A vortex of eigenvalues (Vol. 45 No.4)

Rotating vortex patches behave like clumps of eigenvalues for random matrices.

As the size of a random normal matrix grows, so does the number of its eigenvalues. As this number tends to infinity (with the mutual “repulsion” of eigenvalues reducing simultaneously) the eigenvalues “clump together” into a finite collection of dense, uniform, regions. Here we demonstrate the surprising result that exactly the same phenomenon pertains to rotating equilibrium arrangements of vorticity - so-called “vortex patches” or “V-states” – whose dynamics are governed by the famous Euler equations for ideal fluids. The underlying mathematical structure in these quite distinct areas of physics turns out to be identical.

The connection is made via an inverse moment problem in which the geometry (of either the limiting eigenvalue distribution, or the vortex patches) is dictated by an imposed background potential, but in an indirect way that must be decoded. This analogy is significant not only because it links two erstwhile unconnected areas of study, but also because it affords valuable mathematical “technology transfer”, especially with respect to decoding the shape of what we might now call the limiting “vortex of eigenvalues” in random matrix theory.

D. G. Crowdy, “Vortex patch equilibria of the Euler equation and random normal matrices”, J. Phys. A: Math. Theor., 47, 212002 (2014)