Local Bootstrap Percolation (2009)
Gravner, Janko; University Of California Davis; Gravner@math.ucdavis.edu, Holroyd, Alexander E.; University Of British Columbia, Microsoft Research; Holroyd@math.ubc.ca
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...
A GROWTH MODEL IN A RANDOM ENVIRONMENT 1 (2008)
Janko Gravner, Craig A. Tracy, Harold Widom
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...
Modeling snow crystal growth II: A mesoscopic lattice map with plausible dynamics (2008)
Janko Gravner, David Griffeath
We present a local lattice model for the evolution of snow crystals that combines diffusion-limited aggregation with anisotropic attachment kinetics and an idealized quasi-liquid layer. Despite a...
Local Bootstrap Percolation (2008)
Gravner, Janko, Holroyd, Alexander E.
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...
Modeling snow crystal growth III: three-dimensional snowfakes (2007)
Gravner, Janko, Griffeath, David
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...
Random Threshold Growth Dynamics (2007)
A site in Z 2 becomes occupied with a certain probability as soon as it sees at least a threshold number of already occupied sites in its neighborhood. Such randomly growing sets have the following...
Slow Convergence in Bootstrap Percolation (2007)
Gravner, Janko, Holroyd, Alexander E.
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...
Percolation on fitness landscapes: effects of correlation, phenotype, and incompatibilities (2006)
Gravner, Janko, Pitman, Damien, Gavrilets, Sergey
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...
Random growth models with polygonal shapes (2006)
Gravner, Janko, Griffeath, David
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...
Modeling snow crystal growth I: Rigorous results for Packard’s digital snowflakes (2006)
Janko Gravner, David Griffeath
Digital snowflakes are solidifying cellular automata on the triangular lattice with the property that a site having exactly one occupied neighbor always becomes occupied at the next time. We...
Random growth models with polygonal shapes (2006)
Janko Gravner, David Griffeath
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...
Random growth models with polygonal shapes (2005)
Gravner, Janko, Griffeath, David
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...
A growth model in a random environment (2002)
Gravner, Janko, Tracy, Craig A., Widom, Harold
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...
Mathematical Physics Fluctuations in the Composite Regime of a Disordered Growth Model (2002)
Janko Gravner, Craig A. Tracy, Harold Widom
Abstract: 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...
Fluctuations in the composite regime of a disordered growth model (2001)
Gravner, Janko, Tracy, Craig A., Widom, Harold
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...
Limit theorems for height fluctuations in a class of discrete space and time growth models (2001)
Janko Gravner, Craig A. Tracy, Harold Widom
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...
A growth model in a random environment (2000)
Gravner, Janko, Tracy, Craig A., Widom, Harold
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...
Internal DLA and the Stefan problem (2000)
Gravner, Janko, Quastel, Jeremy
Generalized internal diffusion limited aggregation is a stochastic growth model on the lattice in which a finite number of sites act as Poisson sources of particles which then perform symmetric...
DYNAMICS OF SPECIATION AND DIVERSIFICATION IN A METAPOPULATION (2000)
Sergey Gavrilets, Randal Acton, Janko Gravner
We develop a simple framework for modeling speciation and diversification as a continuous process of accumulation of genetic (or morphological) differences accompanied by species and subpopulation...
Limit Theorems for Height Fluctuations in a Class of Discrete Space and Time Growth Models (2000)
Gravner, Janko, Tracy, Craig A., Widom, Harold
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...
Recurrent ring dynamics in two-dimensional excitable cellular automata (1999)
The Greenberg-Hastings model (GHM) is a simple cellular automaton which emulates two properties of excitable media: excitation by contact and a refractory period. We study two ways in which external...
Multitype threshold growth: convergence to Poisson-Voronoi tessellations (1997)
Gravner, Janko, Griffeath, David
A Poisson-Voronoi tessellation (PVT) is a tiling of the Euclidean plane in which centers of individual tiles constitute a Poisson field and each tile comprises the locations that are closest to a...
Nucleation parameters for discrete threshold growth on {$\bold Z\sp 2$} (1997)
Gravner, Janko, Griffeath, David
Threshold Growth is a cellular automaton on an integer lattice in which the occupied set grows according to a simple local rule: a site becomes occupied if and only if it sees at least a threshold...
Percolation Times in Two-Dimensional Models For Excitable Media (1996)
Gravner, Janko; University Of California, Davis; Gravner@feller.ucdavis.edu
The three-color Greenberg--Hastings model (GHM) is a simple cellular automaton model for an excitable medium. Each site on the lattice $bZ^2$ is initially assigned one of the states 0, 1 or 2. At...
First passage times for threshold growth dynamics on ${\bf Z}\sp 2$ (1996)
Gravner, Janko, Griffeath, David
In the threshold growth model on an integer lattice, the occupied set grows according to a simple local rule: a site becomes occupied iff it sees at least a threshold number of already occupied sites...
Percolation Times In Two-Dimensional Models For Excitable Media (1996)
. The three-color Greenberg--Hastings model (GHM) is a simple cellular automaton model for an excitable medium. Each site on the lattice Z 2 is initially assigned one of the states 0, 1 or 2. At each...
. The three-color Greenberg--Hastings model (GHM) is a simple cellular automaton model for an excitable medium. Each site on the lattice Z 2 is initially assigned one of the states 0, 1 or 2. At each...
Threshold-Range Scaling of Excitable Cellular Automata (1993)
Fisch, Robert, Gravner, Janko, Griffeath, David
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...
Threshold Growth Dynamics (1993)
Gravner, Janko, Griffeath, David
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...
Metastability in the Greenberg-Hastings Model (1993)
Fisch, Robert, Gravner, Janko, Griffeath, David
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...
Asymptotic densities for Packard Box rules (0000)
This paper studies two-dimensional totalistic solidification cellular automata with a Moore neighbourhood such that a single occupied neighbour is sufficient for a site to join the occupied set. We...
MODELING SNOW CRYSTAL GROWTH III: three-dimensional snowfakes
Janko Gravner, David Griffeath
Abstract 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....
Asymptotic densities for Packard Box rules
This paper studies two-dimensional totalistic solidification cellular automata with a Moore neighbourhood such that a single occupied neighbour is sufficient for a site to join the occupied set. We...