  Short period, massive planets, known as hot Jupiters (HJs), have been
discovered around $\sim 1$ percent of local field stars. The inward migration
necessary to produce HJs may be `low eccentricity', due to torques in the
primordial disc, or `high eccentricity' (HEM). The latter involves exciting
high orbital eccentricity, allowing sufficiently close passages with the host
star to raise circularising tides in the planet. We present an analytic
framework for quantifying the role of dynamical encounters in high density
environments during HEM. We show that encounters can enhance or suppress HEM,
depending on the local stellar density and the initial semi-major axis $a_0$.
For moderate densities, external perturbations can excite large eccentricities
that allow a planet to circularise over the stellar lifetime. At extremely high
densities, these perturbations can instead result in tidal disruption of the
planet, thus yielding no HJ. This may explain the apparent excess of HJs in M67
compared with their local field star abundance versus their apparent deficit in
47 Tuc. Applying our analytic framework, we demonstrate that for an initial
massive planet population similar to the field, the expected HJ occurrence rate
in 47 Tuc is $f_\mathrm{HJ}=2.2\times 10^{-3}$, which remains consistent with
present constraints. Future large (sample sizes $\gtrsim 10^5$) or sensitive
transit surveys of stars in globular clusters are required to refute the
hypothesis that the initial planet population is similar to the solar
neighbourhood average. Non-detection in such surveys would have broad
consequences for planet formation theory, implying planet formation rates in
globular clusters must be suppressed across a wide range of $a_0$.
