The Symplectic Schur Process

TL;DR

The paper constructs the symplectic Schur process as a Cartan type C analogue of the Schur process, built from universal symplectic characters $SP_\\lambda$ and a new family of down-up Schur functions $T_{\lambda,\mu}$. It proves the associated point process is determinantal with an explicit double-contour kernel and develops dynamics that preserve the SSP, along with a Berele-insertion based sampling scheme for oscillating variants. Through detailed asymptotic analysis, it derives a limit shape and fluctuations: extended Sine in the bulk, Airy at the edge, GUE corners at tangency, and a novel Pearcey-like kernel near a critical bottom-edge point. The results extend integrable probability techniques to Cartan type C, opening avenues for Macdonald-Koornwinder lifts and alternative sampling frameworks while revealing rich universal limit behaviors across bulk and critical regimes.

Abstract

We define a measure on tuples of partitions, called the symplectic Schur process, that should be regarded as the right analogue of the Schur process of Okounkov-Reshetikhin for the Cartan type C. The weights of our measure include factors that are universal symplectic characters, as well as a novel family of "Down-Up Schur functions" that we define and for which we prove new identities of Cauchy-Littlewood-type. Our main structural result is that the point process corresponding to the symplectic Schur process is determinantal and we find an explicit correlation kernel. We also present dynamics that preserve the family of symplectic Schur processes and explore an alternative sampling scheme, based on the Berele insertion algorithm, in a special case. Finally, we study the asymptotics of the Berele insertion process and find explicit formulas for the limit shape and fluctuations near the bulk and the edge. One of the limit regimes leads to a new kernel that resembles the symmetric Pearcey kernel.

Figures (8)

• Figure 1: Propagation of jump instructions to particles at level $k+1$ from the update of a particle at level $k$.
• Figure 2: Propagation of jump instructions to particles at level $2m$ from the update of leftmost particle at level $2m-1$.
• Figure 3: The random update of a half-triangular array $\mathsf{x}(t)$. Here $\mathsf{G}_{1,t}=1$, $\mathsf{G}_{2,t}=1$, $\mathsf{G}_{3,t}=0$, $\mathsf{G}_{4,t}=1$. On the left panel the jump instruction given to particle $\mathsf{x}_{1,1}$ triggers the instantaneous updates at level above. Similarly, in the central panel, the initial jump instruction if given to particle $\mathsf{x}_{2,1}$. In the right panel, corresponding to the draw $\mathsf{G}_{4,t}=1$, the particle $\mathsf{x}_{4,1}$ receives a right jump instruction.
• Figure 4: Representation of the support of the symplectic Schur process.
• Figure 5: A depiction of the point process $\mathcal{L}(\vec{\lambda})$ constructed through Berele insertion.

Theorems & Definitions (90)

• Theorem 1.1
• Theorem 1.2: See \ref{['thm:det']} in the text
• Remark 1.3
• Theorem 1.4: Limit shape and limiting density
• Theorem 1.5: Limiting fluctuations in the bulk
• Theorem 1.6: Variant of the Pearcey Kernel
• Definition 2.1
• Proposition 2.2: Cauchy-Littlewood identity for symplectic Schur polynomials
• Definition 2.3
• Definition 3.1