The $6-ε$ Expansion for Long-Range Lee--Yang and Percolation Criticality
arXiv:2608.15120
2026
Architecture
1 ideas extracted · analyzed Sep 1, 2026
What the math gives to ML
The paper develops a long-range φ³ renormalization-group expansion near d=6, with fractional kinetic operator (-Δ)^{σ/2}, and identifies a continuous long-range/short-range crossover at σ*=2 rather than an extrapolated discontinuity. Its transferable asset is a quantitative scale-dependent mechanism: nonlocal interactions dominate for σ<2, the effective cubic coupling has canonical distance ε'=3σ-d=ε-3δ from marginality, and the correct architecture should interpolate between fractional and ordinary Laplacian propagation. A practical neural analogue is a spectrally parameterized residual block with a learnable fractional diffusion exponent and an explicit crossover gate, tested for a predicted change in scaling behavior near σ=2.
Ideas from this paper
Unverified
2026
Replace a single local message-passing or convolution operator by a spectrally controlled mixture of fractional and ordinary diffusion. The exponent σ is learned or scheduled, while a crossover gate forces the model to change parameterization near the renormalization-group threshold σ*=2, allowing long-range propagation when useful without retaining an unnecessarily nonlocal operator at short scales.
Useful6/10
Difficulty5/10
Novelty6/10