animation027circle from The grasshopper problem

2017-11-07T10:34:00Z (GMT) by Olga Goulko Adrian Kent
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 <i>d</i>, 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 <i>d</i> > 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 <i>d</i> < <i>π</i><sup>−1/2</sup>, the optimal lawn resembles a cogwheel with <i>n</i> cogs, where the integer <i>n</i> is close to π(arcsin (√<i>πd</i>/2))<sup>−1</sup>. We find transitions to other shapes for <i>d</i> ≳ <i>π</i><sup>−1/2</sup>.