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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 of the model are kept fixed. Inthis paper, we show that the Marcu-Fredenhagen ratio exists in all predictedphases of the model, and show that it indeed undergoes a phase transition. Inthe Higgs phase of the model we do a more careful analysis of the ratio todeduce its first order behaviour and also give an upper bound on its rate ofconvergence. Finally, we also present a short and concise proof of theexponential decay of correlations in the Higgs phase.

The phase transition of the Marcu-Fredenhagen ratio in the abelian lattice Higgs model