RESUMEN
On approaching the dynamical transition temperature, supercooled liquids show heterogeneity over space and time. Static replica theory investigates the dynamical crossover in terms of the free energy landscape (FEL). Two kinds of static approaches have provided a self-consistent equation for determining this crossover, similar to the mode coupling theory for glassy dynamics. One uses the Morita-Hiroike formalism of the liquid state theory, whereas the other relies on the density functional theory (DFT). Each of the two approaches has advantages in terms of perturbative field theory. Here, we develop a replica field theory that has the benefits from both formulations. We introduce the generalized Franz-Parisi potential to formulate a correlation functional. Considering fluctuations around an inhomogeneous density determined by the Ramakrishnan-Yussouf DFT, we find a new closure as the stability condition of the correlation functional. The closure leads to the self-consistent equation involving the triplet direct correlation function. The present field theory further helps us study the FEL beyond the mean-field approximation.
RESUMEN
A phase transition in high-dimensional random geometry is analyzed as it arises in a variety of problems. A prominent example is the feasibility of a minimax problem that represents the extremal case of a class of financial risk measures, among them the current regulatory market risk measure Expected Shortfall. Others include portfolio optimization with a ban on short-selling, the storage capacity of the perceptron, the solvability of a set of linear equations with random coefficients, and competition for resources in an ecological system. These examples shed light on various aspects of the underlying geometric phase transition, create links between problems belonging to seemingly distant fields, and offer the possibility for further ramifications.
RESUMEN
The study of the properties of glass-forming liquids is difficult for many reasons. Analytic solutions of mean-field models are usually available only for systems embedded in a space with an unphysically high number of spatial dimensions; on the experimental and numerical side, the study of the properties of metastable glassy states requires thermalizing the system in the supercooled liquid phase, where the thermalization time may be extremely large. We consider here a hard-sphere mean-field model that is solvable in any number of spatial dimensions; moreover, we easily obtain thermalized configurations even in the glass phase. We study the 3D version of this model and we perform Monte Carlo simulations that mimic heating and cooling experiments performed on ultrastable glasses. The numerical findings are in good agreement with the analytical results and qualitatively capture the features of ultrastable glasses observed in experiments.