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

TL;DR

This paper investigates how finite interaction ranges affect the mosaic picture of the glass transition in a one-dimensional disordered model, highlighting slow convergence issues in measuring the mosaic length.

Contribution

It introduces a detailed analysis of finite-range corrections to the mosaic length in a 1D random energy model, emphasizing the slow convergence to the Kac limit.

Findings

Overlap curves cross near the mosaic length at finite ranges.

Convergence to the Kac limit is very slow.

Measuring the mosaic length in realistic models may be challenging.

Abstract

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.