Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The grasshopper problem   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 $d \gtrsim
\pi^{-1/2}$.

1
The categories of this paper are: cs
33
The categories of this paper are: plasm-ph
18
The categories of this paper are: econ
23
The categories of this paper are: solv-int
17
The categories of this paper are: q-fin
5
The categories of this paper are: gr-qc
36
The categories of this paper are: cmp-lg
7
The categories of this paper are: hep-th
22
The categories of this paper are: chao-dyn
15
The categories of this paper are: nucl-ex
31
The categories of this paper are: ao-sci
9
The categories of this paper are: nlin
26
The categories of this paper are: bayes-an
12
The categories of this paper are: nucl-th
10
The categories of this paper are: q-bio
11
The categories of this paper are: hep-lat