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}$i := {g_i} \cup {g_i^{-1}}$\mathrm{.}Let$\mu{$\mathcal{A}$}^\ell$betheuniformmeasureoverallwordsoflength$\ell$whosealphabetsbelongto$$\mathcal{A}$$. We give probabilistic bounds on the nearness of a heat kernel smoothening of ${\mu }_{\mathcal{A}}^{\mathrm{\ell }}$ 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 ${\mu }_{\mathcal{A}}^{\mathrm{\ell }}$. 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 $S{U}_2$ was reiterated by Bourgain and Gamburd, being first raised by Lubotzky, Philips and Sarnak in 1987 and is still open. In the setting of $S{U}_2$, our results can be viewed as addressing a quantitative version of a weak variant of this question.