A Do we need stochasticity in a discrete spacetime?

Suppose that the spacetime is discrete, with only certain positions being possible for any particle. In this case, the probability distributions of particles have nonzero values at the points on which the wavefunction is defined. Do we need randomness in the transitions of particles in such a spacetime?

My reasoning is, in a discrete space-time there is no continuous function that would allow to transport the distribution of P into the distribution P'. Basically if any point of the space has a deterministic trajectory and is mapped to another certain point, the distribution cannot change. We don't have this problem in a continuum, but what about a discrete space?
 
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Suppose that the spacetime is discrete, with only certain positions being possible for any particle. In this case, the probability distributions of particles have nonzero values at the points on which the wavefunction is defined. Do we need randomness in the transitions of particles in such a spacetime?
This question can't be answered because nobody has an actual theory in which spacetime is discrete. All of our theories are based on spacetime as a continuum.

What we do know is that, in a discrete model, concepts like "probability distribution" are not well-defined; a distribution needs a continuous space. In fact even the concept of a "trajectory" doesn't make sense, since it implies a continuous curve. So even your framing of the question makes implicit assumptions that don't apply to the scenario you are proposing.
 
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in a discrete space-time there is no continuous function
Of course not. (Note that the phrase quoted is true without any other qualifications; you could have just left out the rest of the sentence.) A continuous function requires a continuous domain and range. But this doesn't make your reasoning correct; it makes it inapplicable.
 

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