Let $\mu$ be a measure on a finite group $G$. We define the spectral gap of $\mu$ to be the operator norm of the map that sends $\phi \in L^2(G)^{\circ}$ to $\mu * \phi$. We say that $\mu$ is symmetric if $\mu(x) = \mu(x^{-1})$. Now fix $G = SL_2(\mathbb{Z}/n\mathbb{Z}) \times SL_2(\mathbb{Z}/n\mathbb{Z})$, with $n \in \mathbb{N}$ being arbitrary. Suppose that $\mu$ is a measure on $G$ such that it's pushforwards to the left and right component have spectral gaps lesser than $\lambda_0 < 1$ and $\mu$ takes a minimum of $\alpha_0$ on it's support. Further suppose that the support of $\mu$ generates $G$. Then we show that there are constants $L, \beta > 0$ depending only on $\lambda_0,\alpha_0$ such that $\mu^{(*)L\log |G|}(\Gamma) \leq \frac{1}{|G|^{\beta}}$, where $\Gamma$ is the graph of any automorphism of $SL_2(\mathbb{Z}/n\mathbb{Z})$. We will discuss this result and its implications. This talk is based on joint work with Professor Alireza Salehi-Golsefidy.