The scales of disorder in perfect quasicrystals

What the math gives to ML

The paper identifies a symmetry-controlled disorder-to-order crossover: finite observation windows cannot distinguish a deterministic high-symmetry quasicrystal from randomness until the window exceeds a crossover length \(\lambda_N\). It provides a finite-size scaling law for number variance, with \(\lambda_N\sim N^\nu\) and \(\nu=2\) in two dimensions or \(\nu=1\) in one dimension, while a second scale \(\Lambda_\infty(N)\sim N^\gamma\) controls fluctuation amplitude. A transferable neural-network mechanism is a multiscale detector or training curriculum that deliberately increases receptive-field size past a predicted crossover, rather than forcing local features to identify globally ordered but locally noise-like patterns.