the force along the radial direction is $$\label{14}
f_r \approx \frac{2 c k T}{3} \frac{1}{r} \left(\frac{\rho}{2 r}\right)^{2/3}$$ which decays as $1/r^{5/3}$ at large distances.

Consider now a point-like object subjected to the Casimir force above plus a thermal bath at temperature $T'$. The equilibrium Gibbs measure for the object is ${\cal P}={\cal P}(z,\bar z)^{T/T'}/{\cal Z}$, and the partition sum of polymer plus object is ${\cal Z} = \int {{\mathrm{d}}}^2 z\, {\cal P}(z,\bar z)^{T/T'}$. Here ${\cal P}$ is either Schramm’s probability ${\cal P}_0$ for a single polymer, given above Eq. (\[single\]), or ${\cal P}_m$ in Eq. (\[11\]) for an object caught between two SAWs. For the latter case, equiprobability lines are plotted at the bottom of Fig. \[f:6\] for $T'=T$. Depending on the geometry and $T/T'$, ${\cal Z}$ is either infinite, and the object diffuses to the region where the integral is divergent, or finite and the object is bound. The latter occurs for any $T'<T$ in geometry ${\cal C}$ (top right of Fig. \[f:6\]), since at large $r$, ${\cal P} {{\mathrm{d}}}^2 z\, \sim {{\mathrm{d}}}\theta\, r {{\mathrm{d}}}r (\sin^2(\theta)/r^2)^{T/T'}$. For $T=T'$, a natural choice when the two polymers and the object are in mutual thermal equilibrium, this geometry is critical, hence the 