On the distribution of random words in a compact Lie group

Let $G$ be a compact Lie group. Suppose $g_1, \dots, g_k$ are chosen independently from the Haar measure on $G$. Let $\mathcal{A} = \cup_{i \in [k]} \mathcal{A}_i$, where, $\mathcal{A}_i := \{g_i\} \cup \{g_i^{-1}\}$. Let $\mu_{\mathcal{A}}^\ell$ be the uniform measure over all words of length $\ell$ whose alphabets belong to $\mathcal{A}$. We give probabilistic bounds on the nearness of a heat kernel smoothening of $\mu_{\mathcal{A}}^\ell$ to a constant function on $G$ in $\mathcal{L}^2(G)$. We also give probabilistic bounds on the maximum distance of a point in $G$ to the support of $\mu_{\mathcal{A}}^\ell$. Lastly, we show that these bounds cannot in general be significantly improved by analyzing the case when $G$ is the $n-$dimensional torus. The question of a spectral gap of a natural Markov operator associated with $\mathcal{A}$ when $G$ is $SU_2$ was reiterated by Bourgain and Gamburd, being first raised by Lubotzky, Philips and Sarnak in 1987 and is still open. In the setting of $SU_2$, our results can be viewed as addressing a quantitative version of a weak variant of this question.