  In the 1960s, the technique of using cluster expansion bounds in order to
achieve bounds on the virial expansion was developed by Lebowitz and Penrose
(1964) and Ruelle (1969). This technique is generalised to more recent cluster
expansion bounds by Poghosyan and Ueltschi (2009), which are related to the
work of Procacci (2007) and the tree-graph identity, detailed by Brydges
(1986). The bounds achieved by Lebowitz and Penrose can also be sharpened by
doing the actual optimisation and achieving expressions in terms of the Lambert
W-function. The different bound from the cluster expansion shows some
improvements for bounds on the convergence of the virial expansion in the case
of positive potentials, which are allowed to have a hard core.
