For a reversible Markov chain $X_{t}$ on $\mathbb{R}^{n}$ with transition kernel $K$ and stationary distribution $\pi$, it is well-known that the `spectral gap' (basically, the size of $K$ when restricted to functions orthogonal to $\pi$) of $K$ can be estimated roughly by the following Cheeger constant:

$\Phi = \inf_{0<\pi(S) <0.5} \Phi(S)$

where $\Phi(S) = \frac{\int_{S} K(x,S^{c})dx}{\pi(S)}$ - see e.g. "Applications of geometric bounds to the convergence rate of Markov chains on $\mathbb{R}^{n}$" by W.K. Yuen for this version of the Cheeger constant, and many other places in the Markov chain literature for closely related versions.

There are many excellent papers in which people show that $\Phi$ is `very small', and thus that the spectral gap is also small; this can be done by finding a single bad set $S$. Can anyone give me literature pointers to nontrivial analyses which show that $\Phi$ is`

fairly large'? So far, I have only seen examples, such as the `local' walk on the unit interval studied by Yuen, for which everything can be calculated very explicitly. I don't care too much about the details of e.g. continuous vs discrete times/state spaces.

In case this is helpful, the examples I'm most interested in look a little bit like the n-fold product walk of a local walk on the unit interval, except that the updated coordinate moves a little bit from a point that depends on the other numbers, rather than from its old value - similar to the relationship of the Curie-Weiss model to simple random walk on the hypercube. More precisely, at time $t$ I update the walk on $[0,1]^{n}$ by choosing a coordinate $i \in [n]$ at random, and updating $X_{t+1}[i] = \frac{1}{n-1} X_{t}[i] + \epsilon_{t}$ if that is in $[0,1]$, where $\epsilon_{t} = U[-\epsilon,\epsilon]$; if it isn't in $[0,1]$, set $X_{t+1}[i] = X_{t}[i]$. In any case, for $j \neq i$, set $X_{t+1}[j] = X_{t}[j]$. In terms of parameters, I care a lot more about the dependence on $\epsilon \rightarrow 0$ than about the dependence on $n \rightarrow \infty$, but both are pretty interesting.