Statistical mechanics and disordered systems

1985 ◽  
Vol 28 (4) ◽  
pp. 363-373 ◽  
Author(s):  
Scott Kirkpatrick ◽  
Robert H. Swendsen
Keyword(s):  
Statistical Mechanics ◽  
Disordered Systems

scholarly journals The Statistical Mechanics of Random Set Packing and a Generalization of the Karp-Sipser Algorithm

2014 ◽  
Vol 2014 ◽  
pp. 1-13 ◽  
Author(s):  
C. Lucibello ◽  
F. Ricci-Tersenghi
Keyword(s):  
Statistical Mechanics ◽  
Disordered Systems ◽  
Independent Set ◽  
Maximum Matching ◽  
Cavity Method ◽  
Set Packing ◽  
Leaf Removal ◽  
Symmetric Cavity ◽  
Random Instances ◽  
Analytic Prediction

We analyse the asymptotic behaviour of random instances of the maximum set packing (MSP) optimization problem, also known as maximum matching or maximum strong independent set on hypergraphs. We give an analytic prediction of the MSPs size using the 1RSB cavity method from statistical mechanics of disordered systems. We also propose a heuristic algorithm, a generalization of the celebrated Karp-Sipser one, which allows us to rigorously prove that the replica symmetric cavity method prediction is exact for certain problem ensembles and breaks down when a core survives the leaf removal process. The e-phenomena threshold discovered by Karp and Sipser, marking the onset of core emergence and of replica symmetry breaking, is elegantly generalized to Cs=e/(d-1) for one of the ensembles considered, where d is the size of the sets.

INFORMATION PROCESSING BY A PERCEPTRON

1992 ◽  
Vol 03 (supp01) ◽  
pp. 41-50 ◽  
Author(s):  
J.-P. Nadal ◽  
N. Parga
Keyword(s):  
Information Theory ◽  
Information Processing ◽  
Mutual Information ◽  
Statistical Mechanics ◽  
Disordered Systems ◽  
Information Capacity ◽  
Global System

The information coming into a module, which is part of a global system, will in general require pre-processing that consists in building a representation — expressing it in a new code — convenient to the task to be performed by this particular module. This information, coming either from the environment or from another module in the system, will undergo this operation in what we can call an encoder. In this work we describe our results for a perceptron architecture viewed as an encoder, using encoding principles based on Information Theory. In particular we show how to evaluate the information capacity and the typical mutual information, quantities which are relevant to analize the different criteria used to code the information. Techniques taken from statistical mechanics of disordered systems will be shown to be useful for these calculations.

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