Conjecture that the Hausdorff dimension of the frontier of the SFF random walk approaches 4/3 for chaotic Hamiltonians and 1 for integrable ones, with proofs of Gaussian statistics under Lyapunov conditions on degeneracies and exact moments for unequal-step walks.
For example, for a smooth curve, the fractal dimension is1while it is2 for a plane-filling curve [52]
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Integrability and Chaos via fractal analysis of Spectral Form Factors: Gaussian approximations and exact results
Conjecture that the Hausdorff dimension of the frontier of the SFF random walk approaches 4/3 for chaotic Hamiltonians and 1 for integrable ones, with proofs of Gaussian statistics under Lyapunov conditions on degeneracies and exact moments for unequal-step walks.