Existence and static stability of a capillary free surface appearing in a dewetted Bridgman process. I

This paper present six theoretical results concerning the existence and static stability of a capillary free surface appearing in a dewetted Bridgman crystal growth technique. The results are obtained in an axis symmetric 2D model for semiconductors for which the sum of wetting angle and growth angle is less than 180. Numerical results are presented in case of InSb semiconductor growth. The reported results can help, the practical crystal growers, in better understanding the dependence of the free surface shape and size on the pressure difference across the free surface and prepare the appropriate seed size, and thermal conditions before seeding the growth process.Comment: This is an extended version of the conference paper TIM 19 of 10pages and 9 figure

Relaxed sector condition

In this note we present a new sufficient condition which guarantees martingale approximation and central limit theorem a la Kipnis-Varadhan to hold for additive functionals of Markov processes. This condition which we call the relaxed sector condition (RSC) generalizes the strong sector condition (SSC) and the graded sector condition (GSC) in the case when the self-adjoint part of the infinitesimal generator acts diagonally in the grading. The main advantage being that the proof of the GSC in this case is more transparent and less computational than in the original versions. We also hope that the RSC may have direct applications where the earlier sector conditions don't apply. So far we don't have convincing examples in this direction.Comment: 11 page

Fast graphs for the random walker

Consider the time T_oz when the random walk on a weighted graph started at the vertex o first hits the vertex set z. We present lower bounds for T_oz in terms of the volume of z and the graph distance between o and z. The bounds are for expected value and large deviations, and are asymptotically sharp. We deduce rate of escape results for random walks on infinite graphs of exponential or polynomial growth, and resolve a conjecture of Benjamini and Peres.Comment: 22 page

Brownian beads

We show that the past and future of half-plane Brownian motion at certain cutpoints are independent of each other after a conformal transformation. Like in Ito's excursion theory, the pieces between cutpoints form a Poisson process with respect to a local time. The size of the path as a function of this local time is a stable subordinator whose index is given by the exponent of the probability that a stretch of the path has no cutpoint. The index is computed and equals 1/2.Comment: 24 pages, 1 figur