Two-dimensional percolation with algebraically decaying interactions II: Critical exponents in the long-range regime

What the math gives to ML

The paper provides a quantitative universality diagram for two-dimensional long-range connectivity, with sharp boundaries at \(\sigma=2\), \(\sigma=1\), and the upper-critical threshold \(\sigma=d/3=2/3\). Its transferable mechanism is a distance-decaying random interaction graph whose coarse-grained behavior changes from mean-field/global mixing to non-Gaussian long-range scaling and finally short-range behavior. A neural implementation can impose this kernel as a sparse attention or message-passing prior, then test whether receptive-field, gradient-correlation, and optimization statistics show the predicted crossover near these exponents. The strongest use is a tunable multiscale architecture rather than treating the measured percolation exponents as an unverified loss regularizer.