We are given a simple path graph $P(V,E)$ with vertex set $V$ and edge set $E$, having $n=|V|$ nodes. Given an initial distribution $\mathbf{\mu}$ over $V$, let $d_t(\mathbf{\mu},\pi)$ be defined as $\|\mathbf{\mu}P^t-\mathbf{\pi}\|_2$, where $\pi$ is the stationary distribution and $P$ is the random walk transition matrix $D^{-1}A$, being $D$ the diagonal matrix with $D_{i,i}$ equal to the degree of node $i$ in $P$, and $A$ the adjacency matrix of $P$. Let $t_{\min}(\varepsilon)$ be the minimum integer value $t$ such that $d_t(\mathbf{\mu},\pi)<\varepsilon$ holds for any initial distribution $\mathbf{\mu}$.
Question: How can be prove that, for $n\to\infty$, both $t_{\min}(\epsilon)$ and the mixing time $t_{\mathrm{mix}}(\varepsilon)$ are equal to $\Theta\left(n^2\log\left(\frac{1}{\varepsilon}\right)\right)$ ?
