Evacuation schemes on Cayley graphs and non-amenability of groups

In this paper we introduce a concept of an evacuation scheme on the Cayley graph of an infinite finitely generated group. This is a collection of infinite simple paths bringing all vertices to infinity. We impose a restriction that every edge can be used a uniformly bounded number of times in this scheme. An easy observation shows that existing of such a scheme is equivalent to non-amenability of the group. A special case happens if every edge can be used only once. These scheme are called pure. We obtain a criterion for existing of such a scheme in terms of isoperimetric constant of the graph. We analyze R.\,Thompson's group $F$, for which the amenability property is a famous open problem. We show that pure evacuation schemes do not exist for the set of generators $\{x_0,x_1,\bar{x}_1\}$, where $\bar{x}_1=x_1x_0^{-1}$. However, the question becomes open if edges with labels $x_0^{\pm1}$ can be used twice. Existing of pure evacuation scheme for this version is implied by some natural conjectures.