Random-field Ising-like effective theory of the glass transition. I. Mean-field models

Abstract

In this paper and in the companion one we address the problem of identifying the effective theory that describes the statistics of the fluctuations of what is thought to be the relevant order parameter for glassy systems---the overlap field with an equilibrium reference configuration---close to the putative thermodynamic glass transition. Our starting point is the mean-field theory of glass formation which relies on the existence of a complex free-energy landscape with a multitude of metastable states. In this paper, we focus on archetypal mean-field models possessing this type of free-energy landscape and set up the framework to determine the exact effective theory. We show that the effective theory at the mean-field level is generically of the random-field + random-bond Ising type. We also discuss what are the main issues concerning the extension of our result to finite-dimensional systems. This extension is addressed in detail in the companion paper.