  The cooling of a Bose gas in finite time results in the formation of a
Bose-Einstein condensate that is spontaneously proliferated with vortices. We
propose that the vortex spatial statistics is described by a homogeneous
Poisson point process (PPP) with a density dictated by the Kibble-Zurek
mechanism (KZM). We validate this model using numerical simulations of the
two-dimensional stochastic Gross-Pitaevskii equation (SGPE) for both a
homogeneous and a hard-wall trapped condensate. The KZM scaling of the average
vortex number with the cooling rate is established along with the universal
character of the vortex number distribution. The spatial statistics between
vortices is characterized by analyzing the two-point defect-defect correlation
function, the corresponding spacing distributions, and the random tessellation
of the vortex pattern using the Voronoi cell area statistics. Combining the PPP
description with the KZM, we derive universal theoretical predictions for each
of these quantities and find them in agreement with the SGPE simulations. Our
results establish the universal character of the spatial statistics of
point-like topological defects generated during a continuous phase transition
and the associated stochastic geometry.
