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 disordered interactions, Jij=±1) decays as r-?. We have estimated, using finite size scaling techniques, the infinite volume correlation length and spin glass susceptibility for ?=5/3 and ?=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 ?=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 ?=2.