Take a finite tape Turing machine with N states and tape length T and N^T total possible tape states.

Now consider that you have a probability for each state instead of a definite state. The transitions of the Turing machine induce transitions of the probabilities. These transitions define a Markov chain on a N^T dimensional probability space.

Is this useful? Absolutely not. It's just a trivial rewriting. But it shows that high dimensional spaces are extremely powerful. You can trade off sophisticated transition rules for high dimensionality.