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

Long-range power-law correlated percolation is investigated using Monte Carlo simulations. We obtain several static and dynamic critical exponents as functions 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.

Discretized landscapes can be mapped onto ranked surfaces, where every element (site or bond) has a unique rank associated with its corresponding relative height. By sequentially allocating these elements according to their ranks and systematically preventing the occupation of bridges, namely elements that, if occupied, would provide global connectivity, we disclose that bridges hide a new tricritical point at an occupation fraction p = pc, where pc is the percolation threshold of random percolation. For any value of p in the interval pc < p ≤ 1, our results show that the set of bridges has a fractal dimension dBB ≈ 1.22 in two dimensions. In the limit p → 1, a self-similar fracture is revealed as a singly connected line that divides the system in two domains. We then unveil how several seemingly unrelated physical models tumble into the same universality class and also present results for higher dimensions.

The Gaussian model of discontinuous percolation, recently introduced by Araújo 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 ± 0.005. The largest cluster at the threshold is compact, but its external perimeter is fractal with fractal dimension dA = 2.5 ± 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.

We present a numerical calculation of the total number of disordered jammed configurations Ω of N repulsive, three-dimensional spheres in a fixed volume V. To make these calculations tractable, we increase the computational efficiency of the approach of Xu et al. [Phys. Rev. Lett. 106, 245502 (2011)10.1103/PhysRevLett.106.245502] and Asenjo et al. [Phys. Rev. Lett. 112, 098002 (2014)10.1103/PhysRevLett.112.098002] and we extend the method to allow computation of the configurational entropy as a function of pressure. The approach that we use computes the configurational entropy by sampling the absolute volume of basins of attraction of the stable packings in the potential energy landscape. We find a surprisingly strong correlation between the pressure of a configuration and the volume of its basin of attraction in the potential energy landscape. This relation is well described by a power law. Our methodology to compute the number of minima in the potential energy landscape should be applicable to a wide range of other enumeration problems in statistical physics, string theory, cosmology, and machine learning that aim to find the distribution of the extrema of a scalar cost function that depends on many degrees of freedom.

The BFW model introduced by Bohman, Frieze, and Wormald [Random Struct. Algorithms, 25, 432 (2004)] and recently investigated in the framework of discontinuous percolation by Chen and D'Souza [Phys. Rev. Lett., 106, 115701 (2011)], is studied on the square and simple-cubic lattices. In two and three dimensions, we find numerical evidence for a strongly discontinuous transition. In two dimensions, the clusters at the threshold are compact with a fractal surface of fractal dimension d f = 1.49 ± 0.02. On the simple-cubic lattice, distinct jumps in the size of the largest cluster are observed. We proceed to analyze the tree-like version of the model, where only merging bonds are sampled, for dimension two to seven. The transition is again discontinuous in any considered dimension. Finally, the dependence of the cluster-size distribution at the threshold on the spatial dimension is also investigated.

In the late 1980s, Sam Edwards proposed a possible statisticalmechanical framework to describe the properties of disordered granular materials 1 . A key assumption underlying the theory was that all jammed packings are equally likely. In the intervening years it has never been possible to test this bold hypothesis directly. Here we present simulations that provide direct evidence that at the unjamming point, all packings of soft repulsive particles are equally likely, even though generically, jammed packings are not. Typically, jammed granular systems are observed precisely at the unjamming point since grains are not very compressible. Our results therefore support Edwards' original conjecture. We also present evidence that at unjamming the configurational entropy of the system is maximal.In science, most breakthroughs cannot be derived from known physical laws: they are based on inspired conjectures 2 . Comparison with experiment of the predictions based on such a hypothesis allows us to eliminate conjectures that are clearly wrong. However, there is a distinction between testing the consequences of a conjecture and testing the conjecture itself. A case in point is Edwards' theory of granular media. In the late 1980s, Edwards and Oakeshott 1 proposed that many of the physical properties of granular materials ('powders') could be predicted using a theoretical framework that was based on the assumption that all distinct packings of such a material are equally likely to be observed. The logarithm of the number of such packings was postulated to play the same role as entropy does in Gibbs' statistical-mechanical description of the thermodynamic properties of equilibrium systems. However, statistical-mechanical entropy and granular entropy are very different objects. Until now, the validity of Edwards' hypothesis could not be tested directlymainly because the number of packings involved is so large that direct enumeration is utterly infeasible-and, as a consequence, the debate about the Edwards hypothesis has focused on its consequences, rather than on its assumptions. Here we present results that show that now, at last, it is possible to test Edwards' hypothesis directly by numerical simulation. To our own surprise, we find that the hypothesis appears to be correct precisely at the point where a powder is just at the (un)jamming threshold. However, at higher densities, the hypothesis fails. At the unjamming transition, the configurational entropy of jammed states appears to be at a maximum.The concept of 'ensembles' plays a key role in equilibrium statistical mechanics, as developed by J. Willard Gibbs, well over a century ago 3 . The crucial assumption that Gibbs made to arrive at a tractable theoretical framework to describe the equilibrium properties of gases, liquid and solids was that, at a fixed total energy, every state of the system is equally likely to be observed. The distinction between, say, a liquid at thermal equilibrium and a granular material is that in a liquid, atoms undergo thermal motion whereas...

We show that in the continuum limit watersheds dividing drainage basins are Schramm-Loewner Evolution (SLE) curves, being described by one single parameter κ. Several numerical evaluations are applied to ascertain this. All calculations are consistent with SLEκ, with κ = 1.734 ± 0.005, being the only known physical example of an SLE with κ < 2. This lies outside the well-known duality conjecture, bringing up new questions regarding the existence and reversibility of dual models. Furthermore it constitutes a strong indication for conformal invariance in random landscapes and suggests that watersheds likely correspond to a logarithmic Conformal Field Theory (CFT) with central charge c ≈ −7/2.PACS numbers: 89.75. Da, 64.60.al, 91.10.Jf The possibility of statistically describing the properties of random curves with a single parameter fascinates physicists and mathematicians alike. This capability is provided by the theory of Schramm-Loewner Evolution (SLE), where random curves can be generated from a Brownian motion with diffusivity κ [1]. Once κ is identified, several geometrical properties of the curve are known (e.g. fractal dimension, winding angle, and leftpassage probability) [2,3]. Among the examples of such curves, we find self-avoiding walks [4] and the contours of critical clusters in percolation [5], Q-state Potts model [6], and spin glasses [7], as well as in turbulence [8]. Establishing SLE for such systems has provided valuable information on the underlying symmetries and paved the way to some exact results [5,9,10]. In fact, SLE is not a general property of non-self-crossing walks since many curves have been shown not to be SLE as, for example, the interface of solid-on-solid models [11], the domain walls of bimodal spin glasses [12], and the contours of negative-weight percolation [13].Recently, the watershed (WS) of random landscapes [14][15][16], with a fractal dimension d f ≈ 1.22, was shown to be related to a family of curves appearing in different contexts such as, e.g., polymers in strongly disordered media [17], bridge percolation [14], and optimal path cracks [18]. In the present Letter, we show that this universal curve has the properties of SLE, with κ = 1.734 ± 0.005. κ < 2 is a special limit since, up to now, all known examples of SLE found in Nature and statistical physics models have 2 ≤ κ ≤ 8, corresponding to fractal dimensions d f between 1.25 and 2.Scale invariance and, consequently, the appearance of fractal dimensions have always motivated to apply concepts from conformal invariance to shed light on critical systems. Archetypes of self-similarity are the contours of critical clusters in lattice models. Already back in 1923, Loewner proposed an expression for the evolution of an analytic function which conformally maps the region bounded by these curves into a standard domain [19]. Such an evolution, follows the theory, should only depend on a continuous function of a real parameter, known as driving function. Recently, Schramm argued that to guarantee conformal invariance, and...