Variance of Linear Statistic for Plancherel Young Diagrams

In this paper we compute the precise asymptotics of the variance of linear statistic of descents on a growing interval for Plancherel Young diagrams (following Vershik and Kerov, diagrams are considered rotated by $\pi/4$). We also give an example of a local configuration with linearly growing variance in a fixed regime and prove the central limit theorem for this configuration in the given regime.