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In this article, we illustrate how the qualitative phase diagram of a gaugetheory coupled to matter can be directly proved and how rigorous numericalbounds may be established. Our work reaffirms the seminal result of Fradkin andShenker from another vista. Our main ingredient is the combined use of theself-duality of the three dimensional Z2/Z2 theory and an extended Lee-Yangtheorem. We comment on extensions of these ideas and firmly establish theexistence of a sharp crossover line in the two dimensional Z2/Z2 theory.Comment: 7 pages, 7 figure

Derivation of the Fradkin-Shenker result from duality: Links to spin systems in external magnetic fields and percolation crossovers