RSK linear operators and the Vershik-Kerov-Logan-Shepp curve

TL;DR

The paper proves a conjecture of Stelzer and Yong by showing that, in the RSK linear-operator setting on ${ m Mat}_{1^n,1^n}( N)$, most diagonal entries are zero as $n oty$. It achieves this by linking the zeros to Schensted lateral-bump interactions and leveraging the Vershik–Kerov–Logan–Shepp limit-shape theorem under Plancherel measure to establish that permutations with no lateral bumps become negligible. The argument is self-contained, relying on the Limit Shape Theorem and elementary combinatorics, avoiding dependence on deeper prior results from Romik and Śniady beyond reformulations. The result confirms that the diagonal of the relevant block in the RSK operator concentrates near zero, illuminating the asymptotic structure of the operator in a probabilistic limit regime.

Abstract

Stelzer and Yong (2024) studied the Robinson-Schensted-Knuth (RSK) correspondence as a linear operator on the coordinate ring of matrices. They showed that this operator is block diagonal and conjectured that, in a special block, most diagonal entries vanish. We establish this conjecture by identifying these zeros with certain Schensted insertion interactions and analyzing them probabilistically using the Vershik-Kerov-Logan-Shepp Limit Shape Theorem.

Figures (4)

• Figure 1: The limit shape $\Gamma$ and a $\frac{1}{10}$-rescaled partition $\lambda \in {\sf Par}(100)$.
• Figure 2: Planar set of $\lambda=(2,2,1)$.
• Figure 3: The rooted tree of permutations with no lateral bumps.
• Figure 4: Example $\lambda$ with 2 columns, $k$ and $k+1$, of the same height.

Theorems & Definitions (37)

• Example 1.1
• Theorem 1.2
• Theorem 1.3
• Example 2.1
• Example 2.2
• Example 2.3
• Example 2.4
• Lemma 2.5
• proof
• Corollary 2.6