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 $d \gtrsim
\pi^-1/2$.
%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
}