For random Cayley graphs on groups of order N_k, whp diameter ≤ d when p ≥ [(1+ε) d! log N_k / N_k^{d-1}]^{1/d} and diameter > d when p ≤ [(1-ε) 2^{-d} log N_k / N_k^{d-1}]^{1/d} for d up to roughly sqrt(log N / log log N).
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Diameter Thresholds of Random Cayley Graphs
For random Cayley graphs on groups of order N_k, whp diameter ≤ d when p ≥ [(1+ε) d! log N_k / N_k^{d-1}]^{1/d} and diameter > d when p ≤ [(1-ε) 2^{-d} log N_k / N_k^{d-1}]^{1/d} for d up to roughly sqrt(log N / log log N).