%0 DATA
%A Olga, Goulko
%A Adrian, Kent
%D 2017
%T animation027circle from The grasshopper problem
%U https://rs.figshare.com/articles/animation027circle_from_The_grasshopper_problem/5576806
%R 10.6084/m9.figshare.5576806.v1
%2 https://rs.figshare.com/ndownloader/files/9696988
%K geometric combinatorics
%K spin models
%K Bell inequalities
%K statistical physics
%X 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* < *π*^{−1/2}, the optimal lawn resembles a cogwheel with *n* cogs, where the integer *n* is close to π(arcsin (√*πd*/2))^{−1}. We find transitions to other shapes for *d* ≳ *π*^{−1/2}.