We revisited, by means of numerical simulations, the one dimensional bonddiluted Levy Ising spin glasses outside the limit of validity of mean fieldtheories. 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 volumecorrelation length and spin glass susceptibility for $\rho=5/3$ and $\rho=9/5$.We have obtained strong evidence for divergences of the previous observables ata non zero critical temperature. We discuss the behavior of the criticalexponents, especially when approaching the value $\rho=2$, corresponding to acritical threshold beyond which the model has no phase transition. Finally, wenumerically study the model right at the threshold value $\rho=2$.
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