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Scaling of the largest dynamical barrier in the one-dimensional long-range Ising spin-glass

The long-range one-dimensional Ising spin-glass with random couplings decaying as $J(r) \propto r^{-σ}$ presents a spin-glass phase $T_c(σ)>0$ for $0 \leq σ<1$ (the limit $σ=0$ corresponds to the mean-field SK-model). We use the eigenvalue method introduced in our previous work [C. Monthus and T. Garel, J. Stat. Mech. P12017 (2009)] to measure the equilibrium time $t_{eq}(N)$ at temperature $T=T_c(σ)/2$ as a function of the number $N$ of spins. We find the activated scaling $\ln t_{eq}(N) \sim N^ψ$ with the same barrier exponent $ψ\simeq 0.33$ in the whole region $0\leqσ<1$.