Mosaic length and finite interaction-range effects in a one-dimensional random energy model

In this paper, we study finite interaction-range corrections to the mosaic picture of the glass transition as it emerges from the study of the Kac limit of large interaction range for disordered models. To this aim we consider point-to-set correlation functions, or overlaps, in a one-dimensional random energy model as a function of the range of interaction. In the Kac limit, the mosaic length defines a sharp first-order transition separating a high overlap phase from a low overlap one. Correspondingly, we find that overlap curves as a function of the window size and different finite interaction ranges cross roughly at the mosaic length. Nonetheless, we find a very slow convergence to the Kac limit and discuss why this could be a problem for measuring the mosaic length in realistic models.