arXiv subjects

Janko Gravner

Publications and source records attributed to Janko Gravner.

35 records · Page 2Linked to original sources

arXiv2015

Percolation and disorder-resistance in cellular automata

We rigorously prove a form of disorder-resistance for a class of one-dimensional cellular automaton rules, including some that arise as boundary dynamics of two-dimensional solidification rules. Specifically, when started from a random initial seed on an interval of length $L$, with probability tending to one as $L\to\infty$, the evolution is a replicator. That is, a region of space-time of density one is filled with a spatially and temporally periodic pattern, punctuated by a finite set of other finite patterns repeated at a fractal set of locations. On the other hand, the same rules exhibit provably more complex evolution from some seeds, while from other seeds their behavior is apparently chaotic. A principal tool is a new variant of percolation theory, in the context of additive cellular automata from random initial states.

math.PR↗

arXiv2015

Introducing Lyapunov profiles of cellular automata

In line with the stability theory of continuous dynamical systems, Lyapunov exponents of cellular automata (CAs) have been conceived two decades ago to quantify to what extent their dynamics changes following a perturbation of their initial configuration. More precisely, Lyapunov exponents of CAs have either been understood as the rate by which the resulting defect cone widens as these dynamical systems are evolved from a perturbed initial configuration, or as the rate by which defects accumulate during their evolution. The former viewpoint yields insight into the extent of the affected region, whereas the latter tells us something about the intensity of the defect propagation. In this paper, we will show how these viewpoints can be united by relying on Lyapunov profiles of CAs.

math.DS↗

arXiv2015

Bootstrap percolation on products of cycles and complete graphs

Bootstrap percolation on a graph iteratively enlarges a set of occupied sites by adjoining points with at least $θ$ occupied neighbors. The initially occupied set is random, given by a uniform product measure, and we say that spanning occurs if every point eventually becomes occupied. The main question concerns the critical probability, that is, the minimal initial density that makes spanning likely. The graphs we consider are products of cycles of $m$ points and complete graphs of $n$ points. The major part of the paper focuses on the case when two factors are complete graphs and one factor is a cycle. We identify the asymptotic behavior of the critical probability and show that, when $θ$ is odd, there are two qualitatively distinct phases: the transition from low to high probability of spanning as the initial density increases is sharp or gradual, depending on the size of $m$.

math.PR↗

arXiv2015

Bootstrap percolation on the Hamming torus

The Hamming torus of dimension $d$ is the graph with vertices $\{1,\dots,n\}^d$ and an edge between any two vertices that differ in a single coordinate. Bootstrap percolation with threshold $θ$ starts with a random set of open vertices, to which every vertex belongs independently with probability $p$, and at each time step the open set grows by adjoining every vertex with at least $θ$ open neighbors. We assume that $n$ is large and that $p$ scales as $n^{-α}$ for some $α>1$, and study the probability that an $i$-dimensional subgraph ever becomes open. For large $θ$, we prove that the critical exponent $α$ is about $1+d/θ$ for $i=1$, and about $1+2/θ+Θ(θ^{-3/2})$ for $i\ge2$. Our small $θ$ results are mostly limited to $d=3$, where we identify the critical $α$ in many cases and, when $θ=3$, compute exactly the critical probability that the entire graph is eventually open.

math.PR↗

arXiv2014

Nucleation scaling in jigsaw percolation

Jigsaw percolation is a nonlocal process that iteratively merges connected clusters in a deterministic "puzzle graph" by using connectivity properties of a random "people graph" on the same set of vertices. We presume the Erdos--Renyi people graph with edge probability p and investigate the probability that the puzzle is solved, that is, that the process eventually produces a single cluster. In some generality, for puzzle graphs with N vertices of degrees about D (in the appropriate sense), this probability is close to 1 or small depending on whether pD(log N) is large or small. The one dimensional ring and two dimensional torus puzzles are studied in more detail and in many cases the exact scaling of the critical probability is obtained. The paper settles several conjectures posed by Brummitt, Chatterjee, Dey, and Sivakoff who introduced this model.

math.PR↗

arXiv2010

A sharper threshold for bootstrap percolation in two dimensions

Two-dimensional bootstrap percolation is a cellular automaton in which sites become 'infected' by contact with two or more already infected nearest neighbors. We consider these dynamics, which can be interpreted as a monotone version of the Ising model, on an n x n square, with sites initially infected independently with probability p. The critical probability p_c is the smallest p for which the probability that the entire square is eventually infected exceeds 1/2. Holroyd determined the sharp first-order approximation: p_c \sim π^2/(18 log n) as n \to \infty. Here we sharpen this result, proving that the second term in the expansion is -(log n)^{-3/2+ o(1)}, and moreover determining it up to a poly(log log n)-factor. The exponent -3/2 corrects numerical predictions from the physics literature.

math.PR↗

arXiv2008

Local Bootstrap Percolation

We study a variant of bootstrap percolation in which growth is restricted to a single active cluster. Initially there is a single active site at the origin, while other sites of Z^2 are independently occupied with small probability p, otherwise empty. Subsequently, an empty site becomes active by contact with 2 or more active neighbors, and an occupied site becomes active if it has an active site within distance 2. We prove that the entire lattice becomes active with probability exp[alpha(p)/p], where alpha(p) is between -pi^2/9 + c sqrt p and pi^2/9 + C sqrt p (-log p)^3. This corrects previous numerical predictions for the scaling of the correction term.

math.PR↗

arXiv2007

Modeling snow crystal growth III: three-dimensional snowfakes

We introduce a three-dimensional, computationally feasible, mesoscopic model for snow crystal growth, based on diffusion of vapor, anisotropic attachment, and a semi-liquid boundary layer. Several case studies are presented that faithfully emulate a wide variety of physical snowflakes.

math-ph↗

arXiv2007

Slow Convergence in Bootstrap Percolation

In the bootstrap percolation model, sites in an L by L square are initially infected independently with probability p. At subsequent steps, a healthy site becomes infected if it has at least 2 infected neighbours. As (L,p)->(infinity,0), the probability that the entire square is eventually infected is known to undergo a phase transition in the parameter p log L, occurring asymptotically at lambda = pi^2/18. We prove that the discrepancy between the critical parameter and its limit lambda is at least Omega((log L)^(-1/2)). In contrast, the critical window has width only Theta((log L)^(-1)). For the so-called modified model, we prove rigorous explicit bounds which imply for example that the relative discrepancy is at least 1% even when L = 10^3000. Our results shed some light on the observed differences between simulations and rigorous asymptotics.

math.PR↗

arXiv2006

Percolation on fitness landscapes: effects of correlation, phenotype, and incompatibilities

We study how correlations in the random fitness assignment may affect the structure of fitness landscapes. We consider three classes of fitness models. The first is a continuous phenotype space in which individuals are characterized by a large number of continuously varying traits such as size, weight, color, or concentrations of gene products which directly affect fitness. The second is a simple model that explicitly describes genotype-to-phenotype and phenotype-to-fitness maps allowing for neutrality at both phenotype and fitness levels and resulting in a fitness landscape with tunable correlation length. The third is a class of models in which particular combinations of alleles or values of phenotypic characters are "incompatible" in the sense that the resulting genotypes or phenotypes have reduced (or zero) fitness. This class of models can be viewed as a generalization of the canonical Bateson-Dobzhansky-Muller model of speciation. We also demonstrate that the discrete NK model shares some signature properties of models with high correlations. Throughout the paper, our focus is on the percolation threshold, on the number, size and structure of connected clusters, and on the number of viable genotypes.

q-bio.PE↗

arXiv2006

Random growth models with polygonal shapes

We consider discrete-time random perturbations of monotone cellular automata (CA) in two dimensions. Under general conditions, we prove the existence of half-space velocities, and then establish the validity of the Wulff construction for asymptotic shapes arising from finite initial seeds. Such a shape converges to the polygonal invariant shape of the corresponding deterministic model as the perturbation decreases. In many cases, exact stability is observed. That is, for small perturbations, the shapes of the deterministic and random processes agree exactly. We give a complete characterization of such cases, and show that they are prevalent among threshold growth CA with box neighborhood. We also design a nontrivial family of CA in which the shape is exactly computable for all values of its probability parameter.

math.PR↗

arXiv2002

Fluctuations in the composite regime of a disordered growth model

We continue to study a model of disordered interface growth in two dimensions. The interface is given by a height function on the sites of the one--dimensional integer lattice and grows in discrete time: (1) the height above the site $x$ adopts the height above the site to its left if the latter height is larger, (2) otherwise, the height above $x$ increases by 1 with probability $p_x$. We assume that $p_x$ are chosen independently at random with a common distribution $F$, and that the initial state is such that the origin is far above the other sites. Provided that the tails of the distribution $F$ at its right edge are sufficiently thin, there exists a nontrivial composite regime in which the fluctuations of this interface are governed by extremal statistics of $p_x$. In the quenched case, the said fluctuations are asymptotically normal, while in the annealed case they satisfy the appropriate extremal limit law.

math.PR↗

arXiv2001

A growth model in a random environment

We consider a model of interface growth in two dimensions, given by a height function on the sites of the one--dimensional integer lattice. According to the discrete time update rule, the height above the site $x$ increases to the height above $x-1$, if the latter height is larger; otherwise the height above $x$ increases by 1 with probability $p_x$. We assume that $p_x$ are chosen independently at random with a common distribution $F$, and that the initial state is such that the origin is far above the other sites. We explicitly identify the asymptotic shape and prove that, in the pure regime, the fluctuations about that shape, normalized by the square root of time, are asymptotically normal. This contrasts with the quenched version: conditioned on the environment, and normalized by the cube root of time, the fluctuations almost surely approach a distribution known from random matrix theory.

math.PR↗

arXiv2000

Limit Theorems for Height Fluctuations in a Class of Discrete Space and Time Growth Models

We introduce a class of one-dimensional discrete space-discrete time stochastic growth models described by a height function $h_t(x)$ with corner initialization. We prove, with one exception, that the limiting distribution function of $h_t(x)$ (suitably centered and normalized) equals a Fredholm determinant previously encountered in random matrix theory. In particular, in the universal regime of large $x$ and large $t$ the limiting distribution is the Fredholm determinant with Airy kernel. In the exceptional case, called the critical regime, the limiting distribution seems not to have previously occurred. The proofs use the dual RSK algorithm, Gessel's theorem, the Borodin-Okounkov identity and a novel, rigorous saddle point analysis. In the fixed $x$, large $t$ regime, we find a Brownian motion representation. This model is equivalent to the Seppäläinen-Johansson model. Hence some of our results are not new, but the proofs are.

math.PR↗

arXiv1993

Threshold-Range Scaling of Excitable Cellular Automata

Each cell of a two-dimensional lattice is painted one of k colors, arranged in a "color wheel." The colors advance (0 to k-1 mod k) either automatically or by contact with at least a threshold number of successor colors in a prescribed local neighborhood. Discrete-time parallel systems of this sort in which color 0 updates by contact and the rest update automatically are called Greenberg-Hastings (GH) rules. A system in which all colors update by contact is called a cyclic cellular automaton (CCA). Started from appropriate initial conditions these models generate periodic traveling waves. Started from random configurations the same rules exhibit complex self-organization, typically characterized by nucleation of locally periodic "ram's horns" or spirals. Corresponding random processes give rise to a variety of "forest fire" equilibria that display large-scale stochastic wave fronts. This article describes a framework, theoretically based, but relying on extensive interactive computer graphics experimentation, for investigation of the complex dynamics shared by excitable media in a broad spectrum of scientific contexts. By focusing on simple mathematical prototypes we obtain a better understanding of the basic organizational principles underlying spatially-distributed oscillating systems.

patt-sol↗

arXiv1993

Threshold Growth Dynamics

We study the asymptotic shape of the occupied region for monotone deterministic dynamics in d-dimensional Euclidean space parametrized by a threshold theta, and a Borel set N with positive and finite Lebesgue measure. If A_n denotes the occupied set of the dynamics at integer time n, then A_n+1 is obtained by adjoining any point x for which the volume of overlap between x+N and A_n exceeds theta. Except in some degenerate cases, we prove that A_n converges to a unique limiting "shape" L starting from any bounded initial region that is suitably large. Moreover, L is computed as the polar transform for 1/w, where w is an explicit width function that depends on N and theta. It is further shown that L describes the limiting shape of wave fronts for certain cellular automaton growth rules related to lattice models of excitable media, as the threshold and range of interaction increase suitably. In the case of 2-d box neighborhoods, these limiting shapes are calculated and the dependence of their anisotropy on theta is examined. Other specific two- and three- dimensional examples are also discussed in some detail.

patt-sol↗

arXiv1993

Metastability in the Greenberg-Hastings Model

The Greenberg-Hastings Model (GHM) is a family of multitype cellular automata that emulate excitable media, exhibiting the nucleation and spiral formation characteristic of such complex systems. In this paper we study the asymptotic frequency of nucleation in 2-d GHM dynamics as the number k of types, or colors, becomes large. Starting from uniform product measure over colors, and assuming that the excitation threshold is not too large, the box size needed for formation of a spiral core is shown to grow exponentially in k. By exploiting connections with percolation theory we find that a box of size exp{(.23+-.06)k} is required in the nearest neighbor, threshold 1 case. In contrast, GHM rules obey power law nucleation scaling when started from a suitable non-uniform product measure over the k colors; this effect is driven by critical percolation. Finally, we present some analogous results for a Random GHM, an interacting Markovian system closely related to the epidemic with regrowth of Durrett and Neuhauser.

patt-sol↗