Category: Physics

Summary: Estimating the sparsity exponent where sparse Wigner matrices stop showing Wigner-Dyson eigenvalue statistics and leave the universal regime.

Sparse random matrices interpolate between dense universal ensembles and graph-like systems where level statistics can change sharply. This experiment asks for the critical sparsity exponent at which sparse Wigner matrices stop behaving like GOE spectra and cross out of the universal Wigner-Dyson regime.

The GPU implementation generates symmetric matrices with entry probability p = N^{-gamma} and computes eigenvalue statistics across large sizes. By scanning gamma systematically, it targets the boundary between the rigorously known dense side and the much less settled sparse side.

That makes the project a quantitative threshold study of universality breakdown. Its value is in building a direct numerical map of the crossover exponent rather than relying only on analytic bounds that are widely believed to be non-sharp.

Method: GPU eigenspectrum sweeps for sparse symmetric Wigner matrices with p = N^{-gamma}, comparing spacing statistics across sizes up to 8192.

What is measured: Critical sparsity exponent gamma, eigenvalue-spacing statistics, departure from GOE behavior, finite-size crossover trends, and size dependence.