Statistical mechanics of coupled supercooled liquids in finite dimensions

We study the statistical mechanics of supercooled liquids when system evolves at a temperature $T$ with field $\epsilon$ linearly coupled to its overlap reference configuration same liquid sampled $T_0$. use mean-field theory fully characterize influence $T_0$, and we mainly case fixed, low-$T_0$ value in computer simulations. numerically investigate extended phase diagram $(\epsilon,T)$ plane model glass-forming spatial dimensions $d=2$ $d=3$, relying on umbrella sampling reweighting techniques. For both $2d$ $3d$ cases, similar phenomenology nontrivial thermodynamic fluctuations is observed low temperatures, but detailed finite-size analysis reveals qualitatively distinct behaviors. establish existence first-order transition line for nonzero ending critical point universality class random-field Ising (RFIM) $d=3$. In instead, no found large enough systems least down temperatures below extrapolated calorimetric glass $T_g$. Our results confirm that samples limited size display expected finite undergoing random transition. They also support relevance physics RFIM liquids, which may then explain qualitative difference between glass-formers.