exerted by two polymers is $f_y =  k T \partial _y \ln {\cal P}$, $$\begin{aligned}
{\cal P} &=& \textstyle 24 \pi  \cos(\frac{2 \pi  y}{L})
  (1-\frac{y}{L})+36 \pi (1-\frac{y}{L})+28
   \sin(\frac{2 \pi  y}{L}){\nonumber}\\
   && \textstyle +\sin(\frac{4 \pi 
   y}{L}) \label{18}\end{aligned}$$ Its ratio to the force (\[strip2\]) exerted by a single polymer increases monotonically from $\frac{16}5$ (at $y=0$) to $\frac{7}2$ (at $y=L$). For an interpretation of the first number see below.

We can now compute the force exerted by a single polymer on an object placed on the boundary of the system (e.g. the upper half plane $H$). We use the nice result of [@SchrammSAW] arising from the so-called restriction property obeyed by SAWs. It states that the probability that a SAW (from $0$ to infinity) does not visit a subdomain $A$ is $|g'_A(0)|^{5/8}$, where $g_A$ is the map from $H\setminus A$ to $H$, which removes $A$ and has $g_A(0)=0$, and $g_A(z) \sim z$ at infinity. Note that $H\setminus A$ must be singly connected, hence the object connected to the boundary. For a general domain $D$ and endpoints $a$ and $b$ on the boundary the probability is $|g'_A(a)|^{5/8} |g'_A(b)|^{5/8}$ with $g_A(a)=a$ and $g_B(b)=b$. Note that a similar result holds for a Brownian excursion, i.e. a 