0), (0, . . . , 0) ∞}) is also a monotone function in n when p is close to 1.

Monotonicity and phase diagram for multirange percolation on oriented trees

“…which contradicts (7) applied to e. Now fix an edge e = ⟨u, u ⋅ r⟩ as in (10). Consider the k short edges in the trace of e. By the first statement, we know that at most one of these short edges is in (A).…”

“…Statement (11) is an immediate consequence of (8) and (10). ▪ Again fix ∈ {0, 1} E ,k and A ⊆ V ,k satisfying (9).…”

“…A good example is the following: for Bernoulli bond percolation on the usual graph , prove that the probability of the origin being connected to ( n ,0,…,0) monotone in n . This problem is still open, except when the parameter is close to 0 or 1 . For oriented percolation on , the probability of the origin being connected to ( m − n , m + n ) is decreasing in n ∈ {0,…, m } for fixed m ; this may be obvious but the proof is not straightforward .…”

“…. , 0) monotone in n. This problem is still open, except when the parameter is close to 0 or 1 [LPS15]. For oriented percolation on Z 2 + , the probability of the origin being connected to (m − n, m + n) is decreasing in n ∈ {0, .…”

The randomly fluctuating hyperrectangles are spatially monotone

“…A partial result in the direction of a positive reply is obtained by de Lima et.al. [LPS15]. Whereas occupied site probabilities are monotone for general one-dimensional attractive spin-systems is also in the case of finite initial configurations an important open problem.…”