Fekete points and charge placement on 2D conductors

We have investigated some aspects of the electrostatics of discrete charges in two dimensions. Simply stated, the problem is to find the locations of N equal charges on a 2D domain that minimizes the electrostatic energy of the configuration. The problem is of significant interest because of it's connections to approximation theory [Kor96,DKM98], constructing conformal maps [Pom92] and Integrable systems [WZ00,KKMW⁺01].

The continuum version of this problem is related to the the existence of conformal maps that map an arbitrary domain into the unit circle. We studied the limiting behavior of the distribution of a finit number N charges as N $\rightarrow$ $\infty$. For domains with singularities, namely corners or cusps, the discrete problem has many interesting features, that are not found in the continuum limit. These include nonuniqueness of the energy minima, an assoicated symmetry breaking, and anomalous scaling of the energy with N.

A dynamical version of this problem is also closely related to various Laplacian growth models because of the deep connections between conformal maps and Laplacian growth [HL98,FPD01,ABWZ02]. The dynamics of discrete charges in two dimensions, when they are confined by appropriate potentials [HKD02], is therefore relevant to electronic droplet in Quantum-Hall systems [ABWZ02], and to the Laplacian growth problems including Hele-Shaw [FPD01] and DLA [HL98].

In addition, this problem is also a prototype for the following class of problems - Find the ``optimal'' way to discretize a continuum quantity. As stated, this discrete optimization formulation is much too general, for us to say anything useful about it. Given a specific problem however, there is an appropriate continuum quantity, and an appropriate notion of optimality. As with our problem, these discrete problems can have a much richer structure than the corresponding continuum problem. In particular, discretizing the continuum solution in an obvious manner will not, in general, be a good solution for the discrete problem. One explicitly needs to account for the discreteness in the formulation. Our work on the 2D electrostatics problem reveals surprising universality in the behavior of the ``discrete solutions'', and these ideas might also apply to other such problems.