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I know very little about this topic, but going on what I learned in my one semester of QM, there has to be some Schrodinger-like equation they are using to get the distribution of angles of triangles then using that to run their Monte Carlo simulations of the space-time paths. Hopefully someone who knows more about CDT can understand what I'm asking.

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Since CDT is not yet solved, and the partition function cannot be calculated, its not easy to answer your question.

These quantities are rather measured, depending on the coupling constants and number of simplices and the time extent of the spacetime, you measure different values/distributions. Not mentioning that there are different phases,with significantly different behavior.

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