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

Tid: Ti 2021-10-05 kl 14.15

Föreläsare: Yacin Ameur (Lund)

### 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:

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

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 $$s_n(z)=1+z+\cdots+\frac {z^n}{n!}$$ ($$n\to\infty$$).