Establish a dynamical edge comparison for Jack processes

Establish a dynamical edge comparison between the Jack–Plancherel process, or more generally discrete beta ensembles, and the continuous beta ensembles at the soft edge, in order to identify the multi-time limiting statistics with those of the Airy-beta line ensemble for all beta>0.

Background

The paper proves multi-time soft-edge moment convergence for the Jack–Plancherel process for every beta>0, obtaining an explicit Brownian functional. However, the identification of the resulting multi-time limit with the Airy-beta line ensemble is not completed. The available comparison theorem used in the paper applies only to one-time edge limits and requires beta≥1; a dynamical comparison for the multi-time setting is therefore unresolved.

Resolving this problem would extend the paper’s explicit multi-time Laplace-transform formula from the presently justified regime to all beta>0 and would establish the expected Airy-beta line-ensemble interpretation of the Brownian functional.

References

While the technical condition $\beta\ge 1$ is essential in , and the dynamical edge comparison between discrete and continuous ensembles is not yet established, we expect that for all $\beta>0$, our formula $C_2m\mathbf L_b[\mu_+\mathfrak c_2\boldsymbol s;\boldsymbol\tau]$ gives the second multi-time Laplace transform formula for the Airy$_\beta$ line ensemble after Gorin-Xu-Zhang.

— Airy limit for the Jack process and topological expansion  (2609.03532 - Xu, 3 Sep 2026) in Section 1, subsection “The Brownian functional and the main result”