For an integer $k \geq 2$, the simple random walk on the cycle $\mathbb{Z}_{k}$ proceeds by moving one step clockwise or counterclockwise, each with probability $1/2.$ I'm interested in the random walk on a product of cycles $\mathbb{Z}_{2} \times \mathbb{Z}_{3} \times \cdots \times \mathbb{Z}_{n}$ which moves by first choosing a coordinate $k \in \{2,\dots,n\}$ uniformly at random, and then taking a step according to the simple random walk in $\mathbb{Z}_{k}.$ If we make this walk lazy, it is aperiodic, and it is also easy to verify that it is irreducible and has uniform stationary distribution.
I'm looking for asymptotically correct bounds on the mixing time of this random walk, where mixing is defined in terms of total variation distance as in Markov Chains and Mixing Times by Levin and Peres.
A coordinate-wise coupling argument (à la that given in the solution to exercise 5.4 in MC&MT) gives that for any $\varepsilon>0$, the mixing time $t_{\text{mix}}(\varepsilon)$ is at most $\frac{1}{\log(4)}n^{3}\log(n/\varepsilon)$, where the logs are natural logs. This is of the right order, but I believe that the constant $1/\log(4)$ is too big.
Using a slight generalization of Wilson's Method (see section 13.5 of the text), I obtained that for any $\varepsilon>0$, the mixing time $t_{\text{mix}}(\varepsilon)$ is at least $(1-o(1))\frac{1}{2\pi^{2}}n^{3}\log(n).$ Based on the mixing times of related chains, I think the constant $\frac{1}{2\pi^{2}}$ at least has the right "flavor," if it isn't already correct. At the very least, we know the mixing time is of order $n^{3}\log n.$
Of course, the question I'd like to answer is what is the correct constant? If my hunch about the upper bound being too large is correct, answering this requires obtaining an upper bound with a constant less than $\frac{1}{\log(4)}.$ What method(s) can I use to get a better upper bound? I am guessing that I need some spectral method. Note that the usual spectral upper bound $t_{\text{mix}}(\varepsilon) \leq \frac{1}{\text{spectral gap}}\log\frac{1}{\varepsilon \pi_{\text{min}}}$ (where $\pi_{\text{min}}$ is the smallest probability in the stationary distribution) is not helpful here: the spectral gap is of order $n^{-3}$, but since $\pi_{\text{min}}=1/n!$ the bound is $O(n^{3}\log(n!/\varepsilon)).$ $\mathscr{l}^{2}$ methods like those in section 12.4 of the text seem difficult to apply, since the eigenvalues are complicated.
A final question would be, can the lower bound be improved? Any insight would be appreciated!