I will discuss some recent results regarding the geometry of large, bounded-degree graphs of non-negative Ollivier-Ricci curvature, a notion I will not assume familiarity with. Joint work with Camillo Brena and Florentin Munch.

There has been a recent surge of powerful tools to show rapid mixing of Markov chains, via functional inequalities such as Poincaré inequalities. In many situations, Markov chains fail to mix rapidly from a worst-case initialization, yet are expected to approximately sample from a random initialization. Under such conditions, a Poincaré inequality does not hold, necessitating new tools to prove sampling guarantees. We develop a framework to analyze such initializations, based on establishing so-called "weak Poincaré inequalities". As an application, we prove that "simulated annealing" samples from the Gibbs measure of a spherical spin glass for inverse temperatures up to a natural threshold, matching recent algorithms based on algorithmic stochastic localization. In this talk, we will focus on an application of our techniques to sampling from mixtures of log-concave distributions using data-based initializations. Based on joint work with Brice Huang, Sidhanth Mohanty, and David X Wu. Available at https://arxiv.org/abs/2411.09075.

Aldous’ spectral gap conjecture, proved by Caputo, Liggett and Richthammer, states that, on any weighted graph, the spectral gaps of the interchange process and of the underlying random walk coincide. In this talk, we present an analogue for the Kipnis-Marchioro-Presutti (KMP) model on arbitrary weighted hypergraphs: its spectral gap is not identified with that of one random walk, but with that of a two-particle dual dynamics. This, in particular, settles a recent conjecture of Alon and Puder.

Based on arXiv.2609.10450, joint work with Pietro Caputo and Matteo Quattropani (RomaTre).

Continuum Glauber dynamics is a spatial birth-death process whose stationary distribution is a Gibbs distribution. We establish a spectral gap for Continuum Glauber dynamics applied to Gibbs point processes with repulsive pair potentials, a well-known special case of which is the hard sphere model. For arbitrary-range repulsive pair potentials, we show that a continuous version of Spectral Independence suffices to establish a spectral gap. This extends the regime of activity for which Continuum Glauber dynamics is known to mix, yielding a simple efficient sampling algorithm for arbitrary-range pair potentials that matches the known efficient sampling regime for finite-range pair potentials currently based on specialized algorithms. As a consequence, we also improve the threshold up to which packings of fixed size/density can be efficiently sampled from a bounded domain, the first improvement since Kannan, Mahoney and Montenegro (2003). To prove these results, we develop continuous analogs of Spectral Independence and negative fields localization. We show that a stronger variant of zero-freeness implies Spectral Independence, which in turn allows us to run the localization scheme to boost the spectral gap of Continuum Glauber dynamics from smaller activity to larger activity. While this follows the high-level blueprint of Chen and Eldan (2022) for the discrete setting, we have to address several novel difficulties due to the continuous setting. Notably, we avoid discretization in the algorithm and the analysis and work directly in the continuous setting.