Long-range interactions and Anderson localisation for one-dimensional high-contrast resonator chain

What the math gives to ML

The paper identifies a concrete borderline interaction law, \(\mathcal{C}(n,m)\sim 1/(|n-m|\log^2|n-m|)\), which is absolutely summable but has a much heavier tail than the short-range kernels usually used in neural sequence mixers. This suggests a multiscale neural mixing operator whose distant interactions are retained through logarithmically decaying dyadic scales rather than discarded by a finite receptive field. The key engineering opportunity is to approximate this kernel with range-sum or FFT operations in \(O(L\log L)\) time and \(O(L)\) memory, while obtaining an explicit truncation-error estimate. This is most promising for long-context sequence models, graph networks on chain-like graphs, and lightweight replacements for attention when tasks require long-range but structured interactions.