Quantum mechanics of Plancherel growth

A Unitary Matrix Model for $q$-deformed Plancherel Growth

“…We observe that the no-gap phase of this UMM captures the q-deformed Plancherel growth of Young diagrams. This is an extension of our earlier work on asymptotic growth of Young diagrams under ordinary Plancherel measure [1].…”

“…The limit shape was identified with GWW transition point. A Hilbert space description of Plancherel growth was also discussed in [1]. In this paper, we extend the analysis to q analog of Plancherel growth.…”

“…1. One can write a grand canonical partition function for this process [1,2]. Usually the growth depends on the probability of placing a crystal unit or box at different possible positions.…”

“…Thus, the evolution or growth of Young diagrams follows from the evolution of large N phases of a UMM. It was shown in [1] that the Plancherel growth of Young diagrams is captured by GWW model. Evolution of no-gap phase of GWW model with respect to a parameter of the model exactly matches with the evolution of Young diagrams in the automodel class of Kerov.…”

“…One can exactly calculate the partition function using the normalisation properties of the measure. However, in order to get a handle on the asymptotic behaviour of the Young diagrams we introduce a "fake" N dependence in the problem, following [1]. We restrict the sum over representations to only those Young diagrams which have maximum N number of rows.…”