Let $\Omega_n \subseteq \mathrm{Sym}(n)^4$ be the set of all $4$-tuples $(\sigma_1,\sigma_2,\tau_1,\tau_2)$ of permutations of $\{1,\ldots,n\}$ such that $\sigma_j \tau_k = \tau_k \sigma_j$ for each pair $(j,k) \in \{1,2\}^2$.

Any four permutations determine a $8$-regular edge-labeled directed graph on $\{1,\ldots,n\}$. (Of course, the graph might not be simple.) I am interested in understanding the local structure of this graph when the four permutations are chosen uniformly at random from $\Omega_n$. In particular, I would like to know whether the commutativity constraint enforces any other kind of constraint on the structure of the graph, in the sense that a significant number of short cycles other than commutators need to occur.

A more precise formulation of the problem is as follows. Let $\mathbb{F}_r$ be the free group on $r$ generators and let $G = \mathbb{F}_2 \times \mathbb{F}_2$. Fixing an indexation for the generators of $G$, each element of $\Omega_n$ determines a unique homomorphism from $G$ to $\mathrm{Sym}(n)$. We identify a uniform random element of $\Omega_n$ with the associated random homomorphism, to be denoted $\phi$. Consider the following questions.

(1) Is it the case that for every nontrivial element $g \in G$ and every $\epsilon > 0$ the probability that $\frac{1}{n}|\mathrm{Fix}(\phi(g))| \leq \epsilon$ tends to $1$ as $n \to \infty$? I emphasize that $g$ and $\epsilon$ are fixed before the limit.

(2) Is it the case that for every nontrivial element $g \in G$ the expectation of the random variable $\frac{1}{n}|\mathrm{Fix}(\phi(g))|$ tends to $0$ as $n \to \infty$?

Clearly a positive answer to Question (1) implies a positive answer to Question (2). A positive answer to Question (1) is exactly the assertion that the uniform measures on $\Omega_n$ form a random sofic approximation to $G$. Intuitively, this would mean that a uniform random element of $\Omega_n$ has no 'unnecessary' structure.

Note that residual finiteness of $G$ implies that there exists a sequence $(\phi_n:G \to \mathrm{Sym}(n))_{n=1}^\infty$ of homomorphisms such that for each $g \in G$ the set $\mathrm{Fix}(\phi_n(g))$ is empty for sufficiently large $n$.

A final (vague) question is whether this model can be generated in any way other than its definition. The model given by a uniformly chosen $4$-tuple of permutations with no additional constraints is contiguous with a uniform random $8$-regular graph, and there are multiple ways of generating this. In this case the elementary theory shows that the analog of Question (1) with $G$ replaced by $\mathbb{F}_4$ has a positive answer.