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Consider the long-range percolation model on the integer lattice$\mathbb{Z}^d$ in which all nearest-neighbour edges are present and otherwise$x$ and $y$ are connected with probability $q_{x,y}:=1-\exp(-|x-y|^{-s})$,independently of the state of other edges. Throughout the regime where themodel yields a locally-finite graph, (i.e.\ for $s>d$,) we determine thespectral dimension of the associated simple random walk, apart from at theexceptional value $d=1$, $s=2$, where the spectral dimension is discontinuous.Towards this end, we present various on-diagonal heat kernel bounds, a numberof which are new. In particular, the lower bounds are derived through theapplication of a general technique that utilises the translation invariance ofthe model. We highlight that, applying this general technique, we are able topartially extend our main result beyond the nearest-neighbour setting, andestablish lower heat kernel bounds over the range of parameters $s\in (d,2d)$.We further note that our approach is applicable to short-range models as well.

Spectral dimension of simple random walk on a long-range percolation cluster