The grasshopper problem

We introduce and physically motivate the following problem in geometric combinatorics, originally inspired by analysing Bell inequalities. A grasshopper lands at a random point on a planar lawn of area 1. It then jumps once, a fixed distance d, in a random direction. What shape should the lawn be to maximize the chance that the grasshopper remains on the lawn after jumping? We show that, perhaps surprisingly, a disc-shaped lawn is not optimal for any d > 0. We investigate further by introducing a spin model whose ground state corresponds to the solution of a discrete version of the grasshopper problem. Simulated annealing and parallel tempering searches are consistent with the hypothesis that, for d < p^-1/2, the optimal lawn resembles a cogwheel with n cogs, where the integer n is close to p(arcsin(pd/2))^-1. We find transitions to other shapes for d p^-1/2 .