loop-erased random walk ($\kappa=2$) on an object of arbitrary shape. (ii) from the [*double*]{} left-passage probability of a SAW [@SimmonsCardy] around points $z_1,z_2$ one computes the Casimir interaction between two points. Interestingly, when they are close and away from the boundary the interaction force is [*attractive*]{} and diverges for $y\approx y_1\approx y_2$, $\theta\approx \theta_1\approx \theta_2$ as $|f| \sim k T  A (1-\cos \theta) y^{-2/3} |z_1-z_2|^{-1/3}$ with $A=- \sqrt{3 \pi} \Gamma(5/6)/(3 \Gamma(-2/3))=0.287457...$. Near the boundary for small $y_1=y_2=y$, ${\cal F}_{\mathrm{int}} = -k T y^4/(5 x_1 x_2 (x_1-x_2)^2)+ O(y^6)$, a [*repulsive*]{} interaction for $x_1<x_2/3$.

[![Left: The probability that a polymer does not touch two slits, as a function of $h/a$ for the geometry of figure \[f:2slits\]. Right: The free energy in units of $kT$ (solid line), compared to the sum of the free energies for a slit left and a slit right (dashed line), as a function of $h/a$. The difference is the interaction free energy mediated by the polymer.[]{data-label="f:2slitsprobaN"}](./figures/proba2slits "fig:"){width="0.49\columnwidth"}]{} [![Left: The probability that a polymer does not touch two slits, as a function of $h/a$ for the geometry of figure \[f:2slits\]. Right: The free energy in units of $kT$ (solid line), compared to the sum of the free energies for 