这是一份andrews圣安德鲁斯大学 MT2504作业代写的成功案例
look at the complete words. Thus,
$$
\operatorname{Pr}\left(X_{t}=h \mid \varkappa=t\right)=\frac{1}{N} \cdot \sum_{\omega} \operatorname{Pr}\left(\omega^{\bar{\alpha}}=h\right)
$$
where we sum over the complete words $\omega$ of length $t$ such that no shorter word is a complete word. Therefore from Theorem $1, \omega$ is uniform. So,
$$
\operatorname{Pr}\left(X_{t}=h \mid \varkappa=t\right)=\frac{1}{N} \cdot N \cdot \frac{1}{|G|}=\frac{1}{|G|}
$$
Thus, $x$ is strong uniform.
Note, we can generalise this to any nilpotent group with generators corresponding to our generators, and the mixing time $=\mathrm{O}(|\Lambda| \log |\Lambda|)$.
MT2504 COURSE NOTES ：
Proof:
$$
\rho \geq\left(1-\frac{|B|+1}{D}\right) \cdot \ldots \cdot\left(1-\frac{|B|+k}{D}\right)>\left(1-\frac{2 k}{D}\right)^{k}>1-\frac{2 k^{2}}{D}
$$
Lemma 6 Let
$\xi_{v}(A)=\min {B \subset G,|B|=k} \alpha{v}(A \cup B)$Then $\forall A \subset \Gamma|Z(A, \beta)|>n-\frac{k|A|}{\beta}$.