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). Specifically suppose that to start with every cell is alive with probability $p$ and these probabilities are statistically independent. This question was motivated by a recent talk by Béla Bollobás on bootstrap percolation.

Many thanks for all the answers. 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.

Another example is to consider the following probabilistic variant of the rule of the game itself ($t$ is a small real number):

Any live cell with fewer than two live neighbours dies with probability $1−t$.

Any live cell with two or three live neighbours lives with probability $1−t$ on to the next generation.

Any live cell with more than three live neighbours dies with probability $1−t$.

Any dead cell with exactly three live neighbours becomes a live cell with probability $1−t$.

Following some comments below I asked about the computational power of such a noisy version over here.