Each dot represents an electron that experiences pairwise Coulomb repulsion with every other electron while being confined by an external potential $Q$. The energy of the configuration $z_1, \dots, z_n$ is given by the 2D log-gas Hamiltonian $$ In many different places:
- Gaussian eigenvalues of a random matrix with random entries
- Zeros of polynomials with Gaussian random coefficients
- partial quantum hall effect
- Hele-Shaw/Laplacian growth
- vortices in superconductors
As a result, a large body of research has been devoted to exploring the properties of this family of systems. For example, it was shown in 2017 that the density of particles near the boundary follows an ERFC distribution through a remarkably long proof. Of course, with this simulator we minimize the Hamiltonian, not sample it in a temperature dependent manner. Therefore we estimate a minimum-energy state known as a facet configuration.For more information on the background and context of these systems, I urge you to check out my graduate thesis or this blog post.
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