Supercooled liquids exhibit a pronounced slowdown of their dynamics on cooling 1 without showing any obvious structural or thermodynamic changes 2 . Several theories relate this slowdown to increasing spatial correlations 3-6 . However, no sign of this is seen in standard static correlation functions, despite indirect evidence from considering specific heat 7 and linear dielectric susceptibility 8 . Whereas the dynamic correlation function progressively becomes more non-exponential as the temperature is reduced, so far no similar signature has been found in static correlations that can distinguish qualitatively between a high-temperature and a deeply supercooled glass-forming liquid in equilibrium. Here, we show evidence of a qualitative thermodynamic signature that differentiates between the two. We show by numerical simulations with fixed boundary conditions that the influence of the boundary propagates into the bulk over increasing length scales on cooling. With the increase of this static correlation length, the influence of the boundary decays non-exponentially. Such long-range susceptibility to boundary conditions is expected within the random first-order theory 4,9,10 (RFOT) of the glass transition. However, a quantitative account of our numerical results requires a generalization of RFOT, taking into account surface tension fluctuations between states.Inspired by critical phenomena, it is natural to expect that the slowing down of the dynamics is related to the vicinity of a thermodynamic phase transition, where some kind of long-range order would set in 11 . This is the spirit of different recent theories 4,9,[12][13][14] , but seems at odds with others 5,15 , at least at first sight. In particular, the crucial physical mechanism at the root of random first-order theory 4 (RFOT) is the emergence of long-range amorphous order, the precise definition and quantitative characterization of which is however far from obvious. Dynamic heterogeneities 16 do show a growing dynamic correlation length accompanying the glass transition, both experimentally 17 and numerically 18 . This is certainly a first important step, but not sufficient to prune down-even at a qualitative level-different theories of the glass transition. In particular, it is not clear whether this phenomenon is due to an underlying static or purely dynamic phase transition.The approach followed here is based on the very definition of a thermodynamic phase transition, where the effect of boundary conditions becomes long-ranged. The problem is that for glasses there are no natural boundary conditions, because these should be as 'random' as the bulk amorphous states that they favour. A possible solution is to use equilibrium liquid configurations to define the boundary 19 . In the context of RFOT, this was suggested in ref. 9 (and further discussed in ref. 11), but the scope and some conclusions of this Gedankenexperiment are more general [19][20][21] . Starting from a given equilibrium configuration, we freeze the motion of all particles outside a ...

Collective decision-making in biological systems requires all individuals in the group to go through a behavioural change of state. During this transition fast and robust transfer of information is essential to prevent cohesion loss. The mechanism by which natural groups achieve such robustness, though, is not clear. Here we present an experimental study of starling flocks performing collective turns. We find that information about direction changes propagates across the flock with a linear dispersion law and negligible attenuation, hence minimizing group decoherence. These results contrast starkly with current models of collective motion, which predict diffusive transport of information. Building on spontaneous symmetry breaking and conservation laws arguments, we formulate a new theory that correctly reproduces linear and undamped propagation. Essential to the new framework is the inclusion of the birds’ behavioural inertia. The new theory not only explains the data, but also predicts that information transfer must be faster the stronger the group’s orientational order, a prediction accurately verified by the data. Our results suggest that swift decision-making may be the adaptive drive for the strong behavioural polarization observed in many living groups.