The Ginibre evolution in the large-N limit

We analyse statistics of the real eigenvalues of gl(N,R)-valued Brownian motion (the ‘Ginibre evolution’) in the limit of large $N$. In particular, we calculate the limiting two-time correlation function of spin variables associated with real eigenvalues of the Ginibre evolution. We also show how the formalism of spin variables can be used to compute the fixed time correlation functions of real eigenvalues discovered originally by Forrester and Nagao and Borodin and Sinclair.

💡 Research Summary

The paper investigates the statistical properties of real eigenvalues in the Ginibre evolution, i.e. a Brownian motion on the space of real (N\times N) matrices (\mathfrak{gl}(N,\mathbb{R})), in the limit where the matrix size (N) tends to infinity. Although the Ginibre ensemble is dominated by complex eigenvalues, the rare real eigenvalues carry a rich structure that is relevant for a variety of physical and mathematical problems. The authors introduce a binary “spin’’ variable (\sigma=\pm1) attached to each real eigenvalue, which records the orientation of the eigenvalue as it crosses the real axis. By mapping these spins onto fermionic creation‑annihilation operators, they are able to write the joint probability distribution of spins at two different times as a Pfaffian of a kernel that depends only on the time difference (\Delta t).

In the large‑(N) regime the eigenvalue density follows the circular law, while the density of real eigenvalues on the real axis scales as (N^{-1/2}). Using this scaling, the authors perform a Gaussian integration over the matrix elements and obtain an exact expression for the two‑time spin correlation function
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