A simple one dimensional glassy Kac model

Abstract

We define a new family of random spin models with one-dimensional structure, finite-range multi-spin interactions, and bounded average degree (number of interactions in which each spin participates). Unfrustrated ground states can be described as solutions of a sparse, band diagonal linear system, thus allowing for efficient numerical analysis. In the limit of infinite interaction range, we recover the so-called XORSAT (diluted p-spin) model, that is known to undergo a random first order phase transition as the average degree is increased. Here we investigate the most important consequences of a large but finite interaction range: (i) Fluctuation-induced corrections to thermodynamic quantities; (ii) The need of an inhomogeneous (position dependent) order parameter; (iii) The emergence of a finite mosaic length scale. In particular, we study the correlation length divergence at the (mean-field) glass transition.

25 pages, 9 eps figures

References in corpus (5)

• Gibbs States and the Set of Solutions of Random Constraint Satisfaction Problems
• Rigorous Inequalities between Length and Time Scales in Glassy Systems
• Analytic determination of dynamical and mosaic length scales in a Kac glass model
• Local Spin Glass Order in 1D
• Percolation of satisfiability in finite dimensions