Communication: Evidence for non-ergodicity in quiescent states of periodically sheared suspensions.

We present simulations of an equilibrium statistical-mechanics model that uniformly samples the space of quiescent states of a periodically sheared suspension. In our simulations, we compute the structural properties of this model as a function of density. We compare the results of our simulations with the structural data obtained in the corresponding non-equilibrium model of Corté et al. [Nat. Phys. 4, 420 (2008)]. We find that the structural properties of the non-equilibrium model are very different from those of the equilibrium model, even though the two models have exactly the same set of accessible states. This observation shows that the dynamical protocol does not sample all quiescent states with equal probability. In particular, we find that, whilst quiescent states prepared in a non-equilibrium protocol can be hyperuniform [see D. Hexner and D. Levine, Phys. Rev. Lett. 114, 110602 (2015); E. Tjhung and L. Berthier, Phys. Rev. Lett. 114, 148301 (2015); and J. H. Weijs et al., Phys. Rev. Lett. 115, 108301 (2015)], ergodic sampling never leads to hyperuniformity. In addition, we observe ordering phase transitions and a percolation transition in the equilibrium model that do not show up in the non-equilibrium model. Conversely, the quiescent-to-diffusive transition in the dynamical model does not correspond to a phase transition, nor a percolation transition, in the equilibrium model.This work was supported by ERC Advanced Grant 227758 (COLSTRUCTION), EPSRC Programme Grant EP/I001352/1 and by the Swiss National Science Foundation (Grant No. P2EZP2-152188 and No. P300P2- 161078). K.J.S. acknowledges useful discussions with Nuno Araújo, Tristan Cragnolini, Daphne Klotsa, Erik Luijten, Stefano Martiniani, Anđela Šarić, Iskra Staneva, Jacob Stevenson, and Peter Wirnsberger.This is the author accepted manuscript. It is currently under an indefinite embargo pending publication by AIP

Gaussian model of explosive percolation in three and higher dimensions

The Gaussian model of discontinuous percolation, recently introduced by Ara\'ujo and Herrmann [Phys. Rev. Lett., 105, 035701 (2010)], is numerically investigated in three dimensions, disclosing a discontinuous transition. For the simple-cubic lattice, in the thermodynamic limit, we report a finite jump of the order parameter, $J=0.415 \pm 0.005$. The largest cluster at the threshold is compact, but its external perimeter is fractal with fractal dimension $d_A = 2.5 \pm 0.2$. The study is extended to hypercubic lattices up to six dimensions and to the mean-field limit (infinite dimension). We find that, in all considered dimensions, the percolation transition is discontinuous. The value of the jump in the order parameter, the maximum of the second moment, and the percolation threshold are analyzed, revealing interesting features of the transition and corroborating its discontinuous nature in all considered dimensions. We also show that the fractal dimension of the external perimeter, for any dimension, is consistent with the one from bridge percolation and establish a lower bound for the percolation threshold of discontinuous models with finite number of clusters at the threshold

Recent advances and open challenges in percolation

Percolation is the paradigm for random connectivity and has been one of the most applied statistical models. With simple geometrical rules a transition is obtained which is related to magnetic models. This transition is, in all dimensions, one of the most robust continuous transitions known. We present a very brief overview of more than 60 years of work in this area and discuss several open questions for a variety of models, including classical, explosive, invasion, bootstrap, and correlated percolation

Percolation with long-range correlated disorder

Long-range power-law correlated percolation is investigated using Monte Carlo simulations. We obtain several static and dynamic critical exponents as function of the Hurst exponent $H$ which characterizes the degree of spatial correlation among the occupation of sites. In particular, we study the fractal dimension of the largest cluster and the scaling behavior of the second moment of the cluster size distribution, as well as the complete and accessible perimeters of the largest cluster. Concerning the inner structure and transport properties of the largest cluster, we analyze its shortest path, backbone, red sites, and conductivity. Finally, bridge site growth is also considered. We propose expressions for the functional dependence of the critical exponents on $H$