I am not a mathematician but I want to get some familiarity with percolation theory for an application to my job. I am reading this text, where it is implied on page 5 that the percolation threshold is not well-defined for finite lattices.
For a 2D lattice I can imagine the percolation threshold (i.e. density where the chance you connect top to bottom of your grid becomes 1) equals the total number of points on your grid minus one row divided by the total number of points on your grid image? Adding one more point then has to connect the two halves and the row is the minimal distance to separate the two halves.
I hope someone can help me understand this better, thank you.
Best regards Jan
Percolation theory is mainly concerned with typical phenomena of percolation processes. Because for finite graphs the expected size of percolated clusters is a continuous function of the occupation probability. Discontinuity emerges only when the system size becomes infinite. In literature about percolation processes, 'percolation threshold' is used to refer to the the critical probability $p_\textrm{c}$. Above $p_\textrm{c}$ there is percolation whereas below $p_\textrm{c}$ there is no percolation (where 'no percolation' means the the probability of such event is zero).
As for your example, the argument similar in spirit could be used to estimate the upper bound of percolation threshold. But typically a subgraph produced by a percolation process (e.g., bond/site percolation) would be far different from your example.