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There is a certain consensus that the very fast growth of the relaxation time$\tau$ occurring in glass-forming liquids on lowering the temperature must bedue to the thermally activated rearrangement of correlated regions of growingsize. Even though measuring the size of these regions has defied scientists fora while, there is indeed recent evidence of a growing correlation length $\xi$in glass-formers. If we use Arrhenius law and make the mild assumption that thefree-energy barrier to rearrangement scales as some power $\psi$ of the size ofthe correlated regions, we obtain a relationship between time and length,$T\log\tau \sim \xi^\psi$. According to both the Adam-Gibbs and the RandomFirst Order theory the correlation length grows as $\xi \sim(T-T_k)^{-1/(d-\theta)}$, even though the two theories disagree on the value of$\theta$. Therefore, the super-Arrhenius growth of the relaxation time with thetemperature is regulated by the two exponents $\psi$ and $\theta$ through therelationship $T\log\tau \sim (T-T_k)^{-\psi/(d-\theta)}$. Despite a fewtheoretical speculations, up to now there has been no experimentaldetermination of these two exponents. Here we measure them numerically in amodel glass-former, finding $\psi=1$ and $\theta=2$. Surprisingly, even thoughthe values we found disagree with most previous theoretical suggestions, theygive back the well-known VFT law for the relaxation time, $T\log\tau \sim(T-T_k)^{-1}$.

Numerical determination of the exponents controlling the relationship between time, length and temperature in glass-forming liquids