University of Groningen
Topics in inhomogeneous Bernoulli percolation
Carelos Sanna, Humberto
DOI:
10.33612/diss.150687857
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Publication date: 2020
Link to publication in University of Groningen/UMCG research database
Citation for published version (APA):
Carelos Sanna, H. (2020). Topics in inhomogeneous Bernoulli percolation: A study of two models. University of Groningen. https://doi.org/10.33612/diss.150687857
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Summary
This thesis is an investigation of some mathematical aspects of inhomogeneous Ber-noulli bond percolation in two different graphs G = (V, E). In each of them, we consider a decomposition E0∪E00of the relevant edge set E and, given𝑝, 𝑞 ∈ [0, 1], we
assign parameters𝑝 and 𝑞 to the edges of E0and E00, respectively. In such formulation,
one of the sets, say E00, is regarded as the set of inhomogeneities.
The first graph G = (V, E) we consider is the one induced by the cartesian product of an infinite and connected graph 𝐺 = (𝑉 , 𝐸) and the set of integers Z. We choose an infinite collection C of finite connected subgraphs of 𝐺 and consider the Bernoulli bond percolation model on G which assigns probability 𝑞 of being open to each edge whose projection onto 𝐺 lies in some subgraph of C and probability 𝑝 to every other edge. Here, we focus our attention on the critical percolation threshold, 𝑝𝑐(𝑞 ), defined
as the supremum of the values of 𝑝 for which percolation with parameters 𝑝, 𝑞 does not occur. We show that the function 𝑞 ↦→ 𝑝𝑐(𝑞 )is continuous in (0, 1), provided that
the graphs in C are “suffciciently spaced from each other” on 𝐺 and their vertex sets have uniformly bounded cardinality.
The second graph is the ordinary 𝑑-dimensional hypercubic lattice, L𝑑
= (Z𝑑
,E𝑑), 𝑑 ≥3, where we define the inhomogeneous Bernoulli percolation model in which every edge inside the 𝑠-dimensional subspace Z𝑠
× {0}𝑑−𝑠, 2 ≤ 𝑠 < 𝑑, is open with probability 𝑞 and every other edge is open with probability 𝑝. Defining 𝑞𝑐(𝑝)as the
supremum of the values of 𝑞 for which percolation with parameters𝑝, 𝑞 does not occur and letting 𝑝𝑐 ∈ (0, 1) be the threshold for homogeneous percolation on L
𝑑
, we prove two results: first, the uniqueness of the infinite cluster in the supercritical phase of parameters (𝑝, 𝑞), whenever 𝑝 ≠ 𝑝𝑐; second, we show that, for any 𝑝 < 𝑝𝑐, the critical
point (𝑝, 𝑞𝑐(𝑝))can be approximated by the critical points of slabs, in the spirit of the
classical theorem of Grimmett and Marstrand for homogeneous percolation.
