arXiv subjects

Peter Grassberger

Publications and source records attributed to Peter Grassberger.

At least 19 recordsLinked to original sources

arXiv2025

High-Precision Simulations of the Parity Conserving Directed Percolation Universality Class in 1+1 Dimensions

Next to the directed percolation (DP) universality class, parity conserving directed percolation (pcDP; also called parity conserving branching annihilating random walks, pcBARW) 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 it 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 $\beta$ (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 pcPD clusters. In particular, we find that all exponents which were close to rationals are even closer, and $\beta = 1.000$ with the error in the next digit.

cond-mat.stat-mech↗

arXiv2024

Extreme-value statistics and super-universality in critical percolation?

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.

cond-mat.stat-mech↗

arXiv2024

On three papers by Jurgens \& Crutchfield, and on the basic structure of "computational mechanics"

In a recent paper, Jurgens and Crutchfield [Phys. Rev. E {\bf 104}, 064107 (2021), called ``paper III" in the following] computed what they called the ``ambiguity rate" of hidden Markov processes, a concept supposedly introduced by Claude Shannon. This calculation was based on a ``mixed state" formalism introduced by them in J. Stat. Phys. {\bf 183}, 32 (2021) (``paper I"), and developed further in Chaos, {\bf 31}, 083114 (2021) (``paper II"). We point out that (i) ambiguity rates were {\it not} introduced by Shannon; (ii) their computations in paper III are wrong, because of an error made already in papers I and II; (iii) due to this error (a confusion between open sets and their closures), also many of the ``statistical complexity dimensions" computed in II are wrong; (iv) the ``mixed state" formalism of I is just the well known `forward algorithm' for hidden Markov models; (v) the `causal states' in `$\epsilon$-machines' correspond in general to {\it finite} (as opposed to infinite, as often claimed) histories; and (vi) `$\epsilon$-machines' are always countable, in contrast to frequent claims in the literature. In addition, we propose an alternative complexity measure for models where the forecasting complexity is infinite, and we point out that our results apply also beyond hidden Markov models.

cond-mat.stat-mech↗

arXiv2023

Aftermath Epidemics: Percolation on the Sites Visited by Generalized Random Walks

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 (RW) in 2 dimensions and generalized knight's move RW in 3d). In these walks, the visited sites do not form (as in ordinary RW) a single connected cluster, and thus percolation on them is non-trivial. 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 2-d knight's move RW and Levy flights with Levy parameter $σ\geq 2$. For 3-d generalized knight's move RW, the model is in the universality class of Pacman percolation, and all critical exponents seem to be simple rationals, in particular $β=1$. For 2-d 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 determine unambiguously the critical exponents.

cond-mat.stat-mech↗

arXiv2023

Many universality classes in an interface model restricted to non-negative heights

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 (EW) or in the Kardar-Parisi-Zhang (KPZ) universality class. In addition, there is also a constraint $n(x,t) \geq 0$. 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 of a novel type for pushed fronts, with yet another novel behavior 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 $\langle n(x,t)\rangle \to$ const on one side, and $\to \infty$ 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.

cond-mat.stat-mech↗

arXiv2022

Kardar-Parisi-Zhang type dynamics with periodic tilt dependence of the propagation velocity in 1+1 dimensions

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.

cond-mat.stat-mech↗

arXiv2021

On Generalized Schürmann Entropy Estimators

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. Schuermann, which itself is a generalization of an estimator proposed by myself in arXiv:physics/0307138. For a special set of parameters they are completely free of bias and have a finite variance, something with 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.

cs.IT↗

arXiv2021

Revisiting a Low-Dimensional Model with Short Range Interactions and Mean Field Critical Behavior

In all 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 {\it 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 feed-backs that can occur in non-equilibrium systems.

cond-mat.stat-mech↗

arXiv2021

Trust Me If You Can: Trusted Transformation Between (JSON) Schemas to Support Global Authentication of Education Credentials

Recruiters and institutions around the world struggle with the verification of diplomas issued in a diverse and global education setting. Firstly, it is a nontrivial problem to identify bogus institutions selling education credentials. While institutions are often accredited by qualified authorities on a regional level, there is no global authority fulfilling this task. Secondly, many different data schemas are used to encode education credentials, which represents a considerable challenge to automated processing. Consequently, significant manual effort is required to verify credentials. In this paper, we tackle these challenges by introducing a decentralized and open system to automatically verify the legitimacy of issuers and interpret credentials in unknown schemas. We do so by enabling participants to publish transformation information, which enables verifiers to transform credentials into their preferred schema. Due to the lack of a global root of trust, we utilize a distributed ledger to build a decentralized web of trust, which verifiers can query to gather information on the trustworthiness of issuing institutions and to establish trust in transformation information. Going beyond diploma fraud, our system can be generalized to tackle the generalized problem for other domains lacking a root of trust and agreements on data schemas.

cs.CY↗

arXiv2021

Chase-Escape Percolation on the 2D Square Lattice

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 with non-zero probability, if $p>p_c$ with $p_c<1$. Using Monte Carlo simulations on the square lattice, we estimate the value of $p_c = 0.49451 \pm 0.00001$, and the critical exponents are consistent with the undirected percolation universality class. We define a discrete-time parallel-update version of the model, which brings out the relation between chase-escape and undirected bond percolation. For all $p < p_c$ in $D$-dimensions, the number of predators in the absorbing configuration has a stretched-exponential distribution in contrast to the exponential distribution in the standard percolation theory. We also study the problem starting from the line 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. At $p=1$, the fronts undergo a depinning transition. The fluctuations of the front follow Kardar-Parisi-Zhang scaling both above and below this depinning transition.

cond-mat.stat-mech↗

arXiv2020

Swarming transitions in hierarchical societies

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.

physics.bio-ph↗

arXiv2020

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.

cond-mat.stat-mech↗

arXiv2018

Some comments on computational mechanics, complexity measures, and all that

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.

physics.data-an↗

arXiv2017

Comment on "Inferring Statistical Complexity"

Nearly 30 years ago, J.P. Crutchfield and K. Young proposed in Phys. Rev. Lett. {\bf 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.

cond-mat.stat-mech↗

arXiv2017

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.

cond-mat.stat-mech↗

arXiv2017

How fast does a random walk cover a torus?

We present high statistics simulation data for the average time $\langle T_{\rm cover}(L)\rangle$ that a random walk needs to cover completely a 2-dimensional torus of size $L\times L$. They confirm the mathematical prediction that $\langle T_{\rm cover}(L)\rangle \sim (L \ln L)^2$ for large $L$, but the prefactor {\it seems} to deviate significantly from the supposedly exact result $4/π$ derived by A. Dembo {\it et al.}, Ann. Math. {\bf 160}, 433 (2004), if the most straightforward extrapolation is used. On the other hand, we find that this scaling does hold for the time $ T_{\rm N(t)=1}(L)$ at which the average number of yet unvisited sites is 1, as also predicted previously. This might suggest (wrongly) that $\langle T_{\rm cover}(L)\rangle$ and $T_{\rm N(t)=1}(L)$ scale differently, although the distribution of rescaled cover times becomes sharp in the limit $L\to\infty$. But our results can be reconciled with those of Dembo {\it et al.} by a very slow and {\it non-monotonic} convergence of $\langle T_{\rm cover}(L)\rangle/(L \ln L)^2$, as had been indeed proven by Belius {\it et al.} [Prob. Theory \& Related Fields {\bf 167}, 1 (2014)] for Brownian walks, and was conjectured by them to hold also for lattice walks.

cond-mat.stat-mech↗

arXiv2017

Percolation in Media with Columnar Disorder

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.

cond-mat.dis-nn↗

arXiv2017

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.

cond-mat.stat-mech↗