Quasirandom groups enjoy interleaved mixing

Let $G$ be a group such that any non-trivial representation has dimension at least $d$. Let $X=(X_1,X_2,\dots ,X_t)$ and $Y=(Y_1,Y_2,\dots ,Y_t)$ be distributions over $G^t$. Suppose that $X$ is independent from $Y$. We show that for any $g\in G$ we have $\mathrm{\mid }\mathbb{P}[X_1{Y}_1{X}_2{Y}_2\cdots X_t{Y}_t=g]-1\mathrm{/}\mathrm{\mid }G\mathrm{\mid }\mathrm{\mid }\le \frac{\mathrm{\mid }G{\mathrm{\mid }}^{2t-1}}{d^{t-1}}\sqrt{{\mathbb{E}}_{h\in G^t}X(h{)}^2}\sqrt{{\mathbb{E}}_{h\in G^t}Y(h{)}^2}\mathrm{.}$ Our results generalize, improve, and simplify previous works.