Statistical Mechanics seminars

This talk derives dark-matter halo profiles from their assembly using a coarse-grained Boltzmann–Poisson treatment with a continuous subhalo population. A self-similar solution connects mass-growth rate to density slope, including gravitational scattering and capture. Initial collapse produces a three-halves cusp; hierarchical accretion and stable clustering favour the inner NFW slope; stalled growth approaches its outer slope. The framework ties these familiar profiles to successive dynamical stages rather than imposing them phenomenologically.

A Markov chain can sample effectively from a random or data-informed start even when worst-case initialization mixes slowly. This talk uses weak Poincaré inequalities to analyse that regime, with applications to simulated annealing for spherical spin glasses and to mixtures of log-concave distributions.

Aldous' spectral-gap theorem links the interchange process to a random walk. This talk presents a related result for the Kipnis–Marchioro–Presutti model on weighted hypergraphs, where the correct comparison involves a two-particle dual process. The result resolves a conjecture of Alon and Puder and draws on joint work with Pietro Caputo and Matteo Quattropani.

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.

Classical isoperimetric inequalities on the Boolean hypercube can extend beyond independent product measures. The talk proves KKL, Talagrand and Eldan–Gross inequalities under a Dobrushin-type weak-dependence condition, including a class of Ising models. The method combines semigroup arguments, discrete Bakry–Émery theory and gradient estimates.

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.