The minimum gap on diluted Cayley trees

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Abstract
A new order parameter for percolative systems, the minimum gap (x*), is calculated on diluted Cayley trees. x* is self-averaging and finite for concentrations p below pc and zero above. Numerical work (including finite-size scaling) and analytic arguments show that on approaching pc from below, x* approximately epsilon /(ln(1/ epsilon )+1+ln( alpha -1)), where alpha +1 is the coordination number of the Cayley tree and epsilon =(pc-p)/pc. The behaviour of x* as a function of p (for all pc) is calculated in terms of the solution to a transcendental equation, and as p to 0, x* approximately 1+ln alpha /lnp.

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