Hammersley ’ s Path Process

Hammersley ’ s Path Process Steven Finch

Beta-paths in the Hammersley Process

We study the asymptotics of β-paths in the Hammersley process with sources and sinks, of intensities λ and ρ respectively, introduced by Groeneboom (2002). We derive a strong law of large number for those paths in the regime λρ ≤ 1 and we show that its fluctuation exponent is at most 2/3. Examples of β-paths are the space-time path of a second-class particle in the Hammersley process and also t...

Beardwood–halton–hammersley Theorem for Stationary Ergodic Sequences: a Counterexample by Alessandro Arlotto

We construct a stationary ergodic process X1,X2, . . . such that each Xt has the uniform distribution on the unit square and the length Ln of the shortest path through the points X1,X2, . . . ,Xn is not asymptotic to a constant times the square root of n. In other words, we show that the Beardwood, Halton, and Hammersley [Proc. Cambridge Philos. Soc. 55 (1959) 299–327] theorem does not extend f...

Beardwood-halton-hammersley Theorem for Stationary Ergodic Sequences: Construction of a Counterexample

We construct a stationary ergodic process {X1, X2, X3 . . .} such that each Xt, 1 ≤ t <∞, has the uniform distribution on the unit square and the length Ln of the shortest path through {X1, X2, . . . , Xn} is not asymptotic to a constant times the square root of n. In other words, we show that the Beardwood, Halton and Hammersley theorem does not extend from the case of independent uniformly di...

Beardwood-halton-hammersley Theorem for Stationary Ergodic Sequences: a Counterexample

We construct a stationary ergodic process X1, X2, . . . such that each Xt has the uniform distribution on the unit square and the length Ln of the shortest path through the points X1, X2, . . . , Xn is not asymptotic to a constant times the square root of n. In other words, we show that the Beardwood, Halton and Hammersley theorem does not extend from the case of independent uniformly distribut...