Scaling Limit and Critical Exponents for Two-Dimensional Bootstrap Percolation

Consider a cellular automaton with state space $\{0,1 \}^{{\mathbb Z}^2}$ where the initial configuration $\omega_0$ is chosen according to a Bernoulli product measure, 1's are stable, and 0's become 1's if they are surrounded by at least three neighboring 1's. In this paper we show that the configuration $\omega_n$ at time n converges exponentially fast to a final configuration $\bar\omega$, and that the limiting measure corresponding to $\bar\omega$ is in the universality class of Bernoulli (independent) percolation. More precisely, assuming the existence of the critical exponents $\beta$, $\eta$, $\nu$ and $\gamma$, and of the continuum scaling limit of crossing probabilities for independent site percolation on the close-packed version of ${\mathbb Z}^2$ (i.e., for independent $*$-percolation on ${\mathbb Z}^2$), we prove that the bootstrapped percolation model has the same scaling limit and critical exponents. This type of bootstrap percolation can be seen as a paradigm for a class of cellular automata whose evolution is given, at each time step, by a monotonic and nonessential enhancement.Comment: 15 page

Desarrollo de entorno online de programación para computación natural

Máster en Investigación e Innovación en Tecnologías de la Información y las Comunicaciones (i2- TIC).This work proposes a natural computer programming (for CA and NEPs) environment platform using Blockly. The platform is a web-based tool that provides simulators for two well-known natural computing systems: Cellular Automata (CA) and Network of Evolutionary Processors (NEPs). CA programming blocks presented in this work provide the ability to design and implement several types of CA including Elementary cellular automata, 2D cellular automata, and nD cellular automata. The tool also provides a graphical representation of CA’s grid through projection for any CA that has 3 or more dimensions. A NEPs Blockly programming environment is presented in this work. It provides the ability to design and simulate NEPs. Blocks are used as flexible user interface to enter NEPs specifications. The blocks automatically generate a standard XML configurations code which can be sent to the server side of the simulator for implementation. The tool also provides a graphical representation for the static topology of the system. Both CA and NEPs Blockly programming environments have been tested in several rather academic examples. The work presents an online simulation platform for natural computing algorithm using visual programing tool, namely Blockly. The proposed platform provides software engineering tools for setting up algorithms and also ease of use especially for teaching of these algorithm. The software engineering tools has been implemented on the NEPs as there is much more software tools already presented for cellular automata. The software designed for NEPs are a set of blocks to implement several types of connections between nodes. These blocks reduce time and complexity in setting up NEPs with fully connected nodes, for instance. In the other hand, cellular automata algorithm has been chosen to test the ease of the process of teaching and learning natural computing algorithms as they are much better-known model. The test has been conducted with students, teachers and researchers. Results of the experiment showed that the CA Blockly simulator outperforms traditional manual methods of implementing CA. It also showed that the proposed environment has desired features such as ease of use and decreases learning time. The NEPs part of the system has been tested against several applications. It showed that it provides a flexible designing tool for NEPs. It outperforms traditional XML coding methods in terms of ease of use and designing time. In addition we have designed specific high level constructs that automatize in some way the specific of complex NEPs’ topologies by hand. They could be considered as embryonic software engineering tools to program NEPs. Our tool is considered a generic platform for web-based implementation. It has desired features and wide range of properties that could attract the scientific community to adapt and develop in the future

Modeling tumor cell migration: from microscopic to macroscopic

It has been shown experimentally that contact interactions may influence the migration of cancer cells. Previous works have modelized this thanks to stochastic, discrete models (cellular automata) at the cell level. However, for the study of the growth of real-size tumors with several millions of cells, it is best to use a macroscopic model having the form of a partial differential equation (PDE) for the density of cells. The difficulty is to predict the effect, at the macroscopic scale, of contact interactions that take place at the microscopic scale. To address this we use a multiscale approach: starting from a very simple, yet experimentally validated, microscopic model of migration with contact interactions, we derive a macroscopic model. We show that a diffusion equation arises, as is often postulated in the field of glioma modeling, but it is nonlinear because of the interactions. We give the explicit dependence of diffusivity on the cell density and on a parameter governing cell-cell interactions. We discuss in details the conditions of validity of the approximations used in the derivation and we compare analytic results from our PDE to numerical simulations and to some in vitro experiments. We notice that the family of microscopic models we started from includes as special cases some kinetically constrained models that were introduced for the study of the physics of glasses, supercooled liquids and jamming systems.Comment: Final published version; 14 pages, 7 figure