Let lambda_max^(N)(t) be the largest eigenvalue of an N× N matrix undergoing Dyson's Brownian motion with parameter beta=1. Let cal A_rm 1 denote the Airy one process. GOE Dyson Brownian motion conjecture. The evolution of the largest eigenvalue of N× N matrices for beta=1 Dyson's Brownian Motion converges, in the limit N→∞ and properly rescaled, to the process cal A_rm 1. The conjecture is motivated by the established correspondence between GOE one-point statistics and the top layers of the multilayer flat PNG model, while the full process-level correspondence remains open.

GOE Dyson Brownian motion conjecture for the Airy one process

The conjecture asserts that the properly rescaled largest eigenvalue in $4\beta=1$ Dyson Brownian motion converges to the $4\mathrm{Airy}_1$ process. Ferrari and Prähofer recorded the process-level question as open in 2005.

Known results

Ferrari (2005): fixed-time multilayer point processes converge to the edge-scaled $4\beta=1$ Dyson Brownian-motion point process.
The corresponding $4\beta=2$ matrix-diffusion limit is the $4\mathrm{Airy}_2$ process, unlike the unresolved $4\beta=1$ process-level case.

2008 numerical evidence against the conjecture

Bornemann, Ferrari, and Prähofer compared the $4\mathrm{Airy}_1$ covariance with large- $N$ GOE matrix-diffusion simulations and found a mismatch. They concluded that $4\mathrm{Airy}_1$ is not the limit, but explicitly presented this as numerical evidence rather than an analytic proof.

Current status (as of August 2026): The conjecture has strong numerical evidence against it, but no rigorous proof of falsity or alternative process-level limit is recorded.

GOE Dyson Brownian motion conjecture for the Airy one process

Primary source

Progress summary

Solutions 0