Random non-commutative geometry

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The collection of all Dirac operators for a given fermion space defines its space of geometries.

Formally integrating over this space of geometries can be used to define a path integral and a thus a theory of quantum gravity.

In general this expression is complicated, however for fuzzy spaces a simple expression for the general form of the Dirac operator exists. This simple expression allows us to explore the space of geometries using Markov Chain Monte Carlo methods and thus examine the path integral in a manner similar to that used in CDT.

In doing this we find a phase transition and indications that the geometries close to this phase transition might be manifold like.