%0 Journal Article %1 goulko2017grasshopper %A Goulko, Olga %A Kent, Adrian %D 2017 %K maths qm %T The grasshopper problem %U http://arxiv.org/abs/1705.07621 %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 one. It then jumps once, a fixed distance $d$, in a random direction. What shape should the lawn be to maximise 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 < \pi^-1/2$ the optimal lawn resembles a cogwheel with $n$ cogs, where the integer $n$ is close to $ ( ( d /2 ) )^-1$. We find transitions to other shapes for $d \gtrsim \pi^-1/2$.

@article{goulko2017grasshopper, abstract = {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 one. It then jumps once, a fixed distance $d$, in a random direction. What shape should the lawn be to maximise 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 < \pi^{-1/2}$ the optimal lawn resembles a cogwheel with $n$ cogs, where the integer $n$ is close to $ \pi ( \arcsin ( \sqrt{\pi} d /2 ) )^{-1}$. We find transitions to other shapes for $d \gtrsim \pi^{-1/2}$.}, added-at = {2017-05-23T18:54:14.000+0200}, author = {Goulko, Olga and Kent, Adrian}, biburl = {https://www.bibsonomy.org/bibtex/2da59bdf89994b445c172ecbda9c76ca3/vindex10}, description = {The grasshopper problem}, interhash = {3e7f7e4519b172a6e9307e4794580cb4}, intrahash = {da59bdf89994b445c172ecbda9c76ca3}, keywords = {maths qm}, note = {cite arxiv:1705.07621}, timestamp = {2017-05-26T18:37:32.000+0200}, title = {The grasshopper problem}, url = {http://arxiv.org/abs/1705.07621}, year = 2017 }