the bottom of figure \[f:6\], with the minimum on the circle of radius $L$, passing through $\pm L$ (bold red). This leads to a force $$\begin{aligned}
\vec  f_{{m}} = \frac{2 kT}{r} \frac{(L^4-r^4) \vec n_r + (r^2-L^2)^2 \cot(\theta) \vec n_\theta}{L^4 + r^4 - 2 r^2 L^2 \cos(2 \theta)}\ .\end{aligned}$$ which due to (\[11\]) is the sum of the forces of two independent SAWs. There is now a semi-circle of equilibrium points $r=L$, which is the image of the vertical straight line passing through 0 of geometry $\cal A$. Note that there is no force on the line, thus no [*stable*]{} equilibrium.

(100,35) (0,0)[[![Mapping of self-avoiding walks constrained to pass left and right of disks from geometry $\cal A$ to geometry $\cal C$: all disk drawn correspond to the same probability, i.e. same free energy[]{data-label="f:beau"}](./figures/origcircles "fig:"){width="0.317\columnwidth"}]{}]{} (32,17)[$\to$]{} (37.3,0)[[[![Mapping of self-avoiding walks constrained to pass left and right of disks from geometry $\cal A$ to geometry $\cal C$: all disk drawn correspond to the same probability, i.e. same free energy[]{data-label="f:beau"}](./figures/mappedcircles "fig:"){width="0.627\columnwidth"}]{}]{}]{} (15,21)[$\phi$]{}

We now argue that trapping occurs in two cases (i) a finite-size object, e.g. a small disk; and (ii) a point submitted to a thermal bath. From scale invariance the probability 