I am looking for a reference (or a simple proof) of the fact that a free group is [sofic](https://en.wikipedia.org/wiki/Sofic_group). The preferred dynamical definition of a sofic group seems to be that
there is a sequence of finite sets $V\_n$ with $|V\_n|\to\infty$ and a sequence of maps $\sigma\_n\colon \Gamma\to \text{Sym}(V\_n)$ such that for (1) fixed $g$ and $h$, $\sigma\_n(gh)(v)=\sigma\_n(g)\sigma\_n(h)v$ for most $v\in V\_n$ as $n\to\infty$; and (2) for fixed $g\ne e$, $\sigma\_n(g)v\ne v$ for most $v\in V\_n$ as $n\to\infty$.


Many references that I have seen establish this property for amenable groups (where $V\_n$ is taken to be a Følner sequence in $\Gamma$, and $\sigma\_n(g)$ is defined to be left multiplication by $g$ where this leaves elements of $V\_n$ inside $V\_n$; and defined arbitrarily to make $\sigma\_n(g)$ a bijection for other elements of $V\_n$) and for residually finite groups (where $V\_n$ is taken to be a sequence of quotients of $\Gamma$ by increasing normal subgroups of finite index).


It is quite straightforward to see that $SL(d,\mathbb Z)$, for example, is residually finite.
The case of the free group $F\_n$ is described in the references that I have seen (a number of lectures online, Lewis Bowen's ICM address and the book of Kerr and Li) as an "interesting technical exercise", or is deduced by embedding $F\_n$ in $SL(d,\mathbb Z)$.



> 
> Does anyone know a direct proof that $F\_n$ is sofic, or have a reference to such a proof?
> 



Direct proof that free groups are sofic
There is a very simple probabilistic proof. Begin with a large finite set - its elements are called "parents", and let each parent have $2d$ offspring labelled with the generators of our free group. Now let the kids go to a nightclub, where each of them randomly finds a partner from the opposite sex (or shall I say gender?). Then the resulting labelled graph on the set of parents is a Schreier graph of the free group which converges to the Cayley tree as the number of parents goes to infinity.


