# Linked cluster expansions beyond nearest neighbour interactions: convergence and graph classes

## TL;DR

This paper extends linked cluster expansion techniques to include non-nearest neighbour interactions on hypercubic lattices, demonstrating convergence and providing explicit combinatorial formulas for generalized random walks.

## Contribution

It generalizes linked cluster expansions to non-nearest neighbour interactions, maintaining graph class consistency and deriving new combinatorial expressions for lattice imbedding numbers.

## Key findings

- Graph classes remain unchanged with non-nearest neighbour interactions.
- Explicit combinatorial formulas for generalized random walks are provided.
- Linked cluster series have a non-zero radius of convergence under certain conditions.

## Abstract

We generalize the technique of linked cluster expansions on hypercubic lattices to actions that couple fields at lattice sites which are not nearest neighbours. We show that in this case the graphical expansion can be arranged in such a way that the classes of graphs to be considered are identical to those of the pure nearest neighbour interaction. The only change then concerns the computation of lattice imbedding numbers. All the complications that arise can be reduced to a generalization of the notion of free random walks, including hopping beyond nearest neighbour. Explicit expressions for combinatorical numbers of the latter are given. We show that under some general conditions the linked cluster expansion series have a non-vanishing radius of convergence.