Self-Organized Critical Directed Percolation

We introduce and study a dynamic transport model exhibiting Self-Organized Criticality. The novel concepts of our model are the probabilistic propagation of activity and unbiased random repartition of energy among the active site and its nearest neighbors. For space dimensionality $d\geq 2$ we argue that the model is related to $d+1$ dimensional directed percolation, with time interpreted as the preferred direction.Comment: 4 two-column pages (revtex), 3 ps figures included with epsf, g-zipped, uuencode

Local persistence in directed percolation

We reconsider the problem of local persistence in directed site percolation. We present improved estimates of the persistence exponent in all dimensions from 1+1 to 7+1, obtained by new algorithms and by improved implementations of existing ones. We verify the strong corrections to scaling for 2+1 and 3+1 dimensions found in previous analyses, but we show that scaling is much better satisfied for very large and very small dimensions. For d > 4 (d is the spatial dimension), the persistence exponent depends non-trivially on d, in qualitative agreement with the non-universal values calculated recently by Fuchs {\it et al.} (J. Stat. Mech.: Theor. Exp. P04015 (2008)). These results are mainly based on efficient simulations of clusters evolving under the time reversed dynamics with a permanently active site and a particular survival condition discussed in Fuchs {\it et al.}. These simulations suggest also a new critical exponent $\zeta$ which describes the growth of these clusters conditioned on survival, and which turns out to be the same as the exponent, \eta+\delta in standard notation, of surviving clusters under the standard DP evolution.Comment: 6 pages, including 4 figures; to appear in JSTA

Polymer collapse and crystallization in bond fluctuation models

While the $\Theta$-collapse of single long polymers in bad solvents is usually a continuous (tri-critical) phase transition, there are exceptions where it is preempted by a discontinuous crystallization (liquid $\leftrightarrow$ solid) transition. For a version of the bond-fluctuation model (a model where monomers are represented as $2\times 2\times 2$ cubes, and bonds can have lengths between 2 and $\sqrt{10}$) it was recently shown by F. Rampf {\it et al.} that there exist distinct collapse and crystallization transitions for long but {\it finite} chains. But as the chain length goes to infinity, both transition temperatures converge to the same $T^*$, i.e. infinitely long polymers collapse immediately into a solid state. We explain this by the observation that polymers crystallize in the Rampf {\it et al.} model into a non-trivial cubic crystal structure (the `A15' or `Cr$_3$Si' Frank-Kasper structure) which has many degenerate ground states and, as a consequence, Bloch walls. If one controlls the polymer growth such that only one ground state is populated and Bloch walls are completely avoided, the liquid-solid transition is a smooth cross-over without any sharp transition at all.Comment: 4 page

Morphological transitions in supercritical generalized percolation and moving interfaces in media with frozen randomness

We consider the growth of clusters in disordered media at zero temperature, as exemplified by supercritical generalized percolation and by the random field Ising model. We show that the morphology of such clusters and of their surfaces can be of different types: They can be standard compact clusters with rough or smooth surfaces, but there exists also a completely different "spongy" phase. Clusters in the spongy phase are `compact' as far as the size-mass relation M ~ R^D is concerned (with D the space dimension), but have an outer surface (or `hull') whose fractal dimension is also D and which is indeed dense in the interior of the entire cluster. This behavior is found in all dimensions D >= 3. Slightly supercritical clusters can be of either type in $D=3$, while they are always spongy in D >= 4. Possible consequences for the applicability of KPZ (Kardar-Parisi-Zhang) scaling to interfaces in media with frozen randomness are studied in detail.Comment: 12 pages, including 10 figures; improved data & major changes compared to v

Critical phenomena on k-booklets

We define a `k-booklet' to be a set of k semi-infinite planes with $-\infty < x < \infty$ and $y \geq 0$, glued together at the edges (the `spine') y=0. On such booklets we study three critical phenomena: Self-avoiding random walks, the Ising model, and percolation. For k=2 a booklet is equivalent to a single infinite lattice, for k=1 to a semi-infinite lattice. In both these cases the systems show standard critical phenomena. This is not so for k>2. Self avoiding walks starting at y=0 show a first order transition at a shifted critical point, with no power-behaved scaling laws. The Ising model and percolation show hybrid transitions, i.e. the scaling laws of the standard models coexist with discontinuities of the order parameter at $y\approx 0$, and the critical points are not shifted. In case of the Ising model ergodicity is already broken at $T=T_c$, and not only for $T<T_c$ as in the standard geometry. In all three models correlations (as measured by walk and cluster shapes) are highly anisotropic for small y.Comment: 5 pages, 8 figure

Pair Connectedness and Shortest Path Scaling in Critical Percolation

We present high statistics data on the distribution of shortest path lengths between two near-by points on the same cluster at the percolation threshold. Our data are based on a new and very efficient algorithm. For $d=2$ they clearly disprove a recent conjecture by M. Porto et al., Phys. Rev. {\bf E 58}, R5205 (1998). Our data also provide upper bounds on the probability that two near-by points are on different infinite clusters.Comment: 7 pages, including 4 postscript figure

Percolation transitions in the survival of interdependent agents on multiplex networks, catastrophic cascades, and SOS

The "SOS" in the title does not refer to the international distress signal, but to "solid-on-solid" (SOS) surface growth. The catastrophic cascades are those observed by Buldyrev {\it et al.} in interdependent networks, which we re-interpret as multiplex networks with agents that can only survive if they mutually support each other, and whose survival struggle we map onto an SOS type growth model. This mapping not only reveals non-trivial structures in the phase space of the model, but also leads to a new and extremely efficient simulation algorithm. We use this algorithm to study interdependent agents on duplex Erd\"os-R\'enyi (ER) networks and on lattices with dimensions 2, 3, 4, and 5. We obtain new and surprising results in all these cases, and we correct statements in the literature for ER networks and for 2-d lattices. In particular, we find that $d=4$ is the upper critical dimension, that the percolation transition is continuous for $d\leq 4$ but -- at least for $d\neq 3$ -- not in the universality class of ordinary percolation. For ER networks we verify that the cluster statistics is exactly described by mean field theory, but find evidence that the cascade process is not. For $d=5$ we find a first order transition as for ER networks, but we find also that small clusters have a nontrivial mass distribution that scales at the transition point. Finally, for $d=2$ with intermediate range dependency links we propose a scenario different from that proposed in W. Li {\it et al.}, PRL {\bf 108}, 228702 (2012).Comment: 19 pages, 32 figure

Self-trapping self-repelling random walks

Although the title seems self-contradictory, it does not contain a misprint. The model we study is a seemingly minor modification of the "true self-avoiding walk" (TSAW) model of Amit, Parisi, and Peliti in two dimensions. The walks in it are self-repelling up to a characteristic time $T^*$ (which depends on various parameters), but spontaneously (i.e., without changing any control parameter) become self-trapping after that. For free walks, $T^*$ is astronomically large, but on finite lattices the transition is easily observable. In the self-trapped regime, walks are subdiffusive and intermittent, spending longer and longer times in small areas until they escape and move rapidly to a new area. In spite of this, these walks are extremely efficient in covering finite lattices, as measured by average cover times.Comment: 5 pages main paper + 5 pages supplementary materia