Next to the directed-percolation (DP) universality class, parity-conserving directed-percolation (pcDP; also called parity-conserving branching-annihilating random walks) is the second-most important model with an absorbing state transition. Its distinction from ordinary DP is that particle number is conserved modulo 2, which implies that there are two distinct sectors in systems with a finite initial number of particles: Realizations with even and odd particle numbers show different scaling behaviors, and systems in the odd sector cannot die. An intriguing feature of pcDP is that some of its critical exponents seem to be very simple rational numbers. The most prominent is the one describing the average number of particles (or active sites) in the even sector, which is asymptotically constant. In contrast, the dynamical critical exponent (which is the same in both sectors) seems not close to any simple rational. Finally, the order parameter exponent β (which is also the same in both sectors) is, according to the most precise previous simulations, rather close to 1, but incompatible with it. We present high statistics simulations which clarify this situation, and which indicate several other intriguing properties of pcDP clusters. In particular, we find that all exponents which were close to rationals are even closer, and β = 1.000 with the error in the next digit.

Recently, the number of non-standard percolation models has proliferated. In all these models, there exists a phase transition at which long range connectivity is established, if local connectedness increases through a threshold p_c. In ordinary (site or bond) percolation on regular lattices, this is a well understood second-order phase transition with rather precisely known critical exponents, but there are non-standard models where the transitions are in different universality classes (i.e. with different exponents and scaling functions), or even are discontinuous or hybrid. It was recently claimed that certain scaling functions are in all such models given by extreme-value theory and thus independent of the precise universality class. This would lead to super-universality (even encompassing first-order transitions!) and would be a major break-through in the theory of phase transitions. We show that this claim is wrong.

In a series of three papers, Jurgens and Crutchfield recently proposed a supposedly novel method to compute entropies of hidden Markov models (HMMs), discussed in detail its relationship to iterated function systems, and applied it to compute their ``ambiguity rates", a concept supposedly introduced by Claude Shannon. We point out that the basic formalism is not new (it is the well known ``forward algorithm" for HMMs), and that all three papers have also serious other faults.

We present a simple one-dimensional stochastic model with three control parameters and a surprisingly rich zoo of phase transitions. At each (discrete) site x and time t, an integer n(x,t) satisfies a linear interface equation with added random noise. Depending on the control parameters, this noise may or may not satisfy the detailed balance condition, so that the growing interfaces are in the Edwards-Wilkinson or in the Kardar-Parisi-Zhang universality class. In addition, there is also a constraint n(x,t)≥0. Points x where n>0 on one side and n=0 on the other are called "fronts." These fronts can be "pushed" or "pulled," depending on the control parameters. For pulled fronts, the lateral spreading is in the directed percolation (DP) universality class, while it is in a different universality class for pushed fronts, and another universality class in between. In the DP case, the activity at each active site can in general be arbitrarily large, in contrast to previous realizations of DP. Finally, we find two different types of transitions when the interface detaches from the line n=0 (with 〈n(x,t)〉→const on one side, and →∞ on the other), again with new universality classes. We also discuss a mapping of this model to the avalanche propagation in a directed Oslo rice pile model in specially prepared backgrounds.

We study percolation on the sites of a finite lattice visited by a generalized random walk of finite length with periodic boundary conditions. More precisely, consider Levy flights and walks with finite jumps of length >1 [like Knight's move random walks (RWs) in two dimensions and generalized Knight's move RWs in 3D]. In these walks, the visited sites do not form (as in ordinary RWs) a single connected cluster, and thus percolation on them is nontrivial. The model essentially mimics the spreading of an epidemic in a population weakened by the passage of some devastating agent-like diseases in the wake of a passing army or of a hurricane. Using the density of visited sites (or the number of steps in the walk) as a control parameter, we find a true continuous percolation transition in all cases except for the 2D Knight's move RWs and Levy flights with Levy parameter σ≥2. For 3D generalized Knight's move RWs, the model is in the universality class of pacman percolation, and all critical exponents seem to be simple rationals, in particular, β=1. For 2D Levy flights with 0<σ<2, scale invariance is broken even at the critical point, which leads at least to very large corrections in finite-size scaling, and even very large simulations were unable to unambiguously determine the critical exponents.

We present a new class of estimators of Shannon entropy for severely undersampled discrete distributions. It is based on a generalization of an estimator proposed by T. Schürmann, which itself is a generalization of an estimator proposed by myself.For a special set of parameters, they are completely free of bias and have a finite variance, something which is widely believed to be impossible. We present also detailed numerical tests, where we compare them with other recent estimators and with exact results, and point out a clash with Bayesian estimators for mutual information.

We consider the evolution of interfaces with a diffusive term and a generalized Kardar-Parisi-Zhang (KPZ) non-linearity, which results in a propagation velocity that depends periodically on the tilt of the interface. Using large scale simulations of a model class with these properties in 1+1 dimensions, we show that the fluctuations are in general still in the KPZ universality class, but a new universality class seems to appear in the limit of weak non-linearity. We argue that this is the typical behavior of any interface model with periodic tilt dependence.

Chase-escape percolation is a variation of the standard epidemic spread models. In this model, each site can be in one of three states: unoccupied, occupied by a single prey, or occupied by a single predator. Prey particles spread to neighboring empty sites at rate p, and predator particles spread only to neighboring sites occupied by prey particles at rate 1, killing the prey particle that existed at that site. It was found that the prey can survive forever with non-zero probability, if p > p(c) with p(c) < 1. Earlier simulations showed that p(c) is very close to 1/2. Using Monte Carlo simulations in D = 2, we estimate the value of pc to be 0.49451 +/- 0.00001 and the critical exponents are consistent with the undirected percolation universality class. We check that at p(c), the correlation functions at large length scales are rotationally invariant. We define a discrete-time parallel-update version of the model, which brings out the relation between chase-escape and undirected bond percolation. We further show that for all p < p(c), in D dimensions, the probability that the number of predators in the absorbing configuration is greater than s is bounded from below by exp(-K(p)(-1)s(1/D)), where K is some p-independent constant. This is in contrast to the exponentially decaying cluster size distribution in the standard percolation theory. Even so, the scaling function for the cluster size distribution for p near p(c) decays exponentially: the stretched exponential behavior dominates for s >> s*, but s* diverges near p(c). We also study the problem starting from an initial condition with predator particles on all lattice points of the line y = 0 and prey particles on the line y = 1. In this case, for p(c) < p < 1, the center of mass of the fluctuating prey and predator fronts travel at the same speed. This speed is strictly smaller than the speed of an Eden front with the same value of p, but with no predators. This is caused by the prey sites at the leading edge being eaten up by predators. The fluctuations of the front follow KPZ scaling both above and below the depinning transition at p = 1. (C) 2021 Elsevier B.V. All rights reserved.

In all known local low-dimensional models, scaling at critical points deviates from mean-field behavior -with one possible exception. This exceptional model with "ordinary" behavior is an inherently non-equilibrium model studied some time ago by H.-M. Broker and myself. In simulations, its 2-dimensional version suggested that two critical exponents were mean-field, while a third one showed very small deviations. Moreover, the numerics agreed almost perfectly with an explicit mean-field model. In the present paper we present simulations with much higher statistics, both for 2d and 3d. In both cases we find that the deviations of all critical exponents from their mean-field values are non-leading corrections, and that the scaling is precisely of mean-field type. As in the original paper, we propose that the mechanism for this is "confusion", a strong randomization of the phases of feedbacks that can occur in non-equilibrium systems. Copyright (C) 2022 EPLA

Social hierarchy is central to decision-making in the coordinated movement of many swarming species. Here we propose a hierarchical swarm model in the spirit of the Vicsek model of self-propelled particles. We show that, as the hierarchy becomes important, the swarming transition changes from the weak first-order transition observed for egalitarian populations, to a stronger first-order transition for intermediately strong hierarchies, and finally the discontinuity reduces till vanish, where the order-disorder transition appears to be absent in the extremely despotic societies. Associated to this we observe that the spatial structure of the swarm, as measured by the correlation between the density and velocity fields, is strongly mediated by the hierarchy. A two-group model and vectorial noise are also studied for verification. Our results point out the particular relevance of the hierarchical structures to swarming transitions when doing specific case studies.

We consider the growth of clusters in disordered media at zero temperature, as exemplified by supercritical generalized percolation and by the T = 0 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 similar to R-D is concerned (with D being 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 Kardar-Parisi-Zhang (KPZ) scaling to interfaces in media with frozen pinning centers are studied in detail. In particular, we find-in contrast to KPZ-a weak-coupling phase in 2+1 dimensions.

Social hierarchy is central to decision-making such as the coordinated movement of many swarming species. Here we propose a hierarchical swarm model for collective motion in the spirit of the Vicsek model of self-propelled particles. We show that, as the hierarchy becomes important, the swarming transition changes dramatically from the weak first-order transition observed for egalitarian populations, to a stronger first-order transition for intermediately strong hierarchies, and finally to a second-order phase transition when approaching to the extremely despotic societies. Associated to this we observe that the spatial structure of the swarm, as measured by the correlation between the density and velocity fields, is strongly mediated by the hierarchy. A vectorial network model is developed that provides a correct explanation. A two-group model and vectorial noise are also studied to verify the robustness of the observations. Our results imply that diverse type of swarming transitions is possible, depending on the impact of hierarchy of the species under study.

Based on extensive simulations, we conjecture that critically pinned interfaces in two-dimensional isotropic random media with short-range correlations are always in the universality class of ordinary percolation. Thus, in contrast to interfaces in $>2$ dimensions, there is no distinction between fractal (i.e., percolative) and rough but nonfractal interfaces. Our claim includes interfaces in zero-temperature random field Ising models (both with and without spontaneous nucleation), in heterogeneous bootstrap percolation, and in susceptible-weakened-infected-removed epidemics. It does not include models with long-range correlations in the randomness and models where overhangs are explicitly forbidden (which would imply nonisotropy of the medium).

We study a generalization of site percolation on a simple cubic lattice, where not only single sites are removed randomly, but also entire parallel columns of sites. We show that typical clusters near the percolation transition are very anisotropic, with different scaling exponents for the sizes parallel and perpendicular to the columns. Below the critical point there is a Griffiths phase where cluster size distributions and spanning probabilities in the direction parallel to the columns have power-law tails with continuously varying non-universal powers. This region is very similar to the Griffiths phase in subcritical directed percolation with frozen disorder in the preferred direction, and the proof follows essentially the same arguments as in that case. But in contrast to directed percolation in disordered media, the number of active (“growth”) sites in a growing cluster at criticality shows a power law, while the probability of a cluster to continue to grow shows logarithmic behavior.

We define a "k-booklet" to be a set of k semi-infinite planes with -∞<x<∞ and y≥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, and for k=1 to a semi-infinite lattice. In both these cases the systems show standard critical phenomena. This is not so for k≥3. 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≈0, and the critical points are not shifted. In the 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.

We present simulations of a three-dimensional percolation model studied recently by K. J. Schrenk et al. [Phys. Rev. Lett. 116, 055701 (2016)PRLTAO0031-900710.1103/PhysRevLett.116.055701], obtained with a new and more efficient algorithm. They confirm most of their results in spite of larger systems and higher statistics used in the present Rapid Communication, but we also find indications that the results do not yet represent the true asymptotic behavior. The model is obtained by replacing the isotropic holes in ordinary Bernoulli percolation by randomly placed and oriented cylinders, with the constraint that the cylinders are parallel to one of the three coordinate axes. We also speculate on possible generalizations.

We present high statistics simulation data for the average time 〈T_{cover}(L)〉 that a random walk needs to cover completely a two-dimensional torus of size L×L. They confirm the mathematical prediction that 〈T_{cover}(L)〉∼(LlnL)^{2} for large L, but the prefactor seems to deviate significantly from the supposedly exact result 4/π derived by Dembo et al. [Ann. Math. 160, 433 (2004)ANMAAH0003-486X10.4007/annals.2004.160.433], if the most straightforward extrapolation is used. On the other hand, we find that this scaling does hold for the time T_{N(t)=1}(L) at which the average number of yet unvisited sites is 1, as also predicted previously. This might suggest (wrongly) that 〈T_{cover}(L)〉 and T_{N(t)=1}(L) scale differently, although the distribution of rescaled cover times becomes sharp in the limit L→∞. But our results can be reconciled with those of Dembo et al. by a very slow and nonmonotonic convergence of 〈T_{cover}(L)〉/(LlnL)^{2}, as had been indeed proven by Belius et al. [Probab. Theory Relat. Fields 167, 461 (2017)10.1007/s00440-015-0689-6] for Brownian walks, and was conjectured by them to hold also for lattice walks.

Nearly 30 years ago, J.P. Crutchfield and K. Young proposed in Phys. Rev. Lett. 63, 105 (1989) some supposedly novel measures of time series complexity, and their relations to existing concepts in nonlinear dynamical systems. At that time it seemed that the multiple faults of this paper would make it obsolete soon. Since this has not happened, and these faults still infest the literature on what is now called "computational mechanics", I want here to rectify the situation.

We comment on some conceptual and and technical problems related to computational mechanics, point out some errors in several papers, and straighten out some wrong priority claims. We present explicitly the correct algorithm for constructing a minimal unifilar hidden Markov model ("ϵ-machine") from a list of forbidden words and (exact) word probabilities in a stationary stochastic process, and we comment on inference when these probabilities are only approximately known. In particular we propose minimization of forecasting complexity as an alternative basis for statistical inference of time series, in contrast to the traditional maximum entropy principle. We present a simple and precise way of estimating excess entropy (aka "effective measure complexity". Most importantly, however, we clarify some basic conceptual problems. In particular, we show that there exist simple models (called "totally recurrent graphs") where none of the nodes of the "ϵ-machine" (the "causal states") corresponds to an element of a state (or history) space partition.

We point out a stunning time asymmetry in the short time cross correlations between intra-day and overnight volatilities (absolute values of log-returns of stock prices). While overnight volatility is significantly (and positively) correlated with the intra-day volatility during the \textit{following} day (allowing thus non-trivial predictions), it is much less correlated with the intra-day volatility during the \textit{preceding} day. While the effect is not unexpected in view of previous observations, its robustness and extreme simplicity are remarkable.