A central problem in the theory of glassy systems is the comprehension of metastability in presence of finite range interactions. While in mean field metastable state can have infinite life time, for finite interaction range metastable state must decay on time scales independent of the system size. In this talk I will discuss a recent theoretical approach to metastability based on the analysis of models with long but finite range Kac kind of interactions. These kind of models are well suited to study finite dimensional effects in asymptotic expansions around mean-field and allow to test in a simplified setting phenomenological theories of the glass transition adapting the mean-field theory of “random first order transition” to situations where metastability can only be observed on sufficiently short time scales. I will discuss a detailed picture coming from the analysis of recently proposed correlation functions, that describes in a unified way two different lengths characterizing the growth of correlation in the supercooled phase of these models. I will compare the results of 1D numerical simulations to the asymptotic analysis.

ICTP, Trieste