---
author:
- Pierre Le Doussal and Kay Jörg Wiese
title: 'Fluctuation force exerted by a planar self-avoiding polymer'
---

=1

What is the force exerted by a polymer on a small object, such as a mesoscopic disk or a molecule? Simply because the object cannot be penetrated by the polymer it constrains its thermal fluctuations and feels an entropic force. This question is relevant in view of the recent surge of interest in fluctuation-induced forces, such as Casimir forces, triggered by beautiful experiments in critical systems [@criticalcasimirexp]. Apart from Gaussian fluctuations, calculation of Casimir forces is difficult, and it is useful to obtain exact results for non-trivial theories [@criticalcasimirtheo].

[![Left: Geometry ${\cal A}$: A self-avoiding polymer fixed at the origin and constrained to remain left of the point $z$. Right: Geometry ${\cal B}$: same as ${\cal A}$, the polymer being fixed at the top of a wedge of exterior angle $\phi$.[]{data-label="f:1"}](./figures/geom4N "fig:"){width="\columnwidth"}]{}

[![Geometry ${\cal C}$: A self-avoiding polymer constrained to depart from $x=-L$, reaching $x=L$, and encircling both the origin and point $z$.[]{data-label="f:2"}](./figures/geom2 "fig:"){width="0.65\columnwidth"}]{}

While there are many results available in 2-dimensional critical systems, some recent originating from stochastic Löwner evolution (SLE) [@schramm00] (see [@Cardy2005; @sleBernardBauerRev06] for review), their 