= Solution
Put $u=\lambda v$ and let $n$ now mean number per area. During <time> $T$, the two strips of <impact parameters> between $b$ and $b+db$, on opposite sides of the trajectory, have combined area $2uT\,db$. Thus
$$
\boxed{dn=2nuT\,db=2n\lambda vT\,db.}
$$
The same straight-line <impulse approximation> gives a kick $2Gm/(bu)$. Summing its squared size over these encounters gives
$$
D_v=\frac{\langle|\Delta\mathbf v|^2\rangle}{T}=\frac{8G^2m^2n}{u}\int_{b_{\min}}^{b_{\max}}\frac{db}{b^2}\simeq\frac{8G^2m^2n}{u b_{\min}},
$$
assuming $b_{\max}\gg b_{\min}$. The relevant change is now the small random <speed> $u$, not the full circular <speed> $v$. Setting $D_vt_{\rm rel}\sim u^2$ yields
$$
\boxed{t_{\rm rel}\simeq\frac{\lambda^3v^3b_{\min}}{8G^2m^2n}.}
$$
The <razor-thin disk relaxation> <integral> is dominated by nearby encounters, rather than equally by logarithmic intervals. Because kicks cease to be weak near the cutoff, its numerical coefficient is an estimate.
Here the weak-deflection cutoff must use the relative <speed>: $b_{\min}\sim Gm/u^2=Gm/(\lambda^2v^2)$, which is larger than the spherical cutoff. Thus $t_{\rm rel}\sim u/(Gmn)$. Using $n\sim N/R^2$ and $v^2\sim GNm/R$ from the <virial theorem> gives
$$
\boxed{t_{\rm rel}\sim\lambda\frac Rv=\lambda t_{\rm dyn}.}
$$
For example $n=N/(\pi R^2)$ gives the order-one coefficient $\pi/8$ in this last substitution; the paper's approximate equality suppresses such structural factors. The cancellation of $N$ is the central result.
An unsoftened point-particle system confined to an infinitesimally thin <plane> is therefore \b[not collisionless over orbital times] for $\lambda\sim0.1$ in this encounter model. At fixed total <mass>, increasing $N$ decreases $m$ and the cutoff together; the enhanced importance of close planar encounters cancels the usual benefit of increasing particle number. This is a singular zero-thickness limit. A fixed nonzero <gravitational softening> length or physical thickness restores a different $N$ dependence, so the conclusion is not a prohibition on collisionless numerical models of disks.
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