Abrupt Convergence and Escape Behavior for Birth and Death Chains

We link two phenomena concerning the asymptotical behavior of stochastic processes: (i) abrupt convergence or cut-off phenomenon, and (ii) the escape behavior usually associated to exit from metastability. The former is characterized by convergence at asymptotically deterministic times, while the convergence times for the latter are exponentially distributed. We compare and study both phenomena for discrete-time birth-and-death chains on Z with drift towards zero. In particular, this includes energy-driven evolutions with energy functions in the form of a single well. Under suitable drift hypotheses, we show that there is both an abrupt convergence towards zero and escape behavior in the other direction. Furthermore, as the evolutions are reversible, the law of the final escape trajectory coincides with the time reverse of the law of cut-off paths. Thus, for evolutions defined by one-dimensional energy wells with sufficiently steep walls, cut-off and escape behavior are related by time inversion.Comment: 2 figure

Probing surfaces with thermal He atoms: scattering and microscopy with a soft touch

Helium atom scattering (HAS) is a well established technique, particularly suited for the investigation of insulating and/or fragile materials and light adsorbates including hydrogen. In contrast to other beam techniques based on Xrays or electrons, low energy (typically less than 100 meV) He atoms are scattered by the tail of the electron density distribution which spill out from a surface, therefore HAS is strictly a nonpenetrating technique without any sample damage. HAS has been used to investigate structural properties of crystalline surfaces, including precise determination of atomic step heights, for monitoring thin film growth, to study surface transitions such as surface melting and roughening and for determining the presence and properties of adsorbates. Energy resolved HAS can provide information about surface vibrations (phonons) in the meV range and surface diffusion. This chapter provides a brief introduction to HAS with an outlook on a new, promising surface science technique: Neutral Helium Microscopy