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

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 a disordered interactions, $J_{ij}=\pm 1$) decays as $r^{-\rho}$. We have estimated, using finite size scaling techniques, the infinite volume correlation length and spin glass susceptibility for $\rho=5/3$ and $\rho=9/5$. We have obtained strong evidence for divergences of the previous observables at a non zero critical temperature. We discuss the behavior of the critical exponents, especially when approaching the value $\rho=2$, corresponding to a critical threshold beyond which the model has no phase transition. Finally, we numerically study the model right at the threshold value $\rho=2$.