Growth of Young diagrams, equipped with Plancherel measure, follows the automodel equation of Kerov. Using the technology of unitary matrix model we show that such growth process is exactly same as the growth of gap-less phase in Gross-Witten and Wadia (GWW) model. The limit shape of asymptotic Young diagrams corresponds to GWW transition point. Our analysis also offers an alternate proof of limit shape theorem of Vershik-Kerov and Logan-Shepp. Using the connection between unitary matrix model and free Fermi droplet description, we map the Young diagrams in automodel class to different shapes of two dimensional phase space droplets. Quantising these droplets we further set up a correspondence between automodel diagrams and coherent states in the Hilbert space. Thus growth of Young diagrams are mapped to evolution of coherent states in the Hilbert space. Gaussian fluctuations of large N Young diagrams are also mapped to quantum (large N) fluctuations of the coherent states.

配备了Plancherel测度的Young图的增长遵循Kerov的自动模型方程。使用of矩阵模型技术，我们证明了这种增长过程与Gross-Witten and Wadia（GWW）模型中的无间隙相的增长完全相同。渐近杨氏图的极限形状与GWW过渡点相对应。我们的分析还提供了极限形状的替代证明Vershik-Kerov和Logan-Shepp定理。利用unit矩阵模型与自由费米液滴描述之间的联系，我们将自动模型类中的Young图映射到二维相空间液滴的不同形状。量化这些液滴，我们进一步在希尔伯特空间中的自动模型图和相干状态之间建立了对应关系。因此，杨氏图的增长被映射到希尔伯特空间中相干态的演化。大N杨氏图的高斯涨落也映射到相干态的量子（大N）涨落。