In particular as Gil hints there is a strong connection to bootstrap percolation, and perhaps some of the theorems proven there apply here as well. I didn't see any rigorous arguments in the papers I found but that doesn't mean there aren't any out there. This region is followed by the "steady state" characterized by small fluctuations around an average value. In general in three intervals: First, a region extending from $t =0$ to $t\simeq L^$. They summarize their findings as follows (below $\varphi$ represents some quantity like the number of live clusters or the total number of live sites, etc.):įor initial occupation probabilities satisfying $0.15\leq p \lt 0.75$, each one of the different statistical functions $\varphi$ describing the dynamics of the GL may be divided Here is a plot showing the average number of connected clusters of live sites versus time. Nonlinear dynamics of the cellular-automaton ‘‘game of Life’’. They also investigated the temporal behavior, however I found more detailed simulations and discussion of scaling in this paper: Here's a plot from their paper of results on 256 by 256 toruses (their $\rho_0=p$): In it they attempt a kind of mean-field analysis (which I haven't digested) as well as do some numerical experiments which suggest that the system approaches a nontrivial asymptotic density for all $0\lt p\lt 1$. The first study of the game of life with random initial conditions that I could find was this paper:į. Update: Related question Is there any superstable configuration in the game of life? A related question that I thought about is what is the situation for "noisy" versions of Conway's game of life? For example if in each round a live cell dies with probability $t$ and a dead cell gets life with probability $s$ and both $t$ and $s$ are small numbers and all these probabilities are independent.Īnother example is to consider the following probabilistic variant of the rule of the game itself ($t$ is a small real number):Īny live cell with fewer than two live neighbours dies with probability $1−t$.Īny live cell with two or three live neighbours lives with probability $1−t$ on to the next generation.Īny live cell with more than three live neighbours dies with probability $1−t$.Īny dead cell with exactly three live neighbours becomes a live cell with probability $1−t$.įollowing some comments below I asked about the computational power of such a noisy version over here. This question was motivated by a recent talk by Béla Bollobás on bootstrap percolation. Specifically suppose that to start with every cell is alive with probability $p$ and these probabilities are statistically independent. What is the behavior of Conway's game of life when the initial position is random? - We can ask this question on an infinite grid or on an $n$ by $n$ table (planar or on a torus).
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