Cutoff in separation profile for the flat torus, sphere, and projective spaces

In this paper we show that the cutoff in separation profile for Brownian motion on flat torus T n\,; on spheres S n\,; on real, complex and quaternionic projective space resp. P n pRq, P n pCq and P n pHq, is the tail distribution of some explicit Gumbel distribution. The proof is based on intertwining, dual process together with a representation formula of large moments of the covering time of the dual process.