  We use N-body simulations to quantify how the escape velocity in
cluster-sized halos maps to the gravitational potential in a LambdaCDM
universe. Using spherical density-potential pairs and the Poisson equation, we
find that the matter density inferred gravitational potential profile predicts
the escape velocity profile to within a few percent accuracy for group and
cluster-sized halos (10^13 < M_200 < 10^15 M_sun, with respect to the critical
density). The accuracy holds from just outside the core to beyond the virial
radius. We show the importance of explicitly incorporating a cosmological
constant when inferring the potential from the Poisson equation. We consider
three density models and find that the Einasto and Gamma profiles provide a
better joint estimate of the density and potential profiles than the Navarro,
Frenk and White profile, which fails to accurately represent the escape
velocity. For individual halos, the 1 sigma scatter between the measured escape
velocity and the density-inferred potential profile is small (<5%). Finally,
while the sub-halos show 15% biases in their representation of the particle
velocity dispersion profile, the sub-halo escape velocity profile matches the
dark matter escape velocity profile to high accuracy with no evidence for
velocity bias outside 0.4r_200.
