Long range to short range crossover in one dimension

Abstract

This work investigates the critical behavior of one-dimensional systems with long-range (LR) interactions, focusing on the crossover to short-range (SR) universality. Through large-scale Monte Carlo simulations of self-avoiding L\'evy flights on a 1D lattice, we compute the anomalous dimension \eta, the correlation length exponent \nu, and the susceptibility exponent \gamma across a wide range of LR decay parameters \sigma. Our results provide strong numerical evidence that supports Sak's scenario. They identify the crossover at \sigma^* = 1 and demonstrate the continuity of critical exponents across this point, with strong corrections to scaling. The study also reveals deviations from Flory-type scaling predictions and discusses the limitations of effective dimension approaches in general. These findings clarify the nature of the LR-SR crossover in low-dimensional systems and open avenues for exploring criticality in disordered and complex networks.