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The model of low-energy quantum gravity by the author has the property ofasymptotic freedom at very short distances. The character of transition toasymptotic freedom is studied here. It is shown that this transition is notuniversal, but the one obeys the scaling rule: the range of this transition inunits of $r/E^{1/2}$, where $r$ is a distance between particles and $E$ is anenergy of the screening particle, is the same for any micro-particle. Thisrange for a proton is between $10^{-11} - 10^{-13}$ meter, while for anelectron it is approximately between $10^{-13} - 10^{-15}$ meter.

A non-universal transition to asymptotic freedom in low-energy quantum gravity