1}\right)}\Big]^{-5/8}\ .\qquad
$$

[[![The probability to avoid a wall starting from $(\ell,1)$ to $(\infty,1)$. Inset: the geometry in question.[]{data-label="f:N1"}](./figures/Pparallelwall "fig:"){width="\columnwidth"}]{}]{}

Let us now consider the fluctuation force between two objects, here two identical slits, [*mediated by the polymer*]{}, here in the symmetric position (see figure \[f:2slits\]). Following [@SchwarzChristophel], the map which produces two slits is for $x_1<x_2<x_3$: $
f'(w) =  \frac{w^2-x_2^2}{\sqrt{w^2-x_1^2} \sqrt{w^2-x_3^2}}$, $   f(w)={ \textstyle  E\left(\arcsin ({\frac{w}{x_1})|\frac{x_1^2}{x_3^2}}\right) x_3}
+{F\left(\arcsin(\frac{w}{x_1})|\frac{x_1^2}{x_3^2}\right)(x_2^2-x_3^2)}/{x_3}$, where $E$, $F$ and $K$ (below) are the elliptic $E$, $F$ and $K$ functions, and our choice is $g(0)=0=f(0)$. The condition that $f(x_1)=f(x_3)$, or equivalently that $\Im f(x_3)=0$ yields a non-trivial condition. Define $\alpha:= x_1/x_3$, $\beta:=x_2/x_3$. Then for $0 < \alpha < \beta< 1$: $ 
\beta(\alpha)=\sqrt{\frac{E\left(\alpha ^2\right)-E\left(\arcsin \left(\frac{1}{\alpha }\right)|\alpha
   ^2\right)}{F\left(\arcsin \left(\frac{1}{\alpha }\right)|\alpha ^2\right)-K\left(\alpha
   ^2\right)}+1}
$. The walls have position $\pm a$ and height $h$ (see figure \[f:2slits\]): $$\begin{aligned}
a=f(x_1)&=& \left(E\left(\alpha ^2\right)+\left(\beta ^2-1\right) K\left(\alpha
   ^2\right)\right)x_3 \\
 {h}=\Im {f(x_2)} &=&\Im \Big[ \textstyle E\left(\arcsin(\frac{\beta }{\alpha })|\alpha
   ^2\right) {\nonumber}\\&& \textstyle +\left(\beta ^2-1\right) \textstyle F\left(\arcsin(\frac{\beta }{\alpha
   })|\alpha ^2\right)\Big] x_3\end{aligned}$$

[![Image of the upper half plane, and of lines parallel to the real axis (in thick), resp. imaginary (dotted), under the map $f(w)$ discussed in the text, which creates two slits, with $x_1=1$, $x_2=1.991$, and $x_3$=3. []{data-label="f:2slits"}](./figures/2slits 