scholarly journals Exact Phase Transitions in Random Constraint Satisfaction Problems

2000 ◽  
Vol 12 ◽  
pp. 93-103 ◽  
Author(s):  
K. Xu ◽  
W. Li
Keyword(s):  
Phase Transitions ◽  
Standard Model ◽  
Constraint Satisfaction ◽  
Constraint Satisfaction Problems ◽  
Critical Values ◽  
The Standard Model ◽  
New Type ◽  
Almost All ◽  
Model Rb ◽  
Random Constraint Satisfaction Problems

In this paper we propose a new type of random CSP model, called Model RB, which is a revision to the standard Model B. It is proved that phase transitions from a region where almost all problems are satisfiable to a region where almost all problems are unsatisfiable do exist for Model RB as the number of variables approaches infinity. Moreover, the critical values at which the phase transitions occur are also known exactly. By relating the hardness of Model RB to Model B, it is shown that there exist a lot of hard instances in Model RB.

scholarly journals Theoretical uncertainties for cosmological first-order phase transitions

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
Djuna Croon ◽  
Oliver Gould ◽  
Philipp Schicho ◽  
Tuomas V. I. Tenkanen ◽  
Graham White
Keyword(s):  
Phase Transitions ◽  
Standard Model ◽  
Scale Dependence ◽  
Effective Field ◽  
Bubble Nucleation ◽  
Perturbative Approach ◽  
First Order ◽  
The Standard Model ◽  
First Order Phase ◽  
Renormalisation Scale

Abstract We critically examine the magnitude of theoretical uncertainties in perturbative calculations of fist-order phase transitions, using the Standard Model effective field theory as our guide. In the usual daisy-resummed approach, we find large uncertainties due to renormalisation scale dependence, which amount to two to three orders-of-magnitude uncertainty in the peak gravitational wave amplitude, relevant to experiments such as LISA. Alternatively, utilising dimensional reduction in a more sophisticated perturbative approach drastically reduces this scale dependence, pushing it to higher orders. Further, this approach resolves other thorny problems with daisy resummation: it is gauge invariant which is explicitly demonstrated for the Standard Model, and avoids an uncontrolled derivative expansion in the bubble nucleation rate.

The replica symmetric phase of random constraint satisfaction problems

2019 ◽  
Vol 29 (3) ◽  
pp. 346-422
Author(s):  
Amin Coja-Oghlan ◽  
Tobias Kapetanopoulos ◽  
Noela Müller
Keyword(s):  
Phase Transition ◽  
Constraint Satisfaction ◽  
Broad Class ◽  
Constraint Satisfaction Problems ◽  
Precise Location ◽  
Combinatorial Structures ◽  
Substantial Impact ◽  
Long Range Correlations ◽  
Symmetric Phase ◽  
Random Constraint Satisfaction Problems

AbstarctRandom constraint satisfaction problems play an important role in computer science and combinatorics. For example, they provide challenging benchmark examples for algorithms, and they have been harnessed in probabilistic constructions of combinatorial structures with peculiar features. In an important contribution (Krzakala et al. 2007, Proc. Nat. Acad. Sci.), physicists made several predictions on the precise location and nature of phase transitions in random constraint satisfaction problems. Specifically, they predicted that their satisfiability thresholds are quite generally preceded by several other thresholds that have a substantial impact both combinatorially and computationally. These include the condensation phase transition, where long-range correlations between variables emerge, and the reconstruction threshold. In this paper we prove these physics predictions for a broad class of random constraint satisfaction problems. Additionally, we obtain contiguity results that have implications for Bayesian inference tasks, a subject that has received a great deal of interest recently (e.g. Banks et al. 2016, Proc. 29th COLT).

scholarly journals CRITICALITY AND HETEROGENEITY IN THE SOLUTION SPACE OF RANDOM CONSTRAINT SATISFACTION PROBLEMS

2010 ◽  
Vol 24 (18) ◽  
pp. 3479-3487 ◽  
Author(s):  
HAIJUN ZHOU
Keyword(s):  
Constraint Satisfaction ◽  
Spin Glasses ◽  
Solution Space ◽  
Threshold Value ◽  
Critical Value ◽  
Constraint Satisfaction Problems ◽  
Model Systems ◽  
Dynamical Heterogeneity ◽  
Ergodicity Breaking ◽  
Random Constraint Satisfaction Problems

Random constraint satisfaction problems are interesting model systems for spin-glasses and glassy dynamics studies. As the constraint density of such a system reaches certain threshold value, its solution space may split into extremely many clusters. In this work we argue that this ergodicity-breaking transition is preceded by a homogeneity-breaking transition. For random K-SAT and K-XORSAT, we show that many solution communities start to form in the solution space as the constraint density reaches a critical value αcm, with each community containing a set of solutions that are more similar with each other than with the outsider solutions. At αcm the solution space is in a critical state. The connection of these results to the onset of dynamical heterogeneity in lattice glass models is discussed.

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