One central piece of the problem of dynamic arrest is whether the phenomenology of slow relaxation, increasing dynamical length scales, mild (or dramatic) structural changes are somewhat related to the existence of a zero entropy amorphous state emerging at a non-zero temperature.

A comprehensive theory would need on the one hand to take into account of the well established phenomenon of *dynamical heterogeneities*, *i.e.* the non-homogenous patterns of diffusion that emerge together with the glassy dynamics itself; on the other hand, it should also rationalise the many findings that point (for several model systems) to a dramatic reduction of the so-called configurational entropy as one approaches a finite temperature (sometimes termed Kauzmann temperature) at which also the relaxation times appear to diverge.

In our recent work (*Physical Review X 7, 031028*) Thomas Speck, C. Patrick Royall and I discuss a unified scenario that combines dynamical aspects to structural ones in order to sample very low energy and entropy states, employing dynamical large deviations.

We find that the equilibrium supercooled liquid competes with a secondary metastable amorphous liquid rich in long-lived structural motifs, hidden in the tails of probability distributions in trajectory space. We also show that sampling the tails of such probabilities at a single moderate temperature allows us to retrieve the thermodynamic properties of the ordinary liquid in much wider range of temperatures, down to very low temperatures. We can then draw a diagram for the stable and metastable phase, pointing towards critical-like fluctuations in the region where the Kauzmann temperature is normally located, and allowing us to review currently proposed scenarios from an alternative point of view, rooted in the large deviation theory of metastability.

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