We have simulated, using parallel tempering, the three-dimensional Ising spin glass model with binary couplings in a helicoidal geometry. The largest lattice $\mathrm{}(L=20\mathrm{})\mathrm{}$ has been studied using a dedicated computer (the SUE machine). We have obtained, measuring the correlation length in the critical region, strong evidence for a second-order finite-temperature phase transition, ruling out other possible scenarios like a Kosterlitz-Thouless phase transition. Precise values for the ν and η critical exponents are also presented.

We study the phase diagram of the site-diluted Ising model in a wide dilution range, through Monte Carlo simulations and finite-size scaling techniques. Our results for the critical exponents and universal cumulants turn out to be dilution independent, but only after a proper infinite volume extrapolation, taking into account the leading corrections-to-scaling terms.

Using Finite-Size Scaling techniques we obtain accurate results for critical quanti- ties of the Ising model and the site percolation, in three dimensions. We pay special attention in parameterizing the corrections-to-scaling, what is necessary to put the systematic errors below the statistical ones.

Significance: The Mpemba effect, wherein an initially hotter system relaxes faster when quenched to lower temperatures than an initially cooler system, has attracted much attention. Paradoxically, its very existence is a hot topic. Using massive numerical simulations, we show unambiguously that the Mpemba effect is present in the archetypical model system for complex behavior, spin glasses. We find that the Mpemba effect in spin glasses is due to the aging dynamics of the internal energy, as controlled by the nonequilibrium spin-glass coherence length. Interestingly, the effect is not present when the system remains all of the time in the paramagnetic phase. Therefore, the Mpemba effect suggests itself as an effective probe for a glass transition, potentially useful for other glass-forming systems.

We report a high-precision finite-size scaling study of the critical behavior of the three-dimensional Ising Edwards-Anderson model (the Ising spin glass). We have thermalized lattices up to $L=40$ using the Janus dedicated computer. Our analysis takes into account leading-order corrections to scaling. We obtain ${T}_{\mathrm{c}}=1.1019(29)$ for the critical temperature, $$\nu${}=2.562(42)$ for the thermal exponent, $$\eta${}=$-${}0.3900(36)$ for the anomalous dimension, and $$\omega${}=1.12(10)$ for the exponent of the leading corrections to scaling. Standard (hyper)scaling relations yield $$\alpha${}=$-${}5.69(13)$, $$\beta${}=0.782(10)$, and $$\gamma${}=6.13(11)$. We also compute several universal quantities at ${T}_{\mathrm{c}}$.