We show that the total-variation mixing time of the lamplighter random walk on fractal graphs exhibit sharp cutoff when the underlying graph is transient (namely of spectral dimension greater than two). In contrast, we show that such cutoff can not occur for strongly recurrent underlying graphs (i.e. of spectral dimension less than two).
"Cutoff for lamplighter chains on fractals." Electron. J. Probab. 23 1 - 21, 2018. https://doi.org/10.1214/18-EJP196