Rapid mixing of Swendsen-Wang dynamics in two dimensions

We prove comparison results for the Swendsen-Wang (SW) dynamics, the heat-bath (HB) dynamics for the Potts model and the single-bond (SB) dynamics for the random-cluster model on arbitrary graphs. In particular, we prove that rapid mixing of HB implies rapid mixing of SW on graphs with bounded maximum degree and that rapid mixing of SW and rapid mixing of SB are equivalent. Additionally, the spectral gap of SW and SB on planar graphs is bounded from above and from below by the spectral gap of these dynamics on the corresponding dual graph with suitably changed temperature. As a consequence we obtain rapid mixing of the Swendsen-Wang dynamics for the Potts model on the two-dimensional square lattice at all non-critical temperatures as well as rapid mixing for the two-dimensional Ising model at all temperatures. Furthermore, we obtain new results for general graphs at high or low enough temperatures.Comment: Ph.D. thesis, 66 page

The arctangent law for a certain random time related to a one-dimensional diffusion

For a time-homogeneous, one-dimensional diffusion process $X(t),$ we investigate the distribution of the first instant, after a given time $r,$ at which $X(t)$ exceeds its maximum on the interval $[0,r],$ generalizing a result of Papanicolaou, which is valid for Brownian motion

What general practitioners need to know about patent foramen ovale

A patent foramen ovale (PFO) consists of a hole between the right and left atriums of the heart that did not close the way it should after birth. Twenty five percent of the population have a PFO, but this usually does not cause problems, because the opening is functionally closed by the difference in pressure between the heart and the chest. This study is a literature review about the clinical significance of PFO and its management in three clinical situations: cryptogenic strokes, migraine with aura and scuba divers who sustained a decompression sickness. PFOs had been linked with various medical conditions such as strokes, migraine, and with certain types of decompression sickness (DCS). In general, this association is not very well established. Young patients who sustain a cardiovascular event without a known cause (cryptogenic stroke) have resulted in the tendency to screen these patents becoming the norm and more PFOs are being closed using standard methods and devices. The association of PFOs and migraine attacks is less clear. In the case of scuba divers the risk of suffering from a decompression accident is increased if one has a PFO. The management of these patients remains difficult.peer-reviewe