  There are many difficulties associated with forecasting high-energy solar
particle events at Earth. One issue is understanding why some large solar
eruptive events trigger ground level enhancement (GLE) events and others do
not. In this work we perform 3D test particle simulations of a set of historic
GLEs to understand more about what causes these powerful events. Particular
focus is given to studying how the heliospheric current sheet (HCS) affects
high-energy proton transport through the heliosphere following an event.
Analysis of $\geq$M7.0 flares between 1976$-$2020 shows that active regions
located closer to the HCS ($<$10$^{\circ}$) are more likely to be associated
with a GLE event. We found that modelled GLE events where the source region was
close to the HCS also led to increased heliospheric transport in longitude and
higher count rates (when the Earth was located in the drift direction). In a
model that does not include perpendicular diffusion associated with turbulence,
the HCS is the dominant mechanism affecting heliospheric particle transport for
GLE 42 and 69, and varying other parameters (e.g. a narrow, 10$^{\circ}$, or
wider, 60$^{\circ}$, injection width) causes little change. Overall in our
model, the HCS is relevant in 71$\%$ of our analysed GLEs and including it more
accurately reproduces observed intensities near Earth. Our simulations enable
us to produce model profiles at Earth that can be compared to existing
observations by the GOES satellites and neutron monitors, as well as for use in
developing future forecasting models.
