Full-state convergence and mixing-time bounds for tangent pCN

Establish convergence of the entire tangent pCN state in a common path-space setting and derive a mixing-time bound for the full chain, beyond the stationary joint diffusion limit for the longitudinal and transverse coordinates.

Background

The paper proves a joint diffusion limit only for the centred longitudinal fluctuation and a fixed transverse projection, with the chain started in stationarity. This identifies limiting dynamics for selected observables on the accelerated time scale.

The result does not control the full high-dimensional state as dimension changes, nor does it provide a global mixing-time estimate. The authors explicitly defer both extensions to future work, leaving unresolved whether analogous full-state convergence and quantitative mixing results can be established.

References

It is important to note that it does not establish convergence of the entire state in some common space or provide a mixing-time bound for the entire chain, which is left for future work.

— Optimal scaling and curvature correction for constrained MCMC on spheres  (2610.17946 - Shkeir et al., 13 Oct 2026) in Remark following Theorem 3.4, Section 3.5, “A joint diffusion limit”