The grasshopper problem (1705.07621v3)

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

• The paper finds that a disc is not the optimal shape for a lawn to maximize the probability of a grasshopper staying on it after a random jump for any jump distance d > 0.
• Optimal lawn shapes are not always simple or simply connected and transition from cogwheels for small jump distances to more complex forms as the jump distance increases.
• Using analytical methods, numerical simulations like simulated annealing, and a spin model, the authors provide evidence for their findings and potential applications in fields like spatial optimization and physics.

An Analysis of "The Grasshopper Problem"

In "The Grasshopper Problem," Goulko and Kent present a novel geometric combinatorics challenge originally motivated by investigations into Bell inequalities. The primary question posed involves determining the optimal shape of a lawn on a plane such that a grasshopper, starting at a random point on the lawn and jumping a fixed distance $d$ in a random direction, remains on the lawn with the highest probability.

Problem Formulation and Analysis

The paper rigorously examines the geometry of optimal lawn shapes by formalizing the problem both in the continuous and discrete settings. The authors devise a spin model whose ground state represents the solution in a discretized version of the grasshopper problem. The pivotal finding that challenges prior assumptions is that a disc-shaped lawn is not optimal for any jump distance $d > 0$. This conclusion is substantiated through both analytical and numerical methods, highlighting the inadequacy of isoperimetric arguments in fully capturing the problem's essence.

For small jump distances, $d < \pi^{-1/2}$, the optimal lawn shapes resemble cogwheels with a number of cogs dependent on $d$. As $d$ increases beyond $\pi^{-1/2}$, the shapes transition into more complex forms, including ones that are not simply connected. This indicates a rich structural diversity in optimal designs depending on the jump distance, implying potential insights into achieving balance between perimeter measures and jump probabilities across ranges of $d$.

Numerical Modeling and Verification

The authors utilize simulated annealing and parallel tempering techniques to explore the solution space of this problem numerically. Through various discretization setups and consistency checks, they corroborate their theoretical results by identifying the cogwheel pattern for small jump distances and more fragmented shapes for larger jumps.

These computational methods propose a compelling approximation to otherwise analytically intractable questions, echoing broader implications in classical fields such as optimization within geometric constraints. Their detailed phase diagram captures the transitions and subtle nuances in lawn shape optimality across varying $d$.

Implications and Future Directions

The results presented go beyond theoretical interest in geometric combinatorics by paving new avenues for practical applications in fields where spatial optimization and probabilistic constraints intersect. Moreover, given the problem's conceptual ties to Bell inequalities and quantum information theory, further exploration into higher dimensions or alternative metrics could widen its applicability.

In theoretical physics, this work provides insight that might lead to more profound discoveries regarding optimal configurations in different geometric contexts or under various physical constraints. Additionally, applications may extend to reaction-diffusion models and symmetry operations in natural patterns, hinting at a potential crossing into biological morphology studies, where such optimizations may naturally arise.

Final Remarks

"The Grasshopper Problem" exemplifies the synthesis of geometric intuition with formal computational modeling, challenging traditional assumptions about optimal shapes under probabilistic conditions. The interplay between numerical experimentation and theoretical proofs encourages a reevaluation of strategies in dealing with spatially constrained problems across disciplines. This work points to a broader applicability and stimulates discussions on classical shapes versus complex emergent patterns within optimization paradigms.