Poissonized Plancherel measure

The poissonized Plancherel measure is a measure on partitions \(\lambda=(\lambda_1 \geq \lambda_2 \geq \cdots \geq 0)\in \mathbb{N}^{\mathbb{N}^*}\). Samples from this measure can be obtained in the following way

  • Sample \(N \sim \mathcal{P}(\theta)\)

  • Sample a uniform permutation \(\sigma\in \mathfrak{S}_N\)

  • Compute the sorting tableau \(P\) associated to the RSK (Robinson-Schensted-Knuth correspondence) applied to \(\sigma\)

  • Consider only the shape \(\lambda\) of \(P\).

Finally, the point process formed by \(\{\lambda_i - i + \frac12\}_{i\geq 1}\) is a DPP on \(\mathbb{Z}+\frac12\).

from dppy.exotic_dpps import PoissonizedPlancherel

theta = 500  # Poisson parameter
pp = PoissonizedPlancherel(theta=theta)

(Source code, png, hires.png, pdf)


Fig. 54 Poissonized Plancherel measure

See also