Bounds for diluted mean-fields spin glass models

Panchenko, Dmitry; Talagrand, Michel

doi:10.1007/s00440-004-0342-2

Cited by 124 publications

(174 citation statements)

References 11 publications

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“…There is the analogue of the Guerra upper bound proved by Franz, Leone in [40] (see Panchenko, Talagrand [64] for another exposition). There is a natural analogue of the cavity method (see Panchenko [75]), so proving the matching lower bound reduces to demonstrating the structure of Gibbs distribution predicted by Mézard and Parisi.…”

Section: Multimentioning

confidence: 97%

Introduction to the SK model

Panchenko

2014

Current Developments in Mathematics

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Section: Multimentioning

confidence: 97%

Introduction to the SK model

Panchenko

2014

Current Developments in Mathematics

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“…In particular, it yields the conjecture that the location of the satisfiability threshold is actually exact. From a rigorous point of view it was proven that the 1RSB equations give an upper bound on the satisfiability threshold [FL03,FLT03,PT04].…”

Section: Clustering and Survey Propagationmentioning

confidence: 99%

Statistical physics of hard optimization problems

Zdeborová¹

2009

Acta Physica Slovaca. Reviews and Tutorials

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Statistical physics of hard optimization problemsOptimization is fundamental in many areas of science, from computer science and information theory to engineering and statistical physics, as well as to biology or social sciences. It typically involves a large number of variables and a cost function depending on these variables. Optimization problems in the non-deterministic polynomial (NP)-complete class are particularly difficult, it is believed that the number of operations required to minimize the cost function is in the most difficult cases exponential in the system size. However, even in an NP-complete problem the practically arising instances might, in fact, be easy to solve. The principal question we address in this article is: How to recognize if an NP-complete constraint satisfaction problem is typically hard and what are the main reasons for this? We adopt approaches from the statistical physics of disordered systems, in particular the cavity method developed originally to describe glassy systems. We describe new properties of the space of solutions in two of the most studied constraint satisfaction problems - random satisfiability and random graph coloring. We suggest a relation between the existence of the so-called frozen variables and the algorithmic hardness of a problem. Based on these insights, we introduce a new class of problems which we named "locked" constraint satisfaction, where the statistical description is easily solvable, but from the algorithmic point of view they are even more challenging than the canonical satisfiability.

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“…Let us also chose ξ τ = exp(−βH SK M (τ )). Then, if we call R B (M ) the Boltzmann ROSt just defined, we can prove ( [1]) the following Theorem 6 (Reversed Bound) If we plug the Boltzmann ROSt R B (M ) just defined above into the trial function G N defined by (14), then the Boltzmann trial function G N (R B (M )) provides, in the thermodynamic limit, a lower bound for the SK pressure −βf…”

Section: Disordered Mean Field Models Of Spin Glassesmentioning

confidence: 98%

“…Still, as we showed, the structure of dilute models is similar to the one of the others. See [7,14,4] for recent progress on the subject.…”

Section: Dilute Spin Glassesmentioning

confidence: 99%

Structural Properties of the Disordered Spherical and Other Mean Field Spin Models

Sanctis

2006

J Stat Phys

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We extend the approach of Aizenman, Sims and Starr for the SK-type models to their spherical versions. Such an extension has already been performed for diluted spin glasses. The factorization property of the optimal structures found by Guerra for the SK model, which holds for diluted models as well, is verified also in the case of spherical systems, with the due modifications. Hence we show that there are some common structural features in various mean field spin models. These similarities seem to be quite paradigmatic, and we summarize the various techniques typically used to prove the structural analogies and to tackle the computation of the free energy per spin in the thermodynamic limit.

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Bounds for diluted mean-fields spin glass models

Cited by 124 publications

References 11 publications

Introduction to the SK model

Introduction to the SK model

Statistical physics of hard optimization problems

Structural Properties of the Disordered Spherical and Other Mean Field Spin Models

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