Category: Statistics

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

Sparse Wigner matrices interpolate between dense random-matrix behavior and graph-like regimes where universality can fail. This experiment asks for the critical sparsity exponent gamma at which GOE-style level statistics break down as the probability of a nonzero entry scales like N^{-gamma}.

The script generates symmetric sparse random matrices, bisects gamma, and carries the threshold bracket to larger matrix sizes through iterative deepening. Rather than surveying the full sparsity range independently at each size, it concentrates computation on the suspected transition and builds a finite-size map of where Wigner-Dyson behavior gives way to a different regime.

This matters because rigorous bounds exist but are not believed to be sharp. A systematic numerical threshold map can therefore help clarify where universality appears to fail in practice, with implications for random graph spectra and related probabilistic models.

Method: Dense symmetric eigensolve on sparse Wigner matrices with iterative deepening and bisection on the sparsity exponent gamma across system sizes N = 64 to 2048.

What is measured: Critical sparsity exponent, eigenvalue-spacing statistics, implied universality breakdown point, and threshold bracket width.