Fluctuations of maxima of discrete Gaussian free fields on a class of recurrent graphs

We provide conditions that ensure that the maximum of the Gaussian free field on a sequence of graphs fluctuates at the same order as the field at the point of maximal standard deviation; under these conditions, the expectation of the maximum is of the same order as the maximal standard deviation. In particular, on a sequence of such graphs the recentered maximum is not tight, similarly to the situation in ℤ but in contrast with the situation in ℤ². We show that our conditions cover a large class of "fractal" graphs.