  In this paper we study finite interaction range corrections to the mosaic
picture of the glass transition as 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 lenght. Nonetheless we find very slow convergence to the
Kac limit and we discuss why this could be a problem for measuring the mosaic
lenght in realistic models.
