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Are turing machines with an infinite program tape that is completely random, or another example is a Game of Life simulation on an infinite randomly initialized grid, still turing machines, so to speak?

edit: if this is the wrong place to ask this(?) remove it

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    $\begingroup$ This is the correct place to ask this, but you might want to be a bit more focused and formal about the specific question you want. Currently it borders being opinion-based, since it is not really well-defined what "equivalent to TM with infinite sized random memory" means. I took the interpretation of the question to ask only about TMs. $\endgroup$
    – nir shahar
    Commented Jan 23, 2022 at 23:23

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They are still turing complete. Here is some intuition for why:

You can manually "clear", say, the first $100$ positions by replacing them with a blank. Then, we can "split" the tape into two parts by writing a special character $\#$, for which the first half are all of the tape contents before the first occurrence of $\#$ and the rest of the tape is everything else.

We can now consider the first part of the tape to be a "cleaned out" tape, like a usual TM would have. When we run out of space, we move $\#$ a few cells forward (any constant number you really want) and set the memory it the area it was before to $\sqcup$ (effectively allocating a constant extra memory and then clearing the contents out).

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