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It seems there is an easy solution to the „probabilistic variant of the rule of the game itself”.

If we ask about the situation after the infinite number of steps, the answer is: the system will end up in its absorbing state “all cells dead”.

First observe that in the „probabilistic variant of the rule of the game itself”, the state “all cells dead” is really an absorbing state - no cell has a chance to become live again.

Now, observe that in any current state, there’s always a non-zero probability of the next state being “all cells dead”. Each living cell in step t can be dead in step t+1 and each dead cell in step t can be dead in step t+1 with non-zero probability.

With infinite number of steps, an event with non-zero probability in every step will happen with probability one in some step.

The above is a general idea. To be more precise we should talk about asymptotic density of living cells being zero and not about ending up in a state of “all” cells being dead (see my comments below).

It seems there is an easy solution to the „probabilistic variant of the rule of the game itself”.

If we ask about the situation after the infinite number of steps, the answer is: the system will end up in its absorbing state “all cells dead”.

First observe that in the „probabilistic variant of the rule of the game itself”, the state “all cells dead” is really an absorbing state - no cell has a chance to become live again.

Now, observe that in any current state, there’s always a non-zero probability of the next state being “all cells dead”. Each living cell in step t can be dead in step t+1 and each dead cell in step t can be dead in step t+1 with non-zero probability.

With infinite number of steps, an event with non-zero probability in every step will happen with probability one in some step.

The above is a general idea. To be more precise we should talk about asymptotic density of living cells being zero and not about ending up in a state of “all” cells being dead. Probably, the best way to start the analysis of the infinite grid case is to observe that in any step there’s a non-zero probability of the next state having arbitrarily low density of living cells. For further discussion see the comments below.

It seems there is an easy solution to the „probabilistic variant of the rule of the game itself”.

If we ask about the situation after the infinite number of steps, the answer is: the system will end up in its absorbing state “all cells dead”.

First observe that in the „probabilistic variant of the rule of the game itself”, the state “all cells dead” is really an absorbing state - no cell has a chance to become live again.

Now, observe that in any current state, there’s always a non-zero probability of the next state being “all cells dead”. Each living cell in step t can be dead in step t+1 and each dead cell in step t can be dead in step t+1 with non-zero probability.

As there’s infinite number of steps, an event with non-zero probability in every step will happen with probability one in some step.

The above is a general idea. To be more precise we should talk about asymptotic density of living cells being zero and not about ending up in a state of “all” cells being dead (see my comment below).

It seems there is an easy solution to the „probabilistic variant of the rule of the game itself”.

If we ask about the situation after the infinite number of steps, the answer is: the system will end up in its absorbing state “all cells dead”.

First observe that in the „probabilistic variant of the rule of the game itself”, the state “all cells dead” is really an absorbing state - no cell has a chance to become live again.

Now, observe that in any current state, there’s always a non-zero probability of the next state being “all cells dead”. Each living cell in step t can be dead in step t+1 and each dead cell in step t can be dead in step t+1 with non-zero probability.

With infinite number of steps, an event with non-zero probability in every step will happen with probability one in some step.

The above is a general idea. To be more precise we should talk about asymptotic density of living cells being zero and not about ending up in a state of “all” cells being dead (see my comments below).

It seems there is an easy solution to the „probabilistic variant of the rule of the game itself”.

If we ask about the situation after the infinite number of steps, the answer is: the system will end up in its absorbing state “all cells dead”.

First observe that in the „probabilistic variant of the rule of the game itself”, the state “all cells dead” is really an absorbing state - no cell has a chance to become live again.

Now, observe that in any current state, there’s always a non-zero probability of the next state being “all cells dead”. Each living cell in step t can be dead in step t+1 and each dead cell in step t can be dead in step t+1 with non-zero probability.

As there’s infinite number of steps, an event with non-zero probability in every step will happen with probability one in some step.

The above is a general idea. To be more precise we should talk about asymptotic density of living cells being zero and not about ending up in a state of “all” cells being dead.

It seems there is an easy solution to the „probabilistic variant of the rule of the game itself”.

If we ask about the situation after the infinite number of steps, the answer is: the system will end up in its absorbing state “all cells dead”.

First observe that in the „probabilistic variant of the rule of the game itself”, the state “all cells dead” is really an absorbing state - no cell has a chance to become live again.

Now, observe that in any current state, there’s always a non-zero probability of the next state being “all cells dead”. Each living cell in step t can be dead in step t+1 and each dead cell in step t can be dead in step t+1 with non-zero probability.

As there’s infinite number of steps, an event with non-zero probability in every step will happen with probability one in some step.

The above is a general idea. To be more precise we should talk about asymptotic density of living cells being zero and not about ending up in a state of “all” cells being dead (see my comment below).