Finite Size Corrections in Disordered Systems and Optimization Problems

Abstract

Disordered systems are actively investigated in the statistical physics community, their presence being ubiquitous in material science, biology, finance, information theory. Even the most simple models of disorder systems, e.g. the SherringtonKirkpatrik Spin-Glass, show a remarkably complex phenomenology and stimulated the development of powerful and general mathematical tools, namely the Replica Trick[8] and the Cavity Method[4], well suited to tackle those difficult problems. In many cases results obtained through these methods have been rigorously proven only decades later [12][1]. The scenario that arose in the context of mean field models, was quite surprising: below a certain critical temperature the Gibbs measure decomposes in an exponential number of pure states, organized in a hierarchical structure[8]. It is not quite understood if such a scenario holds in finite dimensional systems, the statistical physics community being sharply clustered around two conflicting hypothesis [2][10], and finding an answer to such a debated question promises to be one of the great challenges of the century. The current analytical techniques are not yet suited to be applied to the finite dimensional problems: the presence of many short sized and interconnected loops breaks the assumptions on which relies the aforementioned Cavity Method while the Replica machinery on finite dimensional systems becomes simply mathematically and numerically intractable. As a first step to tackle these tremendous problem, in my doctoral project I am going to investigate the effects of uncorrelated loops in well understood mean-field models. A preliminary investigation showed that the presence of loops is deeply connected to the O ( 1 N ) corrections to the thermodynamic free energy of these models, a quantity which can be obtained in the Replica framework through the gaussian corrections to the saddle point of the replicated free energy functional. Computing the some quantity using the Cavity Method is a conceptually more involved work and has been done only in a few simple systems[9]. Anyway both method evidence the deep relation between small and strongly correlated topological structures and the O ( 1 N ) corrections to free energy.

Cite this paper

@inproceedings{RicciTersenghi2013FiniteSC, title={Finite Size Corrections in Disordered Systems and Optimization Problems}, author={Federico Ricci-Tersenghi}, year={2013} }