Yacin Ameur: An explicit charge-charge correlation function at the edge of a two-dimensional Coulomb droplet

Abstract

Consider a two-dimensional Coulomb droplet. It is expected that different charges at the edge should be correlated in a relatively strong way. The physical picture is that the screening cloud about a charge at the boundary has a non-zero dipole moment, which gives rise to a slow decay of the correlation function. This phenomenon was studied (on the ”physical” level of rigor) by Forrester and Jancovici in a paper from 1995 for the elliptic Ginibre ensemble. Coincidentally a recent joint work between myself and Joakim Cronvall on reproducing kernels turned out to be closely related to this question.

Indeed, if there are \(n\) particles, we obtain that the order of magnitude of the correlation function \(K_n(z,w)\) is proportional to \(\sqrt{n}\) if \(z,w\) are on the boundary and \(z\ne w\), while \(K_n(z,z)\) is proportional to \(n\). This gives the ”slow decay” of correlations at the boundary. (For comparison, if one of the charges (say \(z\)) is in the bulk, then \(K_n(z,w)\) decays quickly for \(z\ne w\): \(|K_n(z,w)|\lesssim e^{-c\sqrt{n}}\).)

In addition we find that in the limit as \(n\to\infty\), there emerges the following correlation kernel \(K(z,w)\) for \(z,w\) on the (outer) boundary:

\(\begin{equation}(*)\qquad K(z,w)=\frac 1 {\sqrt{2\pi}}(\Delta Q(z)\Delta Q(w))^{\frac 1 4}\frac {\sqrt{\phi'(z)}\overline{\sqrt{\phi'(w)}}}{\phi(z)\overline{\phi(w)}-1}.\end{equation}\)

Here we assumed that the droplet is connected and that \(z,w\) are on the outer boundary curve \(\Gamma\); then \(\phi\) is a Riemann mapping from \(\operatorname{Ext}\Gamma\) to the exterior disc \(\{|z|>1\}\). (Thus it should be understood that \(|\phi(z)|=|\phi(w)|=1\) in \((*)\).) Finally \(Q\) is the (rather arbitrary) external potential used to define the ensemble. For example: \(Q(x+iy)=ax^2+by^2\) in the case of elliptic Ginibre.

The kernel \(S(z,w)=\frac 1 {2\pi}\frac {\sqrt{\phi'(z)}\overline{\sqrt{\phi'(w)}}}{\phi(z)\overline{\phi(w)}-1}\) appearing in \((*)\) can be recognized as the so-called Szegő kernel of the boundary curve \(\Gamma\). (That \(S(z,z)=\infty\) reflects the fact that long-range vs. short-range interactions take place on different scales.)

Our method for deriving these results builds on the technique of full-plane orthogonal polynomials due to Hedenmalm and Wennman (work to appear in Acta Math). Using summation by parts and ”tail-kernel approximation”, we in fact obtain asymptotic results for the canonical correlation kernel in cases beyond the boundary-boundary case; in particular our results extend nicely to the exterior of the droplet.

In the basic case of the Ginibre ensemble, we obtain more precise asymptotics for \(K_n(z,w)\) (an expansion in powers of \(1/n\)) by developing techniques found is Szegő's classical work on the distribution of zeros of partial sums \(s_n(z)=1+z+\cdots+\frac {z^n}{n!}\) (\(n\to\infty\)).