The grasshopper problem

A grasshopper lands at a random point on a planar lawn of area one. It then makes one jump of 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? This easily stated yet hard to solve mathematical problem has intriguing connections to quantum information and statistical physics. A generalized version on the sphere provides insight into a new class of Bell inequalities. In this setup two parties measure spins about randomly chosen axes and obtain correlations for pairs of axes separated by a fixed angle. A discrete version can be modeled by a spin system, representing a new class of statistical models with fixed-range interactions, where the range d can be large. We show that, perhaps surprisingly, there is no d > 0 for which a disc shaped lawn is optimal. If the jump distance is smaller than the radius of the unit disc, the optimal lawn resembles a cogwheel, with transitions to more complex, disconnected shapes at larger d. We will discuss several classes of optimal lawn shapes on the plane and on the sphere with focus on their connection to Bell inequalities that involve random measurement choices.