Infinite volume extrapolation in the one-dimensional bond diluted Levy spin-glass model near its lower critical dimension

Abstract

We revisited, by means of numerical simulations, the one-dimensional bond diluted Levy Ising spin glasses outside the limit of validity of mean-field theories. In these models the probability that two spins at distance $r$ interact (via disordered interactions, ${J}_{ij}=\ifmmode\pm\else\textpm\fi{}1)$ decays as ${r}^{\ensuremath{-}\ensuremath{\rho}}$. We have estimated, using finite size scaling techniques, the infinite volume correlation length and spin glass susceptibility for $\ensuremath{\rho}=5/3$ and $\ensuremath{\rho}=9/5$. We have obtained strong evidence for divergences of the previous observables at a nonzero critical temperature. We discuss the behavior of the critical exponents, especially when approaching the value $\ensuremath{\rho}=2$, corresponding to a critical threshold beyond which the model has no phase transition. Finally, we numerically study the model right at the threshold value $\ensuremath{\rho}=2$.