  We use Monte Carlo methods to explore the set of toric threefold bases that
support elliptic Calabi-Yau fourfolds for F-theory compactifications to four
dimensions, and study the distribution of geometrically non-Higgsable gauge
groups, matter, and quiver structure. We estimate the number of distinct
threefold bases in the connected set studied to be $\sim { 10^{48}}$. The
distribution of bases peaks around $h^{1, 1}\sim 82$. All bases encountered
after "thermalization" have some geometric non-Higgsable structure. We find
that the number of non-Higgsable gauge group factors grows roughly linearly in
$h^{1,1}$ of the threefold base. Typical bases have $\sim 6$ isolated gauge
factors as well as several larger connected clusters of gauge factors with
jointly charged matter. Approximately 76% of the bases sampled contain
connected two-factor gauge group products of the form SU(3)$\times$SU(2), which
may act as the non-Abelian part of the standard model gauge group.
SU(3)$\times$SU(2) is the third most common connected two-factor product group,
following SU(2)$\times$SU(2) and $G_2\times$SU(2), which arise more frequently.

High Energy Physics - Theory