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Suppose we remove all closed edges, and consider the remaining open subgraph of the lattice. Percolation theory is concerned with the geometry of this open graph. Of particular interest are such quantites as the size of the open cluster C x containing a given vertex x, and particularly the probability that C x is infinite. The sample space is the set = {0, 1}E of 0/1-vectors ω indexed by the edgeset; here, 1 represents ‘open’, and 0 ‘closed’. As σ -field we take that generated by the finite-dimensional cylinder sets, and the relevant probability measure is product measure P p with density p.

Suppose we remove all closed edges, and consider the remaining open subgraph of the lattice. Percolation theory is concerned with the geometry of this open graph. Of particular interest are such quantites as the size of the open cluster C x containing a given vertex x, and particularly the probability that C x is infinite. The sample space is the set = {0, 1}E of 0/1-vectors ω indexed by the edgeset; here, 1 represents ‘open’, and 0 ‘closed’. As σ -field we take that generated by the finite-dimensional cylinder sets, and the relevant probability measure is product measure P p with density p.

The law of (C, σ ) is simply the probability that the coloured moves are given appropriately. That is, P (C, σ ) = (c, A) = c∈c e∈c pe− ,e+ α( A), c ∈ C, A ∈ r. Since this factorizes in the form f (c)g( A), the random variables C and σ are independent, and P(σ = A) is proportional to α( A) as required. 3 Weak limits on lattices Let Ld = (Zd , Ed ) be the d-dimensional hypercubic lattice, with d ≥ 2. Let µn be the UST measure on the box B(n) = [−n, n]d . 10) Theorem [170]. The weak limit µ = lim n→∞ µn exists and is a translationinvariant and ergodic3 probability measure.

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3-Quasiperiodic functions on graphs and hypergraphs by Rudenskaya O. G.


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