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 distinct percolation thresholds p c1 and p c2 . The mean cluster size diverges as p approaching to p c1 from below. The system is critical at all the points in the intermediate phase (p c1 < p < p c2 ) and there exist infinitely many infinite clusters. In this phase the corresponding fractal exponent continuously increases with p from zero to unity. Above p c2 the system has a unique infinite cluster.

Statistics of the free volume available to individual particles have previously been studied for simple and complex fluids, granular matter, amorphous solids, and structural glasses. Minkowski tensors provide a set of shape measures that are based on strong mathematical theorems and easily computed for polygonal and polyhedral bodies such as free volume cells (Voronoi cells). They characterize the local structure beyond the two-point correlation function and are suitable to define indices 0 ≤ β a,b ν ≤ 1 of local anisotropy. Here, we analyze the statistics of Minkowski tensors for configurations of simple liquid models, including the ideal gas (Poisson point process), the hard disks and hard spheres ensemble, and the Lennard-Jones fluid. We show that Minkowski tensors provide a robust characterization of local anisotropy, which ranges from β a,b ν ≈ 0.3 for vapor phases to β a,b ν → 1 for ordered solids. We find that for fluids, local anisotropy decreases monotonously with increasing free volume and randomness of particle positions. Furthermore, the local anisotropy indices β a,b ν are sensitive to structural transitions in these simple fluids, as has been previously shown in granular systems for the transition from loose to jammed bead packs.

We propose a generic scaling theory for critical phenomena that includes power-law and essential singularities in finite and infinite dimensional systems. In addition, we clarify its validity by analyzing the Potts model in a simple hierarchical network, where a saddle-node bifurcation of the renormalization-group fixed point governs the essential singularity.

The Frenkel-Kontorova (FK) model is known to exhibit so-called Aubry's transition which is a jamming or frictional transition at zero temperature. Recently we found similar transition at zero and finite temperatures in a superconducting Josephson junction array (JJA) on a square lattice under external magnetic field. In the present paper we discuss how these problems are related.at North Dakota State University on July 6, 2015 http://ptps.oxfordjournals.org/ Downloaded from * ) The FK model with fixed volume (length) and that under fixed pressure (external force) behave completely differently. 11), 12) The former is relevant in the context of friction (jamming) and frustration is in some sense stronger than the latter. In the latter case the response of the system with respect to the increments of the external force exhibits devil's stair case singularities. Note also that the Hookian spring force, which arises due to the harmonic potential, does not endure the natural spacing l by itself.

The evaporation-condensation transition of the Potts model on a square lattice is numerically investigated by the Wang-Landau sampling method. An intrinsically system-size-dependent discrete transition between supersaturation state and phase-separation state is observed in the microcanonical ensemble by changing constrained internal energy. We calculate the microcanonical temperature, as a derivative of microcanonical entropy, and condensation ratio, and perform a finite-size scaling of them to indicate the clear tendency of numerical data to converge to the infinite-size limit predicted by phenomenological theory for the isotherm lattice gas model.