Convergence to extremal processes in random environments and extremal ageing in SK models

This paper extends recent results on ageing in mean field spin glasses on short time scales, obtained by Ben Arous and Gün (Commun Pure Appl Math 65:77–127, 2012) in law with respect to the environment, to results that hold almost surely, respectively in probability, with respect to the environment. It is based on the methods put forward in (Gayrard in Aging in reversible dynamics of disordered systems. II. Emergence of the arcsine law in the random hopping time dynamics of the REM, 2010; Electron J Probab 17(58): 1–33, 2012) and naturally complements (Bovier and Gayrard in Ann Probab, 2012).