We consider instances of long-range percolation on Z and R , where points at distance r get connected by an edge with probability proportional to r −s , for s ∈ ( , 2 ), and study the asymptotic of the graph-theoretical (a.k.a. chemical) distance D(x, y) between x and y in the limit as |x − y| → ∞. For the model on Z we show that, in probability as |x| → ∞, the distance D(0, x) is squeezed between two positive multiples of (log r) Δ , where Δ ∶= 1∕ log 2 (1∕ ) for ∶= s∕(2 ). For the model on R we show that D(0, xr) is, in probability as r → ∞ for any nonzero x ∈ R , asymptotic to (r)(log r) Δ for a positive, continuous (deterministic) function obeying (r ) = (r) for all r > 1. The proof of the asymptotic scaling is based on a subadditive argument along a continuum of doubly-exponential sequences of scales. The results strengthen considerably the conclusions obtained earlier by the first author. Still, significant open questions remain.

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We present results concerning when the joint distribution of an exchangeable sequence is determined by the marginal distributions of its partial sums. The question of whether or not this determination occurs was posed by David Aldous. We then consider related uniqueness problems, including a continuous time analog to the Aldous problem and a randomized univariate moment problem.