prsa.suppl.material.description from The grasshopper problem Olga Goulko Adrian Kent 10.6084/m9.figshare.5576803.v1 https://rs.figshare.com/articles/media/prsa_suppl_material_description_from_The_grasshopper_problem/5576803 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>. 2017-11-07 10:33:56 geometric combinatorics spin models Bell inequalities statistical physics