Uniqueness, analyticity and mixing for Gibbs point processes via spectral gaps

A Gibbs point process models particles interacting in the continuum through a potential. Among the most classical examples is the hard-sphere model, where given an activity parameter $\lambda$, a radius $r$, and a bounded set $\Lambda \subset \mathbb{R}^d$ one samples a Poisson process of intensity $\lambda$ in $\Lambda$ conditioned on the points forming the centers of an $r$-sphere packing. We prove uniqueness of infinite-volume Gibbs measure, analyticity of the pressure, and various notions of spatial and temporal mixing for activities up to what we define as the spectral threshold $\lambda_{spec}$ of the potential. For each fixed dimension $d \geq 2$, this improves the uniqueness and analyticity bounds for the hard-sphere model. As $d \to \infty$, our improvement over the classical bounds grows exponentially. We also prove an optimal mixing time bound for heat bath dynamics for the hard-sphere model up to an expected density of $\Theta(d / 2^d)$, the first result that asymptotically matches the maximum density for rapid mixing predicted by Parisi and Zamponi. We also exhibit repulsive, radial pair potentials for which $\lambda_{spec} = + \infty$, showing that the corresponding Gibbs point processes have no phase transition at any activity $\lambda > 0$. Further, in dimensions $8$ and $24$ we exhibit such a potential with no phase transition for which the work of Cohn-Kumar-Miller-Radchenko-Viazovska proves that the unique ground state at any fixed density is given by the $E_8$ and Leech lattices, respectively. Our work builds upon a 2013 work of Kondratiev-Kuna-Ohlerich that implicitly defined $\lambda_{spec}$ and proved a spectral gap for a Glauber-like continuum birth-death dynamics. Our main work shows that such a spectral gap implies several strong notions of absence of phase transition and analyzes the behavior of $\lambda_{spec}$ for interesting potentials.