Where can I find a detailed write-up of the asymptotic shape of a $q$-weighted Young diagram inside an $a$-by-$b$ box, especially one that uses a variational approach?
Equivalently, we can look at probability distribution on binary sequences of length $a+b$ consisting of $a$ 0's and $b$ 1's, where the probability of a particular sequence is proportional to $q$ to the power of the number of inversions (here an inversion is a pair of indices $i \lt j$ such that the $i$th bit is 1 and the $j$th bit is 0); equivalently, the probability of a particular sequence is proportional to $q$ to the power of the sum of the indices $i$ such that the $i$th bit is 0.
What I mean more precisely is that $a+b$ goes to infinity while $a/b$ converges to some finite non-zero number, and we rescale space by a factor of $a+b$. I looked at this about twenty years ago, and as I recall, if $q$ goes to 1 at the right rate, the picture stabilizes and one gets an asymptotic shape theorem, but I never worked out all the details of the argument. I could try to reconstruct and patch it, but even then I was pretty sure that the result must be "well known to those who know it", and I'm even more confident of that now, given all the work that people are doing on harder problems of a similar flavor. So this must be in the literature, but where? (Maybe in the exclusion-model literature?)