Author ORCID Identifier

0009-0004-7373-2066

Date of Award

8-31-2026

Document Type

Open Access Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Physics, Applied

First Advisor

Olga Goulko

Second Advisor

Rahul Kulkarni

Third Advisor

Christopher Fuchs

Abstract

The Grasshopper Problem asks a simple geometric question. A grasshopper lands on a lawn of fixed area and jumps a fixed distance in a random direction. What shape of lawn maximizes the probability that the grasshopper remains on the lawn after jumping? The jump rule is rotationally symmetric, but the best lawns do not have to be. This dissertation studies how that symmetry breaking occurs, maps the continuum problem to a novel constrained spin system, and uses the spherical Grasshopper Problem to compare quantum singlet correlations with classical local models.

For planar lawns, boundary-integral and perturbative calculations explain why the disk is unstable in two dimensions and why the analogous flat-boundary instability is absent in three and higher dimensions. The three-dimensional numerics produce a sequence that begins with balls and spherical shells, then continues to symmetry-broken and disconnected regions. At finite temperature, the fixed-density Grasshopper Ising model is used to study how these symmetry-breaking states lose anisotropy under thermal fluctuations.

On the sphere, large numerical searches are carried out for complementary antipodal lawns, independent antipodal lawns, and complementary lawns without the antipodal constraint. Comparisons across spherical designs, HEALPix, Goldberg, and Coulomb grids help separate geometric structure from grid artifacts. The optimized states include cogwheels, labyrinths, and stripes.

The spherical lawns can also be used as local hidden-variable models. In this interpretation, the optimized lawn probabilities provide classical anti-correlation benchmarks for random-axis measurements of a singlet state. The optimized curves place the largest fixed-angle quantum–classical gap for general independent local colorings at the Clauser–Horne–Shimony–Holt (CHSH) angle π/4. With perfect anti-correlation imposed at identical axes, the largest gap occurs at π/5. The dissertation also discusses the computational methods and Grasshopper code library used for these calculations.

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