Generalized Lotka-Volterra Equations with Random, Nonreciprocal Interactions: The Typical Number of Equilibria.

We compute the typical number of equilibria of the generalized Lotka-Volterra equations describing species-rich ecosystems with random, nonreciprocal interactions using the replicated Kac-Rice method. We characterize the multiple-equilibria phase by determining the average abundance and similarity between equilibria as a function of their diversity (i.e., of the number of coexisting species) and of the variability of the interactions. We show that linearly unstable equilibria are dominant, and that the typical number of equilibria differs with respect to the average number.

Artificial selection of communities drives the emergence of structured interactions.

Species-rich communities, such as the microbiota or microbial ecosystems, provide key functions for human health and climatic resilience. Increasing effort is being dedicated to design experimental protocols for selecting community-level functions of interest. These experiments typically involve selection acting on populations of communities, each of which is composed of multiple species. If numerical simulations started to explore the evolutionary dynamics of this complex, multi-scale system, a comprehensive theoretical understanding of the process of artificial selection of communities is still lacking. Here, we propose a general model for the evolutionary dynamics of communities composed of a large number of interacting species, described by disordered generalised Lotka-Volterra equations. Our analytical and numerical results reveal that selection for scalar community functions leads to the emergence, along an evolutionary trajectory, of a low-dimensional structure in an initially featureless interaction matrix. Such structure reflects the combination of the properties of the ancestral community and of the selective pressure. Our analysis determines how the speed of adaptation scales with the system parameters and the abundance distribution of the evolved communities. Artificial selection for larger total abundance is thus shown to drive increased levels of mutualism and interaction diversity. Inference of the interaction matrix is proposed as a method to assess the emergence of structured interactions from experimentally accessible measures.

Predicting Dynamic Heterogeneity in Glass-Forming Liquids by Physics-Inspired Machine Learning.

We introduce GlassMLP, a machine learning framework using physics-inspired structural input to predict the long-time dynamics in deeply supercooled liquids. We apply this deep neural network to atomistic models in 2D and 3D. Its performance is better than the state of the art while being more parsimonious in terms of training data and fitting parameters. GlassMLP quantitatively predicts four-point dynamic correlations and the geometry of dynamic heterogeneity. Transferability across system sizes allows us to efficiently probe the temperature evolution of spatial dynamic correlations, revealing a profound change with temperature in the geometry of rearranging regions.

Elasticity, Facilitation, and Dynamic Heterogeneity in Glass-Forming Liquids.

We study the role of elasticity-induced facilitation on the dynamics of glass-forming liquids by a coarse-grained two-dimensional model in which local relaxation events, taking place by thermal activation, can trigger new relaxations by long-range elastically mediated interactions. By simulations and an analytical theory, we show that the model reproduces the main salient facts associated with dynamic heterogeneity and offers a mechanism to explain the emergence of dynamical correlations at the glass transition. We also discuss how it can be generalized and combined with current theories.

Finite-Disorder Critical Point in the Yielding Transition of Elastoplastic Models.

Upon loading, amorphous solids can exhibit brittle yielding, with the abrupt formation of macroscopic shear bands leading to fracture, or ductile yielding, with a multitude of plastic events leading to homogeneous flow. It has been recently proposed, and subsequently questioned, that the two regimes are separated by a sharp critical point, as a function of some control parameter characterizing the intrinsic disorder strength and the degree of stability of the solid. In order to resolve this issue, we have performed extensive numerical simulations of athermally driven elastoplastic models with long-range and anisotropic realistic interaction kernels in two and three dimensions. Our results provide clear evidence for a finite-disorder critical point separating brittle and ductile yielding, and we provide an estimate of the critical exponents in 2D and 3D.

Equilibrium fluctuations in mean-field disordered models.

Mean-field models of glasses that present a random first order transition exhibit highly nontrivial fluctuations. Building on previous studies that focused on the critical scaling regime, we here obtain a fully quantitative framework for all equilibrium conditions. By means of the replica method we evaluate Gaussian fluctuations of the overlaps around the thermodynamic limit, decomposing them in thermal fluctuations inside each state and heterogeneous fluctuations between different states. We first test and compare our analytical results with numerical simulation results for the p-spin spherical model and the random orthogonal model, and then analyze the random Lorentz gas. In all cases, a strong quantitative agreement is obtained. Our analysis thus provides a robust scheme for identifying the key finite-size (or finite-dimensional) corrections to the mean-field treatment of these paradigmatic glass models.

Local Dynamical Heterogeneity in Simple Glass Formers.

We study the local dynamical fluctuations in glass-forming models of particles embedded in d-dimensional space, in the mean-field limit of d→∞. Our analytical calculation reveals that single-particle observables, such as squared particle displacements, display divergent fluctuations around the dynamical (or mode-coupling) transition, due to the emergence of nontrivial correlations between displacements along different directions. This effect notably gives rise to a divergent non-Gaussian parameter, α_{2}. The d→∞ local dynamics therefore becomes quite rich upon approaching the glass transition. The finite-d remnant of this phenomenon further provides a long sought-after, first-principle explanation for the growth of α_{2} around the glass transition that is not based on multiparticle correlations.

Dynamics of liquids in the large-dimensional limit.

In this paper we analytically derive the exact closed dynamical equations for a liquid with short-ranged interactions in large spatial dimensions using the same statistical mechanics tools employed to analyze Brownian motion. Our derivation greatly simplifies the original path-integral-based route to these equations and provides insight into the physical features associated with high-dimensional liquids and glass formation. Most importantly, our construction provides a route to the exact dynamical analysis of important related dynamical problems, as well as a means to devise cluster generalizations of the exact solution in infinite dimensions. This latter fact opens the door to the construction of increasingly accurate theories of vitrification in three-dimensional liquids.

Elastoplasticity Mediates Dynamical Heterogeneity Below the Mode Coupling Temperature.

As liquids approach the glass transition temperature, dynamical heterogeneity emerges as a crucial universal feature of their behavior. Dynamic facilitation, where local motion triggers further motion nearby, plays a major role in this phenomenon. Here we show that long-ranged, elastically mediated facilitation appears below the mode coupling temperature, adding to the short-range component present at all temperatures. Our results suggest deep connections between the supercooled liquid and glass states, and pave the way for a deeper understanding of dynamical heterogeneity in glassy systems.

Properties of Equilibria and Glassy Phases of the Random Lotka-Volterra Model with Demographic Noise.

We study a reference model in theoretical ecology, the disordered Lotka-Volterra model for ecological communities, in the presence of finite demographic noise. Our theoretical analysis, valid for symmetric interactions, shows that for sufficiently heterogeneous interactions and low demographic noise the system displays a multiple equilibria phase, which we fully characterize. In particular, we show that in this phase the number of locally stable equilibria is exponential in the number of species. Upon further decreasing the demographic noise, we unveil the presence of a second transition like the so-called "Gardner" transition to a marginally stable phase similar to that observed in the jamming of amorphous materials. We confirm and complement our analytical results by numerical simulations. Furthermore, we extend their relevance by showing that they hold for other interacting random dynamical systems such as the random replicant model. Finally, we discuss their extension to the case of asymmetric couplings.

Amorphous Order and Nonlinear Susceptibilities in Glassy Materials.

We review 15 years of theoretical and experimental work on the nonlinear response of glassy systems. We argue that an anomalous growth of the peak value of nonlinear susceptibilities is a signature of growing "amorphous order" in the system, with spin-glasses as a case in point. Experimental results on supercooled liquids are fully compatible with the random first-order transition (RFOT) prediction of compact "glassites" of increasing volume as temperature is decreased, or as the system ages. We clarify why such a behavior is hard to explain within purely kinetic theories of glass formation, despite recent claims to the contrary.

Revisiting the concept of activation in supercooled liquids.

In this work, we revisit the description of dynamics based on the concepts of metabasins and activation in mildly supercooled liquids via the analysis of the dynamics of a paradigmatic glass former between its onset temperature [Formula: see text] and mode-coupling temperature [Formula: see text]. First, we provide measures that demonstrate that the onset of glassiness is indeed connected to the landscape, and that metabasin waiting time distributions are so broad that the system can remain stuck in a metabasin for times that exceed [Formula: see text] by orders of magnitude. We then reanalyze the transitions between metabasins, providing several indications that the standard picture of activated dynamics in terms of traps does not hold in this regime. Instead, we propose that here activation is principally driven by entropic instead of energetic barriers. In particular, we illustrate that activation is not controlled by the hopping of high energetic barriers and should more properly be interpreted as the entropic selection of nearly barrierless but rare pathways connecting metabasins on the landscape.

Mean-Field Caging in a Random Lorentz Gas.

The random Lorentz gas (RLG) is a minimal model of both percolation and glassiness, which leads to a paradox in the infinite-dimensional, d → ∞ limit: the localization transition is then expected to be continuous for the former and discontinuous for the latter. As a putative resolution, we have recently suggested that, as d increases, the behavior of the RLG converges to the glassy description and that percolation physics is recovered thanks to finite-d perturbative and nonperturbative (instantonic) corrections [Biroli et al. Phys. Rev. E 2021, 103, L030104]. Here, we expand on the d → ∞ physics by considering a simpler static solution as well as the dynamical solution of the RLG. Comparing the 1/d correction of this solution with numerical results reveals that even perturbative corrections fall out of reach of existing theoretical descriptions. Comparing the dynamical solution with the mode-coupling theory (MCT) results further reveals that, although key quantitative features of MCT are far off the mark, it does properly capture the discontinuous nature of the d → ∞ RLG. These insights help chart a path toward a complete description of finite-dimensional glasses.

Interplay between percolation and glassiness in the random Lorentz gas.

The random Lorentz gas (RLG) is a minimal model of transport in heterogeneous media that exhibits a continuous localization transition controlled by void space percolation. The RLG also provides a toy model of particle caging, which is known to be relevant for describing the discontinuous dynamical transition of glasses. In order to clarify the interplay between the seemingly incompatible percolation and caging descriptions of the RLG, we consider its exact mean-field solution in the infinite-dimensional d→∞ limit and perform numerics in d=2...20. We find that for sufficiently high d the mean-field caging transition precedes and prevents the percolation transition, which only happens on timescales diverging with d. We further show that activated processes related to rare cage escapes destroy the glass transition in finite dimensions, leading to a rich interplay between glassiness and percolation physics. This advance suggests that the RLG can be used as a toy model to develop a first-principle description of particle hopping in structural glasses.

Searching for the Gardner Transition in Glassy Glycerol.

We search for a Gardner transition in glassy glycerol, a standard molecular glass, measuring the third harmonics cubic susceptibility χ_{3}^{(3)} from slightly below the usual glass transition temperature down to 10 K. According to the mean-field picture, if local motion within the glass were becoming highly correlated due to the emergence of a Gardner phase then χ_{3}^{(3)}, which is analogous to the dynamical spin-glass susceptibility, should increase and diverge at the Gardner transition temperature T_{G}. We find instead that upon cooling |χ_{3}^{(3)}| decreases by several orders of magnitude and becomes roughly constant in the regime 100-10 K. We rationalize our findings by assuming that the low temperature physics is described by localized excitations weakly interacting via a spin-glass dipolar pairwise interaction in a random magnetic field. Our quantitative estimations show that the spin-glass interaction is twenty to fifty times smaller than the local random field contribution, thus rationalizing the absence of the spin-glass Gardner phase. This hints at the fact that a Gardner phase may be suppressed in standard molecular glasses, but it also suggests ways to favor its existence in other amorphous solids and by changing the preparation protocol.

Complex interactions can create persistent fluctuations in high-diversity ecosystems.

When can ecological interactions drive an entire ecosystem into a persistent non-equilibrium state, where many species populations fluctuate without going to extinction? We show that high-diversity spatially heterogeneous systems can exhibit chaotic dynamics which persist for extremely long times. We develop a theoretical framework, based on dynamical mean-field theory, to quantify the conditions under which these fluctuating states exist, and predict their properties. We uncover parallels with the persistence of externally-perturbed ecosystems, such as the role of perturbation strength, synchrony and correlation time. But uniquely to endogenous fluctuations, these properties arise from the species dynamics themselves, creating feedback loops between perturbation and response. A key result is that fluctuation amplitude and species diversity are tightly linked: in particular, fluctuations enable dramatically more species to coexist than at equilibrium in the very same system. Our findings highlight crucial differences between well-mixed and spatially-extended systems, with implications for experiments and their ability to reproduce natural dynamics. They shed light on the maintenance of biodiversity, and the strength and synchrony of fluctuations observed in natural systems.

Attractive versus truncated repulsive supercooled liquids: The dynamics is encoded in the pair correlation function.

We compare glassy dynamics in two liquids that differ in the form of their interaction potentials. Both systems have the same repulsive interactions but one has also an attractive part in the potential. These two systems exhibit very different dynamics despite having nearly identical pair correlation functions. We demonstrate that a properly weighted integral of the pair correlation function, which amplifies the subtle differences between the two systems, correctly captures their dynamical differences. The weights are obtained from a standard machine learning algorithm.

Jamming transition as a paradigm to understand the loss landscape of deep neural networks.

Deep learning has been immensely successful at a variety of tasks, ranging from classification to artificial intelligence. Learning corresponds to fitting training data, which is implemented by descending a very high-dimensional loss function. Understanding under which conditions neural networks do not get stuck in poor minima of the loss, and how the landscape of that loss evolves as depth is increased, remains a challenge. Here we predict, and test empirically, an analogy between this landscape and the energy landscape of repulsive ellipses. We argue that in fully connected deep networks a phase transition delimits the over- and underparametrized regimes where fitting can or cannot be achieved. In the vicinity of this transition, properties of the curvature of the minima of the loss (the spectrum of the Hessian) are critical. This transition shares direct similarities with the jamming transition by which particles form a disordered solid as the density is increased, which also occurs in certain classes of computational optimization and learning problems such as the perceptron. Our analysis gives a simple explanation as to why poor minima of the loss cannot be encountered in the overparametrized regime. Interestingly, we observe that the ability of fully connected networks to fit random data is independent of their depth, an independence that appears to also hold for real data. We also study a quantity Δ which characterizes how well (Δ<0) or badly (Δ>0) a datum is learned. At the critical point it is power-law distributed on several decades, P_{+}(Δ)∼Δ^{θ} for Δ>0 and P_{-}(Δ)∼(-Δ)^{-γ} for Δ<0, with exponents that depend on the choice of activation function. This observation suggests that near the transition the loss landscape has a hierarchical structure and that the learning dynamics is prone to avalanche-like dynamics, with abrupt changes in the set of patterns that are learned.

Gardner physics in amorphous solids and beyond.

One of the most remarkable predictions to emerge out of the exact infinite-dimensional solution of the glass problem is the Gardner transition. Although this transition was first theoretically proposed a generation ago for certain mean-field spin glass models, its materials relevance was only realized when a systematic effort to relate glass formation and jamming was undertaken. A number of nontrivial physical signatures associated with the Gardner transition have since been considered in various areas, from models of structural glasses to constraint satisfaction problems. This perspective surveys these recent advances and discusses the novel research opportunities that arise from them.

Dynamics around the site percolation threshold on high-dimensional hypercubic lattices.

Recent advances on the glass problem motivate reexamining classical models of percolation. Here we consider the displacement of an ant in a labyrinth near the percolation threshold on cubic lattices both below and above the upper critical dimension of simple percolation, d_{u}=6. Using theory and simulations, we consider the scaling regime and obtain that both caging and subdiffusion scale logarithmically for d≥d_{u}. The theoretical derivation, which considers Bethe lattices with generalized connectivity and a random graph model, confirms that logarithmic scalings should persist in the limit d→∞. The computational validation employs accelerated random walk simulations with a transfer-matrix description of diffusion to evaluate directly the dynamical critical exponents below d_{u} as well as their logarithmic scaling above d_{u}. Our numerical results improve various earlier estimates and are fully consistent with our theoretical predictions.