Szegő type correlations for two-dimensional outpost ensembles

Abstract

We consider two-dimensional Coulomb systems for which the coincidence set contains an outpost in the form of a suitable Jordan curve. We study asymptotics for correlations along the union of the outpost and the outer boundary of the droplet. These correlations turn out to have a universal character and are given in terms of the reproducing kernel for a certain Hilbert space of analytic functions, generalizing the Szegő type edge correlations obtained recently by Ameur and Cronvall. There are several additional results, for example on the effect of insertion of an exterior point charge in the presence of an outpost.

Figures (4)

• Figure 1: The coincidence set $S^*=S\cup C_2$ for the outpost potential described in Section \ref{['mex']} with $\alpha=-0.5$, $r_1=0.735$ and $r_2=1$.
• Figure 2: Outpost potential modeled on an elliptic Ginibre potential, cf. Section \ref{['mex']}
• Figure 3: The total mass of the measure $b^{(2)}_z$ as a function of $r=\phi_2(z)$, $r\ge \frac{r_2}{r_1}$, with $r_1=1, \, r_2=1.5, \, c=1$.
• Figure 4: Droplets corresponding to the potential \ref{['belem']} with $\alpha = -0.5, \, r_1\in\{0.225, 0.375, 0.525, 0.6, 0.675, 0.712, 0.731\}$ and critical value $r_{1*}=0.75.$

Theorems & Definitions (24)

• Lemma 1.1
• Theorem 1.2: AC, Theorem 1.2
• Theorem 1.3
• Corollary 1.4
• Corollary 1.5
• Theorem 1.6
• Theorem 1.7
• Corollary 1.8
• Theorem 1.9
• Remark 1