Establish a weak Harris spectral gap for the projective process

Establish the additional global contraction estimate in the projective-process metric required, together with the local d-smallness result, to apply a weak Harris theorem and prove a spectral gap.

Background

The paper proves local d-smallness for the Markov semigroup of the projective process near the prepared state. Specifically, a prepared neighborhood is shown to satisfy a uniform transport contraction estimate for a truncated metric.

The authors explain that local d-smallness alone is insufficient for a weak Harris spectral-gap argument: one also needs a suitable contraction estimate controlling the dynamics away from the prepared neighborhood. They state that obtaining this global contraction estimate remains beyond the present methods and identify it as a direction for future work.