Abstract. We define a new family of random spin models with one-dimensional structure, finite-range multispin 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.

Key words: disordered systems (theory); structural glasses (theory); exact results

E-print number: 0705.0054
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Received 7 May 2007, accepted for publication 11 June 2007
Published 1 August 2007