Extending log-smooth/log-concave proof strategies to Poincaré or log-Sobolev targets

Determine whether convergence proof strategies for the random-scan Gibbs sampler that rely on log-smoothness and log-concavity—such as entropy contraction via triangular transport maps and axis-disjoint three-set isoperimetric inequalities—extend to target distributions on R^d that only satisfy an L^q-Poincaré inequality or an L^q-log-Sobolev inequality, without assuming log-smoothness or log-concavity.

Background

Recent results for random-scan Gibbs under log-smooth and strongly log-concave targets (e.g., entropy contraction via triangular transport and axis-disjoint three-set isoperimetry) crucially use these structural assumptions.

The paper targets broader regimes where the distribution satisfies Poincaré or log-Sobolev inequalities without log-concavity, raising the question whether the aforementioned techniques remain applicable.