page_title: Percolation threshold
section_title: Thresholds on hyperbolic, hierarchical, and tree lattices
hierarchical_section_title: Percolation threshold / Thresholds in different dimensional spaces / Thresholds on hyperbolic, hierarchical, and tree lattices
caption_attribution_description: English: triangular hyperbolic
context_page_description: The percolation threshold is a mathematical concept in percolation theory that describes the formation of long-range connectivity in random systems. Below the threshold a giant connected component does not exist; while above it, there exists a giant component of the order of system size. In engineering and coffee making, percolation represents the flow of fluids through porous media, but in the mathematics and physics worlds it generally refers to simplified lattice models of random systems or networks, and the nature of the connectivity in them. The percolation threshold is the critical value of the occupation probability p, or more generally a critical surface for a group of parameters p₁, p₂, ..., such that infinite connectivity first occurs.
context_section_description: In these lattices there may be two percolation thresholds: the lower threshold is the probability above which infinite clusters appear, and the upper is the probability above which there is a unique infinite cluster.
  Note: {m,n} is the Schläfli symbol, signifying a hyperbolic lattice in which n regular m-gons meet at every vertex
  For bond percolation on {P,Q}, we have by duality .  For site percolation,   because of the self-matching of triangulated lattices.
  Cayley tree (Bethe lattice) with coordination number z:   pc = 1 / (z − 1)
  Cayley tree with a distribution of z with mean , mean-square    pc= 
  (site or bond threshold)
caption: Visualization of a triangular hyperbolic lattice {3,7} projected on the Poincaré disk (red bonds).  Green bonds show dual-clusters on the {7,3} lattice[213]
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