E l e c t r o n i c J o u r n a l o f P r o b a b i l i t y Electron. AbstractThe subcritical contact process seen from the rightmost infected site has no invariant measures. We prove that nevertheless it converges in distribution to a quasi-stationary measure supported on finite configurations.

Given a countable set of sites and a collection of flip rates at each site, we give a sufficient condition on the long-range dependancies of the flip rates ensuring the well-definedness of the corresponding spin system. This hypothesis has already been widely used ([1], [2], [3], [4]) but our construction brings a new insight to understand why it is natural. The process is first constructed as a limit of finite spin systems. Then we identify its generator and give a simple criterion for a measure to be invariant with respect to it.

E l e c t r o n i c J o u r n a l o f P r o b a b i l i t y Electron. AbstractWe study a crystal growth Markov model proposed by Gates and Westcott ([1], [2]). This is an aggregation process where particles are packed in a square lattice accordingly to prescribed deposition rates. This model is parametrized by three values (βi, i = 0, 1, 2) corresponding to deposition rates on three different types of loci.The main problem is to determine, for the shape of the crystal, when recurrence and when ergodicity do occur. In [3] and [4] sufficient conditions are given both for ergodicity and transience. We establish some improved conditions and give a precise description of the asymptotic behavior in a special case.

No abstract