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Zero-One Law examples

  1. Feb 12, 2014 #1
    There are various zero-one laws (e.g. Kolmogorov's) that assure us that certain measurable events have probability zero or one in a given context.

    Does anybody know any good examples of events (preferably "naturally" occurring elsewhere in math) which are covered by such a theorem, but for which it's currently an open question whether the probability is zero or one?
     
  2. jcsd
  3. Feb 13, 2014 #2
    http://mathoverflow.net/questions/2...e-law-gives-probability-0-or-1-but-hard-to-de

    A simple example: consider the 3-dimensional cubic grid, and connect each point to the 6 adjacent ones by edges. Then delete each edge with probability p.

    Define a "cluster" to be a connected component of the resulting graph.

    Let f(p) be the probability that there exists an infinite cluster. By the zero-one law, f(p) is either 0 or 1, since the existence of an infinite cluster does not depend on the edges in any finite box. Also, it should be clear that increasing p cannot increase the probability of an infinite cluster, so that f is nonincreasing. Therefore, the only question is at which critical probability f switches from 1 to 0. The exact value is unknown (though there are good estimates).
     
  4. Sep 27, 2015 #3
    Any others?
     
  5. Oct 10, 2015 #4
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