Topic: efficient sampling

Continuum Glauber dynamics samples Gibbs point processes with repulsive interactions, including hard spheres. This talk develops continuous spectral-independence and localization tools to establish spectral gaps and efficient sampling for arbitrary-range repulsive potentials. The approach improves fixed-density packing thresholds and works directly in the continuum rather than discretizing the model.

Correlation decay for graph colourings is central to efficient sampling but remains difficult to prove near conjectured thresholds. This talk establishes strong spatial mixing on trees of maximum degree Delta for at least Delta plus three colours, when Delta is at least three. It then obtains optimal Glauber-dynamics mixing for fixed degree and sufficiently large girth under the same colour bound.