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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 $\\mathcal{L}^2(G)$. 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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 $\\mathcal{L}^2(G)$. 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