Free-energy variations for determinantal 2D plasmas with holes

We study the Gibbs equilibrium of a classical 2D Coulomb gas in the determinantal case = 2. The external potential is the sum of a quadratic term and the potential generated by individual charges pinned in several extended groups. This leads to an equilibrium measure (droplet) with flat density and macroscopic holes. We consider ‘‘correlation energy’’ (free energy minus its mean-field approximation) expansions, for large particle number . Under the assumptions that the holes are sufficiently small, separated, and far from the droplet’s outer boundary, we prove that (i) the correlation energy up to order 1 is independent of the holes’ locations and orientations, and (ii) the difference between the correlation energies of systems differing by their number of holes essentially consists of ``topological’’ $O(\log N)$ and $O(1)$ terms.

💡 Research Summary

The paper investigates the large‑N asymptotics of the free energy for a two‑dimensional classical Coulomb gas (plasma) in the determinantal case β = 2. The particles interact via the logarithmic Coulomb potential and are confined by an external potential that consists of a quadratic term (providing a neutralizing background) plus the potential generated by a collection of “pinned” charges. These pinned charges are arranged in several well‑separated clusters; each cluster fills a compact region H_j (a screening or quadrature domain) so that the total charge of the cluster exactly cancels the background charge inside H_j. Consequently, the equilibrium (droplet) measure is uniform on a domain Σ that is a disk of radius R_n with n macroscopic holes H_1,…,H_n removed. The holes are assumed to be small, mutually far apart, and far from the outer boundary of the droplet.

The main object of study is the “correlation energy” (also called the free‑energy correction)
 F_corr(N,V) = F_{N,V} − N² E_MF,
where F_{N,V}=−½ log Z_{N,V} is the full free energy and E_MF is the mean‑field energy obtained by minimizing the functional (1.5). The authors prove two striking results under the above geometric assumptions:

1. Location‑independence up to order 1. The term of order N (the first sub‑leading term after the N² mean‑field contribution) does not depend on the precise positions or orientations of the holes, provided the holes remain small and well separated from each other and from the outer boundary. In other words, the leading correction to the free energy is a universal functional of the hole sizes and numbers, not of their detailed geometry.

2. Topological log N and O(1) contributions. When the number of holes changes from n − 1 to n, the difference in correlation energies is given by a sum of a universal N log N term, an N term, a log N term, and a constant term: \