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?)

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    $\begingroup$ Kerov's thesis (Asymptotic representation theory of the symmetric group) contains a nice description of a q-deformed hook walk, and I think he cites a paper of Wilf. I don't know if this is the same as what you're describing. $\endgroup$ – John Wiltshire-Gordon Jan 12 '12 at 17:42

This question is treated as well as the fluctuation problem in a paper I wrote with Dan Beltoft and Cédric Boutillier http://arxiv.org/abs/1008.0846 to appear in Moscow Math Journal. The approach is based on the use of $q$-Gauss polynomial and the proof is a $q$-analog of the Moivre Laplace proof of the CLT. Fluctuations are given by the bridge of a Ornstein-Uhlenbeck process (tightness is the most delicate point of the paper). I am sure the variational approach is efficient for the more elementary limit shape problem since the functional you have to minimize is the same as in Vershik problem of the limit shape of Tableaux diagrams having a prescribed (large) surface (leading to "Vershik's curve" e^(-x)+e^(-y)=1 ). The only difference comes from the boundary conditions. As a result, the limit shape is nothing but the restriction of Vershik's infinite curve to some interval of R (there is only one possible choice of interval which will match with the prescribed boundary conditions). I don't know if I was completely clear, but there is a discussion of this kind at the end of our paper as well as some recent reference of F. Petrov considering the limit shape problem.

  • $\begingroup$ As it happens, I got to hear Boutillier give a talk about this paper the day after I posted the question on MathOverflow! I felt like the universe was being not just exceedingly kind in answering my question but also extraordinarily prompt. $\endgroup$ – James Propp Jan 15 '12 at 7:44
  • $\begingroup$ +1 to the universe! $\endgroup$ – Johan Wästlund Jan 16 '12 at 9:01

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