In this work we study the stability of the equilibria reached by ecosystems formed by a large number of species. The model we focus on are Lotka-Volterra equations with symmetric random interactions. Our theoretical analysis, confirmed by our numerical studies, shows that for strong and heterogeneous interactions the system displays multiple equilibria which are all marginally stable. This property allows us to obtain general identities between diversity and single species responses, which generalize and saturate May's stability bound. By connecting the model to systems studied in condensed matter physics, we show that the multiple equilibria regime is analogous to a critical spin-glass phase. This relation suggests new experimental ways to probe marginal stability.

We study the effect of freezing the positions of a fraction c of particles from an equilibrium configuration of a supercooled liquid at a temperature T . We show that within the random first-order transition theory pinning particles leads to an ideal glass transition for a critical fraction c ¼ c K ðT Þ even for moderate supercooling; e.g., close to the Mode-Coupling transition temperature. First we derive the phase diagram in the T − c plane by mean field approximations. Then, by applying a real-space renormalization group method, we obtain the critical properties for jc − c K ðT Þj → 0, in particular the divergence of length and time scales, which are dominated by two zero-temperature fixed points. We also show that for c ¼ c K ðT Þ the typical distance between frozen particles is related to the static point-to-set length scale of the unconstrained liquid. We discuss what are the main differences when particles are frozen in other geometries and not from an equilibrium configuration. Finally, we explain why the glass transition induced by freezing particles provides a new and very promising avenue of research to probe the glassy state and ascertain, or disprove, the validity of the theories of the glass transition.glass materials | amorphous order | slow dynamics | ideal glass phase | thermodynamic phase transition A lmost any liquid shows an impressive growth of the relaxation time when supercooled below the melting point: indeed typical time scales increase from picoseconds to hours in a rather restricted temperature window. This increase is so steep that below a certain temperature, T g , it is not possible to equilibrate the system, which then freezes in an amorphous solid, called glass (1). Although T g is not a critical temperature and only indicates a cross-over point, a recurrent question in the literature is whether a true thermodynamic phase transition to an ideal glass state takes place (at a temperature T K < T g ). The answer to this question would provide an explanation for the fast growth of the relaxation time and several other phenomena observed experimentally. Nevertheless, other scenarios based on purely dynamical transitions at zero (or finite) temperature or even no transition at all, might also provide alternative explanations (2, 3). In contrast with usual critical phenomena, the study of the glass transition is hampered by the much more severe slowing down of the dynamics. Recent work on growing length scales (4) suggests that even if a transition is present the corresponding critical length that can be observed in experiments is not larger than a few interparticle distances, meaning that experimental systems can be equilibrated only rather far from the putative transition point. This fact makes it quite difficult to prove (or disprove) the existence of the transition and also to contrast theories solely on the basis of their critical properties. Here we propose a way to bypass these problems: we show that an ideal glass transition † can be induced by a suitable random perturbation even...

We study rough high-dimensional landscapes in which an increasingly stronger preference for a given configuration emerges. Such energy landscapes arise in glass physics and inference. In particular we focus on random Gaussian functions, and on the spiked-tensor model and generalizations. We thoroughly analyze the statistical properties of the corresponding landscapes and characterize the associated geometrical phase transitions. In order to perform our study, we develop a framework based on the Kac-Rice method that allows to compute the complexity of the landscape, i.e. the logarithm of the typical number of stationary points and their Hessian. This approach generalizes the one used to compute rigorously the annealed complexity of mean-field glass models. We discuss its advantages with respect to previous frameworks, in particular the thermodynamical replica method which is shown to lead to partially incorrect predictions.