Symmetry Breaking in 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 problem has intriguing connections to quantum information and statistical physics. 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. Surprisingly, there is no d > 0 for which a disc shaped lawn is optimal. For jump distances smaller than the radius of a unit disc, the optimal lawn shapes resemble cogwheels that break rotational symmetry. When the problem is generalized to higher dimensions, rotational symmetry is restored in the optimal lawn shapes. We will discuss numerical results as well as an analysis of why and how rotational symmetry is broken.