We generalize the technique of linked cluster expansions on hypercubiclattices to actions that couple fields at lattice sites which are not nearestneighbours. We show that in this case the graphical expansion can be arrangedin such a way that the classes of graphs to be considered are identical tothose of the pure nearest neighbour interaction. The only change then concernsthe computation of lattice imbedding numbers. All the complications that arisecan be reduced to a generalization of the notion of free random walks,including hopping beyond nearest neighbour. Explicit expressions forcombinatorical numbers of the latter are given. We show that under some generalconditions the linked cluster expansion series have a non-vanishing radius ofconvergence.
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