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The Marcu-Fredenhagen ratio is a quantity used in the physics literature todifferentiate between phases in lattice Higgs models. It is defined as thelimit of a ratio of expectations of Wilson line observables as the length ofthese lines go to infinity while the parameters are kept fixed. In this paper,we utilize a cluster expansion of the lattice Higgs model to prove theexistence of the Marcu-Fredenhagen ratio, show that it is non-zero in the Higgsphase of the model, and give an upper bound on its rate of convergence. Also,we present a short and concise proof of the exponential decay of correlationsin the Higgs phase.

Convergence of the Marcu-Fredenhagen ratio and decay of correlations in the abelian lattice Higgs model