Comparison of bootstrap percolation models

Domain convexification: A simple model for invasion processes

We propose an invasion model where domains grow up to their convex hulls and merge when they overlap. This model can be seen as a continuum and isotropic counterpart of bootstrap percolation models. From numerical investigations of the model starting with randomly deposited overlapping disks on a plane, we find an invasion transition that occurs via macroscopic avalanches. The disk concentration threshold and the width of the transition are found to decrease as the system size is increased. Our results are consistent with a vanishing threshold in the limit of infinitely large system sizes. However, this limit could not be investigated by simulations. For finite initial concentrations of disks, the cluster size distribution presents a power-law tail characterized by an exponent that varies approximately linearly with the initial concentration of disks. These results at finite initial concentration open novel directions for the understanding of the transition in systems of finite size. Furthermore, we find that the domain area distribution has oscillations with discontinuities. In addition, the deviation from circularity of large domains is constant. Finally, we compare our results to experimental observations on de-adhesion of graphene induced by the intercalation of nanoparticles.

Finite-density effects in the Fredrickson-Andersen and Kob-Andersen kinetically-constrained models

We calculate the corrections to the thermodynamic limit of the critical density for jamming in the Kob-Andersen and Fredrickson-Andersen kinetically-constrained models, and find them to be finite-density corrections, and not finite-size corrections. We do this by introducing a new numerical algorithm, which requires negligible computer memory since contrary to alternative approaches, it generates at each point only the necessary data. The algorithm starts from a single unfrozen site and at each step randomly generates the neighbors of the unfrozen region and checks whether they are frozen or not. Our results correspond to systems of size greater than 10(7) × 10(7), much larger than any simulated before, and are consistent with the rigorous bounds on the asymptotic corrections. We also find that the average number of sites that seed a critical droplet is greater than 1.

Jamming transition of kinetically constrained models in rectangular systems

We theoretically calculate the average fraction of frozen particles in rectangular systems of arbitrary dimensions for the Kob-Andersen and Fredrickson-Andersen kinetically constrained models. We find the aspect ratio of the rectangle's length to width, which distinguishes short, square-like rectangles from long, tunnel-like rectangles, and show how changing it can effect the jamming transition. We find how the critical vacancy density converges to zero in infinite systems for different aspect ratios: For long and wide channels it decreases algebraically v(c) ~ W(-1/2) with the system's width W, while in square systems it decreases logarithmically vc) ~ 1/lnL with length L. Although derived for asymptotically wide rectangles, our analytical results agree with numerical data for systems as small as W ≈ 10.

Tricritical point in heterogeneous k-core percolation

k-core percolation is an extension of the concept of classical percolation and is particularly relevant to understanding the resilience of complex networks under random damage. A new analytical formalism has been recently proposed to deal with heterogeneous k-cores, where each vertex is assigned a local threshold k(i). In this Letter we identify a binary mixture of heterogeneous k-cores which exhibits a tricritical point. We investigate the new scaling scenario and calculate the relevant critical exponents, by analytical and computational methods, for Erdős-Rényi networks and 2D square lattices.

Generic two-phase coexistence, relaxation kinetics, and interface propagation in the quadratic contact process: simulation studies

The quadratic contact process is formulated as an adsorption-desorption model on a two-dimensional square lattice. It involves random adsorption at empty sites and correlated desorption requiring diagonally adjacent pairs of empty neighbors. We assess the model behavior utilizing kinetic Monte Carlo simulations. One finds generic two-phase coexistence between a low-coverage active steady state and a completely covered or "poisoned" absorbing steady state; i.e., both states are stable over a finite range of adsorption rates or "pressures." This behavior is in marked contrast to that for equilibrium phase separation. For spatially homogeneous systems, we provide a comprehensive characterization of the kinetics of relaxation to the steady states. We analyze rapid poisoning for higher pressures above an effective spinodal point terminating a metastable active state, nucleation-mediated poisoning in the metastable region, the dynamics of poisoned droplets within the two-phase coexistence region, and behavior reminiscent of bootstrap percolation dynamics for lower pressures. For spatially inhomogeneous systems, we analyze the propagation of planar interfaces between active and absorbing states, fully characterizing an orientation dependence which underlies the generic two-phase coexistence.

Culling avalanches in bootstrap percolation

We study the culling avalanches which occur after the "death" of a single randomly chosen site in a network where sites are unstable, and are culled, if they have coordination less than an integer parameter m. Avalanche distributions are presented for triangular and cubic lattices for values of m where the associated bootstrap transitions are either first or second order. In second order cases, the culling avalanche distribution is found to be exponential, while in first order cases it follows a power law. We present an exact relation between culling avalanches and conventional bootstrap percolation and show that a relation proposed by Manna [Physica A 261, 351 (1998)] can be a good approximation for strongly first order bootstrap transitions but not for continuous bootstrap transitions.