In their paper *[Computing Systems of Hecke Eigenvalues Associated to Hilbert Modular Forms](http://arxiv.org/abs/0904.3908)*, Greenberg and Voight remark that


...it is a folklore conjecture that if one orders totally real fields by their discriminant, then a (substantial) positive proportion of fields will have strict class number 1.


I've tried searching for more details about this, but haven't found anything. 


Is this conjecture based solely on calculations, or are there heuristics which explain why this should be true? 



Strict Class Numbers of Totally Real Fields
One heuristic is the following: if one imagines that the residue at $s = 1$ of the $\zeta$-function doesn't grow too rapidly, then the value is a combination of the regulator and the class number. I don't know any reason for the regulator not to also grow (there
are a lot of units, after all!), and hence one can imagine that the class number then stays
small.


This is part of a general heuristic that in random number fields there tends to be a trade-off between units and class number, so especially in the totally real case, when there are so many units, the class number should often be 1. 


I learnt some of these heuristics from a colleague of mine who regards it more-or-less as an axiom that a random number field has very small class number. I think this view was formed through a mixture of back-of-the-envelope ideas of the type described above, together with a lot of experience computing with random number fields. So the answer to your question might be that it is a mixture of heuristics and computations.


Incidentally, in the real quadratic case, it is compatible with Cohen--Lenstra, but I think it goes back to Gauss. Also, there are generalizations of Cohen--Lenstra to the
higher degree context, and I'm pretty sure that they are compatible with the class group/unit group trade-off heuristic described above.


