Mixing time and isoperimetry of long-range percolation in the diffusive regime

We consider long-range percolation (LRP) on ${\mathbb{Z}}^d,d\ge 1$ with connection probabilities $\mathbf{p}(x,y)\approx \frac{\beta }{\mathrm{\parallel }x-y{\mathrm{\parallel }}^s}$ for $\beta >0$ and $s>\min (2d,d+2)$. Our main result is that the mixing time of the random walk on the giant component of supercritical LRP inside a box of side length $n$ is of order $n^2$. To establish the upper bound, we show that a $d$-dimensional isoperimetric inequality holds almost surely simultaneously for all connected sets inside the giant component that are large enough.