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Magnetic ordering at low temperature for Ising ferromagnets manifests itself within the associated Fortuin-Kasteleyn (FK) random cluster representation as the occurrence of a single positive density percolating network. In this paper we investigate the percolation signature for Ising spin glass ordering -- both in short-range (EA) and infinite-range (SK) models -- within a two-replica FK representation and also within the different Chayes-Machta-Redner two-replica graphical representation. Based on numerical studies of the $pm J$ EA model in three dimensions and on rigorous results for the SK model, we conclude that the spin glass transition corresponds to the appearance of {it two} percolating clusters of {it unequal} densities.
Understanding the physics of glass formation remains one of the major unsolved challenges of condensed matter science. As a material solidifies into a glass, it exhibits a spectacular slowdown of the dynamics upon cooling or compression, but at the s
We study in detail the quantum Sherrington-Kirkpatrick (SK) model, i.e. the infinite-range Ising spin glass in a transverse field, by solving numerically the effective one-dimensional model that the quantum SK model can be mapped to in the thermodyna
It has been shown recently that predictions from Mode-Coupling Theory for the glass transition of hard-spheres become increasingly bad when dimensionality increases, whereas replica theory predicts a correct scaling. Nevertheless if one focuses on th
In this paper we introduce a new algorithm to study some NP-complete problems. This algorithm is a Markov Chain Monte Carlo (MCMC) inspired by the cavity method developed in the study of spin glass. We will focus on the maximum clique problem and we
We perform Monte-Carlo simulations to study the Bernoulli ($p$) bond percolation on the enhanced binary tree which belongs to the class of nonamenable graphs with one end. Our numerical results show that the system has two different percolation thres