What's this wire behind refrigerator?
If the Dometic that you are installing has the 12-volt element for general operation, you safest bet is to run a fused, heavy lead from your 12-volt distribution panel to the refrigerator. According to the technician who installed the 3-way Dometic in my '64 Overlander this is one of the keys to successful 12-volt operation. The 3-way Refrigerator in my Overlander works as well on 12-volt as it does AC electric or gas - - the one requirement for 12-volt operation is pre-cooling with either AC electric or gas; once that has happened, the unit operates perfectly on 12-volt. If all you need is 12-volt for the AES panel, then it probably isn't necessary (but would be advisable IMHO) to run an additional heavy lead from the service panel if you find a nearby circuit that you can splice into.
Good luck with your new refrigerator!
P.S.: The wire that you have located could be for any number of things, including something added by a previous owner. Among the possibilities that jump into my mind - - a lead for the 12-volt booster for the factory installed (optional) antenna (mine was routed through the refrigerator vent), radio antenna or power lead (this placement varied greatly by year and model), or previous owner added feature such as lead for telephone jack or TV cable.
Kevin D. Allen
WBCCI (Lifetime Member)/VAC/Free Wheelers #6359
1964 Overlander International/1999 GMC K2500 Suburban (7400 VORTEC/4.11 Differentials)
1978 Argosy Minuet 6.0 Metre/1975 Cadillac Eldorado Convertible (8.2 Liter V8/2.70 Final Drive)