Hammersley ’ s Path Process Steven Finch

Hammersley ’ s Path Process

Hammersley ’ s Path Process 2

where each ( ) ∈ , 1 ≤  ≤ , and  is arbitrary. For convenience, define (0 0) = (0 0) and (+1 +1) = (1 1). Define such a point sequence to be an up/right path if, for any  ≥ 1, we have −1 ≤  and −1 ≤ . Hence an up/right path joins points of  in a continuous, piecewise linear manner with line segments of slope , 0 ≤  ≤ ∞, attaching (−1 −1) and ( ) for al...

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...

Nash ’ s Inequality Steven Finch

Shortest Path through Random Points

Let (M, g1) be a compact d-dimensional Riemannian manifold. Let Xn be a set of n sample points in M drawn randomly from a Lebesgue density f . Define two points x, y ∈ M . We prove that the normalized length of the power-weighted shortest path between x, y passing through Xn converges to the Riemannian distance between x, y under the metric gp = f g1, where p > 1 is the power parameter. The res...