GREM-like K processes on trees with infinite depth

Abstract

We take up the issue of deriving the limit as $n\to \infty$ of the GREM-like K process on a tree with $n$ levels. Under specific conditions on the parameters of the process, implying the martingality of a modification of the underlying clock process sequences, we obtain infinite level clock processes as nontrivial limits of the finite level clocks, and use them to construct a process on a suitable product space which is then shown to be the limit of the $n$ level K processes as $n\to \infty$. Some properties of the limiting, infinite level K process are established, like an expression for the asymptotic empirical measure of cylinders, giving information on the prospective equilibrium measure of the infinite level dynamics.

42 pages, 2 figures

References in corpus (1)

• K-processes, scaling limit and aging for the trap model in the complete graph