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We investigate the relationship between the gap between the energy of theground state and the first excited state and the decay of correlation functionsin harmonic lattice systems. We prove that in gapped systems, the exponentialdecay of correlations follows for both the ground state and thermal states.Considering the converse direction, we show that an energy gap can follow fromalgebraic decay and always does for exponential decay. The underlying latticesare described as general graphs of not necessarily integer dimension, includingtranslationally invariant instances of cubic lattices as special cases. Anylocal quadratic couplings in position and momentum coordinates are allowed for,leading to quasi-free (Gaussian) ground states. We make use of methods ofderiving bounds to matrix functions of banded matrices corresponding to localinteractions on general graphs. Finally, we give an explicit entanglement-arearelationship in terms of the energy gap for arbitrary, not necessarilycontiguous regions on lattices characterized by general graphs.Comment: 26 pages, LaTeX, published version (figure added

Correlations, spectral gap and entanglement in harmonic quantum systems on generic lattices