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Finding the mean of the total number N_{tot} of critical points for N-dimensional random energy landscapes is reduced to averaging the absolute value of characteristic polynomial of the corresponding Hessian. For any finite N we provide the…
We present a model of spheres moving in a high-dimensional compact space. We relate it to a mixed matrix model with a $O(N)$ invariant model plus a $P(N)$ invariant perturbation. We then study the low pressure regime by performing a…
We consider theoretically ultracold interacting bosonic atoms confined to quasi-one-dimensional ladder structures formed by optical lattices and coupled to the field of an optical cavity. The atoms can collect a spatial phase imprint during…
We analyze models of spin glasses on the two-dimensional square lattice by exploiting symmetry arguments. The replicated partition functions of the Ising and related spin glasses are shown to have many remarkable symmetry properties as…
We review connections between phase transitions in high-dimensional combinatorial geometry and phase transitions occurring in modern high-dimensional data analysis and signal processing. In data analysis, such transitions arise as abrupt…
We propose a novel computational strategy to study the glass transition of molecular fluids. Our approach combines the construction of simple yet realistic models with the development of Monte Carlo algorithms to accelerate equilibration…
Mean field-like approximations (including naive mean field, Bethe and Kikuchi and more general Cluster Variational Methods) are known to stabilize ordered phases at temperatures higher than the thermodynamical transition. For example, in…
We present a new method which uses Feynman-like diagrams to calculate the statistical quantities of embedded many-body random matrix problems. The method provides a promising alternative to existing techniques and offers many important…
By means of diffusion Monte Carlo calculations, we investigated the quantum phase transition between a superfluid and a Mott insulator for a system of hard-sphere bosons in a quasi one-dimensional optical lattice. For this continuous…
As one increases the concentration of a colloidal suspension, the system exhibits a dramatic increase in viscosity. Structurally, the system resembles a liquid, yet motions within the suspension are slow enough that it can be considered…
I describe in these notes the physical properties of one dimensional interacting quantum particles. In one dimension the combined effects of interactions and quantum fluctuations lead to a radically new physics quite different from the one…
The dynamics of supercooled liquid and glassy systems are usually studied within the Lagrangian representation, in which the positions and velocities of distinguishable interacting particles are followed. Within this representation,…
We propose to realize one-dimensional topological phases protected by SU($N$) symmetry using alkali or alkaline-earth atoms loaded into a bichromatic optical lattice. We derive a realistic model for this system and investigate it…
Ultracold atoms in optical lattices undergo a quantum phase transition from a superfluid to a Mott insulator as the lattice potential depth is increased. We describe an approximate theory of interacting bosons in optical lattices which…
In the spherical Poisson Boolean model, one takes the union of random balls centred on the points of a Poisson process in Euclidean $d$-space with $d \geq 2$. We prove that whenever the radius distribution has a finite $d$-th moment, there…
The phenomenom of emerging regular spectral features from random interactions is addressed in the context of the interacting boson model. A mean-field analysis links different regions of the parameter space with definite geometric shapes.…
States of thermal equilibrium of an infinite system of interacting particles in a Euclidean space are studied. The particles bear 'unbounded' spins with a given symmetric a priori distribution. The interaction between the particles is…
We use dynamic light scattering and computer simulations to study equilibrium dynamics and dynamic heterogeneity in concentrated suspensions of colloidal hard spheres. Our study covers an unprecedented density range and spans seven decades…
We study a colloidal suspension confined between two quasi-parallel walls as a model system for glass transitions in confined geometries. The suspension is a mixture of two particle sizes to prevent wall-induced crystallization. We use…
One of the most actively debated issues in the study of the glass transition is whether a mean-field description is a reasonable starting point for understanding experimental glass formers. Although the mean-field theory of the glass…