Re: What is the spatial distribution of rats in North America?

First, Norway rats (which is I assume the species you are talking about)
can indeed live anywhere humans who live like Modern european people now
live. However, without people, their range would be more restricted. For
example, I'd doubt there were any Norway rats in Tundra areas outside of
human habitation- so there'd be rats expected in Nunavit etc. but only in
and around the villages or other buildings.

I think the question you are asking is considerably different from asking
what might be their real range.  If you are asking about what the range of
 R. norvegicus was before people of the modern age, I'm not sure that's
answerable and the best answer would be only an estimate. From the
following passage, I'd guess their northern limit would be somewhere in
Siberia, and thus similarly a high latitude in North America. 

Geographical range
Native range : Believed to have originated in NE China.
Known introduced range : Rattus norvegicus was not present in western
Europe until 1700-1716. It is now present in the United Kingdom, France,
Switzerland, Portugal, Denmark, Belgium, USA, Mexico, Canada, Chile,
Caribbean, Galapagos Islands, Mauritius, Seychelles, Reunion, Japan, New
Zealand, Falkland Islands, Pacific Islands, South Georgia and South
Sandwich Islands, US minor outlying islands, and French Southern
Territories. There is ongoing risk of invasion of new islands.

Here's a reference that might help:

Jackson, W. B. 1982. Norway rat and allies. Pages 1077-1088. In: J.A.
Chapman and G.A. Feldhamer (eds.). Wild Mammals of North America.
Baltimore: The Johns Hopkins University Press.