Search

Let

where Xi, 1 ≦ i ≦ n, are i.i.d. and uniformly distributed in [0, 1]2. It is proved that Mn ∽ cn1–p/2 a.s. for 1 ≦ p <2. This result is motivated by recent developments in the theory of algorithms and the theory of subadditive processes as well as by a well-known problem of H. Steinhaus.

A generalization of first-passage percolation theory proves that the fundamental convergence theorems hold provided only that the time coordinate distribution has a finite moment of a positive order. The existence of a time constant is proved by considering first-passage times between intervals of sites, rather than the usual point-to-point and point-to-line first-passage times. The basic limit theorems for the related stochastic processes follow easily by previous techniques. The time constant is evaluated as 0 when the atom at 0 of the time-coordinate distribution exceeds½.

The propagation rate of the one-dimensional stochastic simple epidemic converges almost surely to a front velocity for the epidemic. Percolation methods are used in the proof.

Search Results

Refine search

Refine search

Actions for selected content:

3 results

Steinhaus's geometric location problem for random samples in the plane

Weak moment conditions for time coordinates in first-passage percolation models

The front velocity of the simple epidemic

Search Results

Refine search

Refine search

Actions for selected content:

Save Search

3 results

Steinhaus's geometric location problem for random samples in the plane

Weak moment conditions for time coordinates in first-passage percolation models

The front velocity of the simple epidemic