  Planet-planetesimal interactions cause a planet to migrate, manifesting as a
random walk in semi-major axis. In models for Neptune's migration involving a
gravitational upheaval, this planetesimal-driven migration is a side-effect of
the dynamical friction required to damp Neptune's orbital eccentricitiy. This
migration is noisy, potentially causing Trans Neptunian Objects (TNOs) in mean
motion resonance to be lost. With Nbody simulations, we validate a
previously-derived analytic model for resonance retention and determine unknown
coefficients. We identify the impact of random-walk (noisy) migration on
resonance retention for resonances up to fourth order lying between 39 au and
75 au. Using a population estimate for the weak 7:3 resonance from the
well-characterized Outer Solar System Origins Survey (OSSOS), we rule out two
cases: (1) a planetesimal disk distributed between 13.3 and 39.9 au with
$\gtrsim$ 30 Earth masses in today's size distribution and $T_{\rm mig}
\gtrsim$ 40Myr and (2) a top-heavy size distribution with $\gtrsim$ 2000
Pluto-sized TNOs and $T_{\rm mig} \gtrsim$ 10Myr, where $T_{\rm mig}$ is
Neptune's migration timescale. We find that low-eccentricity TNOs in the
heavily populated 5:2 resonance are easily lost due to noisy migration.
Improved observations of the low-eccentricity region of the 5:2 resonance and
of weak mean motion resonances with Rubin Observatory's Legacy Survey of Space
and Time (LSST) will provide better population estimates, allowing for
comparison with our model's retention fractions and providing strong evidence
for or against Neptune's random interactions with planetesimals.
