  The hard-sphere model is one of the most extensively studied models in
statistical physics. It describes the continuous distribution of spherical
particles, governed by hard-core interactions. An important quantity of this
model is the normalizing factor of this distribution, called the partition
function. We propose a Markov chain Monte Carlo algorithm for approximating the
grand-canonical partition function of the hard-sphere model in $d$ dimensions.
Up to a fugacity of $\lambda < \text{e}/2^d$, the runtime of our algorithm is
polynomial in the volume of the system. This covers the entire known
real-valued regime for the uniqueness of the Gibbs measure.
  Key to our approach is to define a discretization that closely approximates
the partition function of the continuous model. This results in a discrete
hard-core instance that is exponential in the size of the initial hard-sphere
model. Our approximation bound follows directly from the correlation decay
threshold of an infinite regular tree with degree equal to the maximum degree
of our discretization. To cope with the exponential blow-up of the discrete
instance we use clique dynamics, a Markov chain that was recently introduced in
the setting of abstract polymer models. We prove rapid mixing of clique
dynamics up to the tree threshold of the univariate hard-core model. This is
achieved by relating clique dynamics to block dynamics and adapting the
spectral expansion method, which was recently used to bound the mixing time of
Glauber dynamics within the same parameter regime.

Probability