Critical points and singularities are encountered in the study of critical phenomena in probability and physics. We present recent results concerning the values of such critical points and the nature of the singularities for two prominent probabilistic models, namely percolation and the more general random-cluster model. The main topic is the statement and proof of the criticality and universal...

The formation of topological defects in a second order phase transition in the early universe is an out-of-equilibrium process. Condensed matter experiments seem to support Zurek’s mechanism, in which the freezing of thermal fluctuations close to the critical point (critical slowing down) plays a crucial role. We discuss how this picture can be extrapolated to the early universe, pointing out t...

The nature of the enigmatic pseudogap region of the phase diagram is the most important and intriguing unsolved puzzle in the field of high transitiontemperature (Tc) superconductivity. This region, the temperature range above Tc and below a characteristic temperature T*, is characterized by highly anomalous magnetic, charge transport, thermodynamic and optical properties [1-2]. Associated with...

The Random Cluster Model offers an interesting reformulation of the Ising and Potts Models in the language of percolation theory. In one regime, the model obeys Positive Association, which has broad implications. Another prominent property of the Random Cluster Model is the existence of a critical point, separating two phases with and without infinite clusters, however much is still unknown or ...

We report grand canonical Monte Carlo simulations of the critical point properties of homopolymers within the Bond Fluctuation model. By employing Configurational Bias Monte Carlo methods, chain lengths of up to N = 60 monomers could be studied. For each chain length investigated, the critical point parameters were determined by matching the ordering operator distribution function to its univer...

A nonlinear response theory is provided by use of the transient linearization method in the spatially one-dimensional Vlasov systems. The theory inclusively gives responses to external fields and to perturbations for initial stationary states, and is applicable even to the critical point of a second-order phase transition. We apply the theory to the Hamiltonian mean-field model, a toy model of ...

Morse theory was originally due to Marston Morse [5]. It gives us a method to study the topology of a manifold using the information of the critical points of a Morse function defined on the manifold. Based on the same idea, Morse homology was introduced by Thom, Smale, Milnor, and Witten in various forms. This paper is a survey of some work in this direction. The first part of the paper focuse...