October 02, 2025
We study the Gibbs equilibrium of a classical 2D Coulomb gas in the determinantal case \(\beta =2\). 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 \(N\). 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 involves “topological” \(O(\log N)\) and \(O(1)\) terms.
The 2D classical Coulomb gas1, on top of being an emblematic statistical physics model in its own right, is widely studied for its many connections with different fields of physics and mathematics [1]–[6]. Of chief interest is the model’s behavior for large particle numbers \(N\), in particular effects beyond mean-field (MF) theory. Indeed, in the setting of our interest below, the leading order behavior is dictated by a non-linear effective one-particle theory, setting the macroscopic distribution of charges (the droplet). After zooming in on the microscopic inter-particles scale, a thermodynamic limit emerges as a local density approximation (LDA) of the original problem, where the “local density” is given by mean-field theory. Fluctuations beyond that are governed by a gaussian free field (GFF) emerging from the LDA. Recent years have seen this picture confirmed in great generality, we refer to [5], in particular Section 9 therein for extensive review and references to the literature. Closest to our setting below, see in particular [7]–[10].
The behavior beyond LDA remains elusive, contrarily to the corresponding question for related 1D models (1D log-gases [11], [12]). Predictions from the physics literature [13]–[15] pointing to further signatures of the emergent GFF and topological effects have so far been mathematically vindicated only in special determinantal cases (and thus, for a specific temperature choice): on Riemann surfaces without boundaries [16]–[18], in a radial context [19]–[22], for a model with at most one hole in the droplet [23], for special models leading to disconnected droplets [24], [25] etc ...
Our purpose is to investigate some of the signatures of the conjectured free-energy expansions [13]–[15] in a special (determinantal) model where the droplet is non-radial and can have several holes. We cannot provide a full free-energy expansion, but we obtain clear signatures of the “topological” \(\log N\) terms of the expansion 2, and some of the expected invariance features of the \(O(1)\) terms.
Consider \(N\) particles in the plane of coordinates \(\mathbf{X}_N = (\mathbf{x}_1, \ldots,\mathbf{x}_N) \in \mathbb{R}^{2N}\) with energy \[\label{eq:Hamil} H_N (\mathbf{x}_1,\ldots,\mathbf{x}_N) := \frac{1}{2}\sum_{j=1}^N N V(\mathbf{x}_j) - \sum_{j<k} \log |\mathbf{x}_j - \mathbf{x}_k|\tag{1}\] and consider the Gibbs state in the determinantal case (inverse temperature \(\beta = 2\)) \[\begin{align} \label{eq:Gibbs} \rho_{N,V} (\mathbf{x}_1,\ldots,\mathbf{x}_N) &= \frac{1}{\mathcal{Z}_{N}^V} \exp\left( -2 H_N (\mathbf{x}_1,\ldots,\mathbf{x}_N)\right)\nonumber \\ &= \frac{1}{\mathcal{Z}_{N}^V} \prod_{1\leq j < k \leq N} |\mathbf{x}_j - \mathbf{x}_k| ^2 e^{-N\sum_{j=1}^N V(\mathbf{x}_j)}. \end{align}\tag{2}\] The logarithmic pairwise interaction corresponds to 2D Coulomb forces, and \(V:\mathbb{R}^2 \mapsto \mathbb{R}\) is an external trapping potential, e.g. generated by a fixed charge distribution interacting with the \(\mathbf{x}_j\). By definition \(\rho_{N,V}\) minimizes the free-energy functional \[\label{eq:intro32free32func} \mathcal{F}_{N,V} [\mu] := \int_{\mathbb{R}^{2N}} H_N (\mathbf{X}_N) \mu (\mathbf{X}_N) d\mathbf{X}_N + \frac{1}{2} \int_{\mathbb{R}^{2N}} \mu (\mathbf{X}_N) \log \mu (\mathbf{X}_N) d\mathbf{X}_N\tag{3}\] amongst probability measures \(\mu\) on \(\mathbb{R}^{2N}\) (in practice, amongst positive \(L^1\)-normalized functions). The corresponding infimum is \[\label{eq:intro32free32ener} F_{N,V} = - \frac{1}{2} \log \mathcal{Z}_{N}^V\tag{4}\] and we are interested in large \(N\) expansions thereof. Define, for a probability measure \(\sigma\) on \(\mathbb{R}^2\), the mean-field energy functional \[\label{eq:intro32MFE} \mathcal{E}^{\rm MF}[\sigma] := \frac{1}{2}\int_{\mathbb{R}^2} V(\mathbf{x}) \sigma (\mathbf{x}) d\mathbf{x}- \frac{1}{2}\iint_{\mathbb{R}^2 \times \mathbb{R}^2}\varrho(\mathbf{x}) \log |\mathbf{x}-\mathbf{y}| \varrho(\mathbf{y}) d\mathbf{x}d\mathbf{y}\tag{5}\] obtained by inserting an uncorrelated ansatz \(\mu = \sigma ^{\otimes N}\) in 3 and neglecting the entropy term. Under very mild assumptions, the above has a minimum, denoted \(E^{\rm MF}\), and a minimizer \(\mu_{\rm eq}\), called the equilibrium measure. Bearing in mind that, in great generality (see the aforementioned references) we have \[F_{N,V} = N^2 E^{\rm MF}(1+o(1))\] for large \(N\), we are chiefly interested in the behavior of the “correlation energy” \[\label{eq:corr32ener} F^{\rm Corr}_{N,V} := F_{N,V} - N^2 E^{\rm MF}.\tag{6}\] The Euler-Lagrange equation for 5 leads to \[\mu_{\rm eq}= \frac{1}{4\pi} \Delta V \mathbb{1}_\Sigma\] for a set \(\Sigma \subset \mathbb{R}^2\) called the droplet. We only consider the case where \[\label{eq:intro32mueq} \mu_{\rm eq}= \frac{1}{\pi} \mathbb{1}_\Sigma\tag{7}\] so that we deal with a system whose density is to leading order flat on the droplet. The latter can however be multiply connected, and this shall be our chief concern. In this particular case, the Zabrodin-Wiegman prediction [15] (corrected to take multiple-connectedness into account [13]) reads \[\begin{gather} \label{eq:intro32ZabWieg} F^{\rm Corr}_{N,V} = -\frac{1}{4} N \log N - \frac{1}{2}\left( \frac{\log 2\pi}{2} - 1 \right) N - \frac{6-\chi}{24} \log N \\ - \frac{\log (2\pi)}{4} - \chi \frac{\zeta' (-1)}{2} +\frac{1}{4} \log \mathrm{det} _\zeta (\Delta_{\mathbb{R}^2 \setminus \Sigma}) + o_N (1). \end{gather}\tag{8}\] We refer to [5] or [2] for a more detailed account. The leading mean-field term, forcing the charge distribution to follow \(\mu_{\rm eq}\) has already been subtracted. As regards the rest of the expansion:
\(\bullet\) The \(O(N \log N)\) term comes about because a Coulomb self-energy of each individual charge, cut-off at the natural inter-particle distance \(\sim N^{-1/2}\) arises when zooming in. This leads to an energy \(N \log \left(N^{-1/2} \right)\), to be multiplied by the temperature factor \(1/2\) from 4 .
\(\bullet\) The \(O(N)\) term is dictated by local density approximation. It can be recovered from an integral over \(\mathbf{x}\in \Sigma\) of the free-energy density of a jellium at density \(\mu_{\rm eq}(\mathbf{x})\). For a constant density, and at temperature \(\beta =2\) (the Ginibre case), this leads to the claimed expresssion. Minimizing this term is what gives rise to the gaussian free field fluctuations. This can be guessed [13] by writing an electrostatic energy in terms of the potential \(\phi\), the field \(\nabla \phi\) and the charge distribution \(-\Delta \phi\) (according to Laplace’s equation) \[-\int_{\mathbb{R}^2} \phi \Delta \phi = \int_{\mathbb{R}^2} \left|\nabla \phi\right|^2\] and replacing the usual partition function expressed in terms of charge density by a (formal) functional integral \[\label{eq:FFintegral} \int e^{- \int_{\mathbb{R}^2} \left|\nabla \phi\right|^2} D\phi.\tag{9}\]
\(\bullet\) The \(\log N\) term has a purely topological origin, in that its prefactor involves only the Euler characteristic of the droplet \[\chi := 2 - b = 1 - n\] where \(b\) is the number of boundaries3, \(n\) the number of holes, and the equality holds for a connected droplet (hence, a single outer boundary) that we shall restrict to shortly. Noteworthily, the occurence of such a term in the expansion was conjectured [13] in analogy with the gaussian free field [26]. Similar terms occur in spectral invariants of the Laplacian on a domain4, naturally connected to the formal integral 9 .
\(\bullet\) Amongst the \(O(1)\) terms, another topological one involving \(\chi\) occurs (with the derivative of the Riemann \(\zeta\) function as prefactor), but the most interesting is the (\(\zeta\)-regularized) spectral determinant of the Laplacian in the exterior of \(\Sigma\), connected to 9 , which is formally the product of Laplacian eigenvalues.
Some interesting terms are absent of the above expansion: for a multi-component droplet there are extra oscillatory terms [19]–[22], [24], [25], and, for other values of the inverse temperature \(\beta\) there is a \(O\left(N^{1/2}\right)\) term corresponding to a contribution of the droplet’s outer boundary. That this terms vanishes at \(\beta = 2\) is a remarkable prediction of [15].
In this paper we are particularly interested in getting indications of the topological \(\log N\) terms. We cannot however expand directly the free energy with the desired precision, even for the particular model we will define shortly. To make some progress, we instead observe some remarkable consequences of Conjecture 8 .
Let external potentials \(V_{1\to n}\) and \(V_n\) be chosen so that the corresponding droplets are \[\begin{align} \label{eq:intro32droplet} \Sigma_{1\to n} &= D(0,R_{1\to n}) \setminus \cup_{k=1} ^n H_k \nonumber \\ \Sigma_{n} &= D(0,R_n) \setminus H_n \end{align}\tag{10}\] where \(D(0,R)\) is the disk of center \(0\) and radius \(R\) and \(H_k,k=1\ldots n\) are \(n\) holes puncturing it. Since the total charge is fixed in 7 we must have \[\begin{align} \label{eq:area} R_{1\to n} &= \sqrt{1 + \pi ^{-1} \sum_{k=1}^n |H_k|} \nonumber\\ R_n &=\sqrt{1 + \pi ^{-1} |H_n|} \end{align}\tag{11}\] Then we should have \[\begin{align} \label{eq:spec32det32holes} \log \mathrm{det} _\zeta (\Delta_{\mathbb{R}^2 \setminus \Sigma_{1\to n}}) &= \log \mathrm{det} _\zeta (\Delta_{\mathbb{R}^2 \setminus D(0,R_n)}) + \sum_{k=1}^n \log \mathrm{det} _\zeta (\Delta_{H_k})\nonumber \\ &= \frac{1}{3} \log R_{1\to n} + \sum_{k=1}^n \log \mathrm{det} _\zeta (\Delta_{H_k}) \end{align}\tag{12}\] where the expression of the contribution of the exterior of \(D(0,R_{1\to n})\) is taken from [15] (see also [23]) and the contributions from the holes is, by translation invariance of the GFF, independent of the locations of the holes within the droplet. From 8 we infer that
(i) \(F^{\rm Corr}_{N,V_n}\) is, up to order \(o_N(1)\), independent of the location of the hole \(H_n\), as long as it stays away from the boundary of \(D(0,R_n)\).
(ii) The change in correlation energy when adding a hole in the droplet is \[\begin{gather} \label{eq:ZabWiegplus1} F^{\rm Corr}_{N,V_{1\to n}} - F^{\rm Corr}_{N,V_{1\to n-1}} - F^{\rm Corr}_{N,V_{n}} = \frac{N \log N}{4} + \frac{1}{2}\left( \frac{\log 2\pi}{2} - 1 \right) N \\ + \frac{5\log N}{24} + \frac{\zeta'(-1)}{2} + \frac{1}{12} \log \frac{R_{1\to n}}{R_{1\to n-1} R_n} + o_N (1) \end{gather}\tag{13}\]
(iii) In particular, if we assume an expansion \[\begin{align} F^{\rm Corr}_{N,V_{1\to n}} &= -\frac{1}{4} N \log N - \frac{1}{2}\left( \frac{\log 2\pi}{2} - 1 \right) N + c_{1\to n} \log N +o (\log N)\\ F^{\rm Corr}_{N,V_{n}}&= -\frac{1}{4} N \log N - \frac{1}{2}\left( \frac{\log 2\pi}{2} - 1 \right) N + c_{n} \log N + o (\log N) \end{align}\] i.e. that the leading correction after the rigorously known terms is of order \(\log N\), and if all holes are identically shaped, then it follows that \[c_{1\to n} = n c_n + \frac{5(n-1)}{24}\] hence the \(\log N\) is indeed topological in nature. Further assuming that \(c_n = -1/4\), in analogy with what is rigorously known for some particular models [23], then it must be that \[c_{1\to n} = - \frac{n-5}{24} = - \frac{6-\chi}{24}\] as expected.
These are the consequences of 8 that we manage to prove, in a particular model with sufficiently small and separated holes. We punch the holes in the droplet as in [28] by filling them with a suitable distribution of \(M\) unit pinned charges. Our potential \(V\) is the sum of a quadratic \(|\mathbf{x}|^2\) term (corresponding to a neutralizing “jellium” background and setting the constant value of the density in 7 ) and the Coulomb potential generated by these pinned charges.
The model we obtain this way benefits from a very useful exact formula [29]–[31]: its free-energy is proportional to the reduced \(M\) particles density of the Ginibre ensemble (i.e. the same model, but without pinned charges) with \(N+M\) particles, evaluated at the locations of the pinned charges. In this representation the properties above translate to
(i) Said reduced \(M\)-particles density is, to the desired precision, translation-invariant. This we prove by controling the error made by replacing, in suitable determinantal expressions, the finite \(N+M\) Ginibre correlation kernel by the limiting correlation kernel of the Ginibre process.
(ii) If the \(M\) pinned charges are split in two well-separated groups of \(M_1\) and \(M_2\) charges (with \(M=M_1 + M_2\)), the reduced \(M\)-particles density factorizes (clustering due to the fast decay of the Ginibre correlation kernel) into the individual contributions of the two groups, involving the reduced \(M_1\)-particles and \(M_2\)-particles densities, respectively.
For both properties, the main difficulty is to obtain reliable estimates with large \(M\propto N\), for this is necessary to punch macroscopic holes in the droplet, and thus set the problem in the regime of conjectured applicability of 8 .
Acknowledgments: This work benefited from insightful conversations with Alice Guionnet, Gaultier Lambert, Thomas Leblé and Sylvia Serfaty.
We turn to a precise description of our model, the assumptions corresponding to our previous vague statements, and our main results.
In essence we need the pinned charges to be “evenly distributed in several sufficiently small and separated clusters”. Since we are defining a very particular toy model on which to check some consequences of 8 , we do not aim at over-optimizing the conditions below.
For two measures \(\mu,\nu\) we define their Coulomb interaction energy \[\label{eq:Coul32int} D(\mu,\nu) = - \iint_{\mathbb{R}^2 \times \mathbb{R}^2} \mu(\mathbf{x}) \log |\mathbf{x}-\mathbf{y}| \nu (\mathbf{y}) d\mathbf{x}d\mathbf{y}.\tag{14}\] For \(n\in \mathbb{N}\) and \(j=1 \ldots n\) let \(\left(\mathbf{w}_{j,k}\right)_{k= 1 \ldots M_j}\) be \(n\) sets of points in the plane. We shall denote \[\label{eq:fraction} c_j = \frac{M_j}{N}, \quad M = \sum_{j=1} ^n M_j, \quad c = \sum_{j=1} ^n c_j\tag{15}\] and assume that each \(M_j\) is of order \(N\), so that \(c_j\) is of order \(1\) when \(N\to \infty\). One of our key assumptions will be that \(c\) is a small enough constant.
The following notion will be useful
Definition 1 (Screening region).
We say that \(H\subset\mathbb{R}^2\) is a screening region* for a set of points \(\mathbf{w}_k\in \mathbb{R}^2, k= 1 \dots M\) if \[\label{eq:screening}
-\log |\,.\,|\star \left(\frac{1}{\pi} \mathbb{1}_{H}- \frac{1}{N}\sum_{k=1} ^M \delta_{\mathbf{w}_k} \right) \begin{cases} = 0onH^c \\ \leq 0onH. \end{cases}\tag{16}\] In particular, it must be that
\[\label{eq:screening32area}
|H| = \pi \frac{M}{N}\tag{17}\] and that \(\mathbf{w}_k \in H\) for all \(k=1\ldots M\).*
That the above definition is non-empty follows from the arguments in [32]. Screening regions are also known as subharmonic quadrature domains [33]–[35], see the discussion in [6] for further references. We will show in Section 3 below that the screening regions correspond to the holes in the droplet.
Assumption 1 (Each cluster of charges evenly fills its screening region).
For all \(j=1\ldots n\), denote \(H_j\) the screening region that Definition 1 associates with the set
of points \(\left(\mathbf{w}_{j,k}\right)_{k= 1 \ldots M_j}\). We demand
**(i) separation of charges. For fixed constants \(C_1,C_2 >0\) \[\label{eq:ref32space} C_1 M^{-1/2} \leq |\mathbf{w}_{j,k} - \mathbf{w}_{j,k'}| \leq C_2 M^{-1/2}.\qquad{(1)}\] where \(\mathbf{w}_{j,k'}\) is the nearest neighbor of \(\mathbf{w}_{j,k}\) within \(\left(\mathbf{w}_{j,k}\right)_{k= 1 \ldots M_j}\).
**(ii) reasonable Coulomb energy. For large \(N\) \[\begin{align} \label{eq:ref32energy} \mathcal{H}_{N} \left(\mathbf{w}_{j,1}, \ldots, \mathbf{w}_{j,M_j}\right) &:= \frac{N}{2} \sum_{k=1} ^{M_j} |\mathbf{w}_{j,k}| ^2 - \sum_{1\leq k < \ell \leq M_j} \log |\mathbf{w}_{j,k} - \mathbf{w}_{j,\ell}| \nonumber \\ &= \frac{N^2}{2\pi} \int_{H_j} |\mathbf{x}|^2 d\mathbf{x}+ \frac{N ^2}{2\pi ^2} D \left(\mathbb{1}_{H_j},\mathbb{1}_{H_j}\right) - \frac{1}{2} M_j \log M_j + O (M) \end{align}\qquad{(2)}\] where \(|O (M)| \leq C M\) for a fixed constant \(C>0\).
Item (i) ensures that we may always think of the pinned charges as individual ones. As for Item (ii), it means that the empirical density \[\label{eq:explain32asum} \rho_{M_j} ^{(1)}:= \sum_{j=1} ^{M_j} \delta_{\mathbf{w}_{j,k}} \simeq \frac{N}{\pi} \mathbb{1}_{H_j}\tag{18}\] in the sense of Coulomb energies. The local value of the density is the equilibrium one for the minimization of \[\frac{N}{2} \int_{\mathbb{R}^2} |\mathbf{x}|^2 \rho (\mathbf{x}) + \frac{1}{2} D (\rho,\rho)\] and hence the density of points we choose is at equilibrium with/screens a harmonic background potential in \(H_j\).
We assume a matching of Coulomb energies only up to order \(N\log N\), which fits squarely within the range of known estimates: recall that 8 is known rigorously up to order \(N\) for all \(\beta\), including \(\beta = \infty\). The existing technology suffices to show that, for example, a regular lattice filling \(H_j\) will satisfy both assumptions. At the level of precision demanded in ?? , the apparent cyclicty in first defining a screening region associated to the charges, and then assuming that the latter fill it evenly, will not be a concern. For example, if one aims at a roughly disk-shaped \(H_j\), a ground state configuration for \(\mathcal{H}_{N}\), suitably translated, will also satisfy our assumptions.
Next we turn to
Assumption 2 (Clusters of charges are well-separated).
For all \(j=1\ldots n\), with the same notation as above, we demand that there be a disk \(D_j\) of radius \(R_j\) such that \[\label{eq:disks}
H_j \subset D_jand\mathbf{w}_{j,k} \in D_j \, for allk= 1 \ldots M_j.\qquad{(3)}\] We impose \[\label{eq:asumscales}
R_j \leq r_1 \min_{j,j'} \mathrm{dist} \left(D_j, D_{j'}\right) \leq r_1 r_2 \min_{j} \mathrm{dist} \left(D_j, D(0,R_n)\right)\qquad{(4)}\] with \(r_1,r_2\) two sufficiently small constants and \[\label{eq:radius32drop}
R_n := \sqrt{1 + \sum_{j=1}^n c_j}.\qquad{(5)}\]
From 17 we have that \[|H_j| = \pi c_j\] and thus, for disjoint holes, \(R_n\) above is the outer radius of the droplet, ensuring a fixed total charge: \[\frac{1}{\pi} \left| D(0,R_n) \setminus \cup_j H_j \right| = 1.\] The above assumptions thus mean that the size of the holes must be sufficiently smaller than their mutual distance, which itself must be sufficiently smaller than their distance to the droplet’s outer boundary.
We come to our results, vindicating the consequences of Conjecture 8 we have been discussing in the introduction, for the particular model we just defined. Namely, we look at the partition function appearing in 2 - 4 where the Hamiltonian 1 is set as \[\label{eq:choice32pot} V(\mathbf{x}) := |\mathbf{x}| ^2 - \frac{2}{N} \sum_{j=1} ^n \sum_{k=1} ^{M_j} \log \left|\mathbf{x}- \left(\mathbf{w}_{j,k} + \mathbf{a}_j\right)\right|.\tag{19}\] The first term is the usual quadratic potential for the Ginibre ensemble, leading to a flat local density. The second term is the Coulomb potential generated by several sets of pinned charges as described above. The vectors \(\mathbf{a}_1, \ldots, \mathbf{a}_n\) are translations that can act on each of the cluster of pinned charges, to investigate the effect of moving holes around. For convenience we regard the reference sets of points \(\left(\mathbf{w}_{j,k}\right)_{k= 1 \ldots M_j}\) as fixed, and only vary the translation vectors \(\mathbf{a}_1, \ldots, \mathbf{a}_n\). Our running assumption will always be that \[\label{eq:runasum} \boxed{the point configurations\left(\mathbf{w}_{j,k} +\mathbf{a}_j \right)_{k= 1 \ldots M_j}satisfy Asumptions~\ref{asum:onehole} and~\ref{asum:multhole} }\tag{20}\] which can be achieved by asking that the assumptions are satisfied for \(\mathbf{a}_1, \ldots, \mathbf{a}_n = 0\) and then only considering variations with \(|\mathbf{a}_j|\) small enough for all \(j= 1 \ldots n\).
The partition functions we look at are thus in the form \[\begin{gather} \label{eq:def32part} \mathcal{Z}_N (\mathbf{a}_1,\ldots,\mathbf{a}_n) := \int_{\mathbb{R}^{2N}}\prod_{1\leq j < k \leq N} |z_j-z_k|^2 e^{-N\sum_{j=1}^N |z_j|^2} \\\times \prod_{\ell = 1} ^N \prod_{j=1} ^n \prod_{k=1}^{M_j} |z_\ell - w_{j,k} - a_j|^2 dz_1 \ldots dz_N \end{gather}\tag{21}\] where we identify vectors \(\mathbf{w}_{j,k},\mathbf{a}_j\) with complex numbers \(w_{j,k},a_j\). Following the introductions this leads to the free energies and correlation energies \[\begin{align} \label{eq:def32free} F_N (\mathbf{a}_1,\ldots,\mathbf{a}_n) &= -\frac{1}{2}\log \mathcal{Z}_N (\mathbf{a}_1,\ldots,\mathbf{a}_n)\nonumber\\ F^{\rm Corr}_N (\mathbf{a}_1,\ldots,\mathbf{a}_n)&= F_N (\mathbf{a}_1,\ldots,\mathbf{a}_n) - N^2 E^{\rm MF}(\mathbf{a}_1,\ldots,\mathbf{a}_N) \end{align}\tag{22}\] where the mean-field energies \(E^{\rm MF}(\mathbf{a}_1,\ldots,\mathbf{a}_N)\) are defined by inserting 19 in 5 .
Note that one may think of the above model as en enlarged Ginibre ensemble (no pinned charges, only quadratic external potential) of \(N+M\) particles, conditioned on fixing \(M\) particles as decribed above. In this setting, our assumptions are events of large probability for the enlarged ensemble.
Our first result investigates the correlation energy \(F^{\rm Corr}_N (\mathbf{a})\) for a single hole/cluster of pinned charges. The prediction of 8 in this case is that there is no dependence on \(\mathbf{a}\) up to order \(o_N (1)\). Hence the only variations of \(F_N (\mathbf{a})\) occur at the macroscopic/mean-field level \(N^2\) of the expansion, see Section 3 below.
Theorem 1 (Moving a single hole around the droplet).
Let \(n=1\), i.e. pick \(\mathbf{w}_1,\ldots,\mathbf{w}_M\) fixed points satisfying Assumption 1 and set
\[c = \frac{M}{N}\] and/or \(|\mathbf{a}|\) small enough (which guarantees ?? in this case). Then, with the above notation, \[\label{eq:resultonehole}
\left| F^{\rm Corr}_N (\mathbf{a}) - F^{\rm Corr}_N (0)\right| \leq o_N (1)\qquad{(6)}\] in the \(N\to \infty\) limit.
Although we explictly consider only translations the hole/cluser of points, note the following:
Remark 2 (Rotating the hole).
It is clear from 21 that \(F^{\rm Corr}_N (0)\) is invariant under a joint rotation of \(\mathbf{w}_1,\ldots,\mathbf{w}_N\) around the origin. Using the
theorem above to translate an arbitrary rotation center to the origin and back to its original location, one deduces that \(F^{\rm Corr}_N (\mathbf{a})\) is also, up to \(o_N (1)\),
invariant under a joint rotation of all the pinned charges around any center, as long as Assumption 1 and ?? hold all along the rotation.\(\diamond\)
In spirit Theorem 1 is reminiscent of [23], which corresponds to the case where all pinned points are collapsed into a single one, leading to a disk-shaped hole. In as much as the two results can be compared, we work under much more restrictive assumptions on the total pinned charge and its location, but allow for an arbitrarily shaped hole.
Next we turn to the case of mutiple holes:
Theorem 3 (Punching multiples holes in the droplet).
Pick \(n\) configurations of points and \(n\) translation vectors so that Assumptions 1 and 2 hold for the translated point clusters \(\mathbf{w}_{j,k}+\mathbf{a}_j,k=1 \ldots M_j\). Then \[\begin{align}
\label{eq:resultmultipleeholes}
F^{\rm Corr}_N (\mathbf{a}_1, \ldots,\mathbf{a}_N) &- \sum_{j=1} ^n F^{\rm Corr}_N (\mathbf{a}_j)= \frac{n-1}{4}N\log N + \frac{1}{2} \left( \frac{\log 2\pi}{2} - 1\right) (n-1) N \nonumber\\
&+ \frac{5(n-1)}{24} \log N + (n-1) \frac{\zeta' (-1)}{2} + (n-1) \frac{\log 2\pi}{4} \nonumber\\
&+ \frac{1}{24}\left(\log (1+c) - \sum_{j=1} ^n \log (1+c_j) \right)+ o_N (1)
\end{align}\qquad{(7)}\] where the charges \(c\) and \(c_j,j=1\ldots n\) are as in 15 .
Combining with Theorem 1 and Remark 2 shows that, at least as long as the holes are sufficiently small and separated, the free energy depends on their locations and relative orientations only through the mean-field term. Our estimate ?? is an iterated version of 13 , we refer to the discussion in the introduction for the relationship between these findinds and the original Conjecture 8 . Note in particular the appearance of topological terms on the second line of the right-hand side. As regards the last line, to compare with 11 and 12 , recall from ?? that \(\sqrt{1+c}\) and \(\sqrt{1+c_j}\) are the outer radii of the droplets will all holes present, respectively only the \(j\)-th one.
The rest of the paper, containing the proofs of Theorems 1 and 3, is organized as follows:
In Section 3 we set up preliminary estimates on the mean-field approximation of the problem. This will permit to realize that the variations we will later find in \(F_N(\mathbf{a}_1,\ldots,\mathbf{a}_N)\) are indeed all accounted for by those of the mean-field energy.
In Section 4 we prove Theorem 1. In particular, we recap the representation of the partition function in terms of a Ginibre correlation function. Our assumption ?? implies useful a priori bounds on the later, that will enter all subsequent estimates. In particular when replacing finite area Ginibre correlation functions by infinite area ones, which is the next big task of the section.
In Section 5 we prove Theorem 3. Following on the representation just mentioned, this boils down to a clustering estimate for Ginibre correlation functions, and a careful computation to identify constant terms in expansions. We rely heavily on the determinal structure for the clustering estimate.
For the convenience of the reader, Appendix 6 recalls known facts about the Ginibre partition function and correlation functions.
Here we study the mean-field approximation of the model described above. In particular we investigate how the ground state energy depends on movements of a cluster of pinned charges and/or the addition of a cluster. This will be useful later, in comparison with the behavior of the full many-body problem, to reconstruct the desired behavior of the correlation energy.
Let \(\mathbf{w}_1,\ldots,\mathbf{w}_M\) be \(M\) points in the plane. We consider the mean-field energy functional \[\begin{gather} \label{eq:MFE32bis} \mathcal{E}^{\rm MF}[\sigma] := \frac{NJ}{2}\int_{\mathbb{R}^2} \left(|\mathbf{x}|^2 - \frac{2}{N} \sum_{k=1}^M \log |\mathbf{x}- \mathbf{w}_k| \right) \sigma (\mathbf{x}) d\mathbf{x}\\ - \frac{J^2}{2}\iint_{\mathbb{R}^2 \times \mathbb{R}^2}\varrho(\mathbf{x}) \log |\mathbf{x}-\mathbf{y}| \varrho(\mathbf{y}) d\mathbf{x}d\mathbf{y} \end{gather}\tag{23}\] for parameters \(N>0,J>0,M\in \mathbb{N}\) and pinned charges \(\mathbf{w}_k\in \mathbb{R}^2, k= 1 \dots M\). The associated minimization problem is \[\label{eq:MFE32min} E^{\rm MF}= \inf \left\{ \mathcal{E}^{\rm MF}[\sigma], \sigma \in L^2 (\mathbb{R}^2), \sigma \geq 0, \int_{\mathbb{R}^2} \sigma = 1 \right\}.\tag{24}\] The extra parameter \(J\) will be helpful because we will need later to consider ensembles with the same background charge density (set by the real parameter \(N\) in front of the \(|\mathbf{x}|^2\) term from 1 - 19 ) but different particle numbers (set by the number of terms in the sums of 1 ).
Regarding the minimization of the mean-field energy 23 we will need the following
Theorem 4 (The mean-field problem).
(i) equilibrium measure.* Assume that the screening region \(H\) associated to \(\mathbf{w}_k\in \mathbb{R}^2, k= 1 \dots M\) by Definition 1 satisfies \[\label{eq:screening32subset}
H \subset D(0,R),withR = \sqrt{\frac{J}{N} + \frac{M}{N}}.\tag{25}\] Then the unique solution \(\mu_{\rm eq}\) of 24 is given by \[\label{eq:MFsol}
\mu_{\rm eq}= \frac{N}{\pi J} \mathbb{1}_{D(0,R) \setminus H}\tag{26}\] and the associated minimal energy is \[\begin{align}
\label{eq:MFsol2}
E^{\rm MF}&= \frac{1}{2} C_R - \frac{J^2}{2} D(\mu_{\rm eq},\mu_{\rm eq}) \nonumber\\
C_R &= NJ R^2 - 2NJ R^2 \log R
\end{align}\tag{27}\] (ii) translating the pinned charges. Let \(\mathbf{a}\in \mathbb{R}^2\) and denote \(E^{\rm MF}(\mathbf{a})\) the minimal energy
corresponding to the points \(\mathbf{w}_1+ \mathbf{a}, \dots, \mathbf{w}_M + \mathbf{a}\). As long as 25 holds for the associated screening region \(H(\mathbf{a})\) we have that \[\label{eq:grad32MF}
\nabla_\mathbf{a}E^{\rm MF}(\mathbf{a}) = - N \sum_{j=1} ^M \left( \mathbf{w}_j + \mathbf{a}\right)\tag{28}\] (iii) adding a cluster of pinned charges. Assume in addition that the points \(\mathbf{w}_k\in \mathbb{R}^2, k= 1 \dots M\) can be split into two groups of \(M_1\) points \(\mathbf{w}_{1,j}, j=1 \ldots M_1\) and \(M_2\) points \(\mathbf{w}_{2,j}, j=1 \ldots M_2\), with screening regions \(H_1,H_2\) respectively.*
Assume that \(H_1 \cap H_2 = \varnothing\). Let \(E^{\rm MF}_{12},E^{\rm MF}_1,E^{\rm MF}_2\) denote the infima of 23 with all the points taken into account, and with respectively only the points of the first or second group. Let correspondingly \(R_{12},R_1,R_2,C_{R_{12}},C_{R_1},C_{R_2}\) be defined as above. Then \[\begin{align} \label{eq:split32MFE} E^{\rm MF}_{12} - E^{\rm MF}_1 - E^{\rm MF}_2 &= \frac{1}{2}\left(C_{R_{12}} -C_{R_{1}} - C_{R_{2}}\right) \nonumber\\ &+ \sum_{j=1}^{M_1} \sum_{k=1}^{M_2} \log |\mathbf{w}_{1,j} - \mathbf{w}_{2,k}|\nonumber\\ &-\frac{N^2}{2}\left(\frac{R_{12}^4}{4} -R_{12}^4 \log R_{12} - \frac{R_{1}^4}{4} + R_{1}^4 \log R_{1}- \frac{R_{2}^4}{4}+ R_{2}^4 \log R_{2}\right)\nonumber\\ &- M_1 N \left(R_{12}^2 \log R_{12} - \frac{R_{12}^2}{2} - R_{1}^2 \log R_{1} + \frac{R_{1}^2}{2}\right) \nonumber\\ &- {M_2 N} \left(R_{12}^2 \log R_{12} - \frac{R_{12}^2}{2} - R_{2}^2 \log R_{2} + \frac{R_{2}^2}{2}\right). \end{align}\qquad{(8)}\]
Proof. Existence and uniqueness of a minimizer \(\mu_{\rm eq}\) is standard for this convex functional, see e.g. [36] or [5]. The Euler-Lagrange equation takes the form \[\begin{align} \label{eq:ELeq} NJ |\mathbf{x}|^2 -2 J \log |\, . \,| \star \left(J \mu_{\rm eq}- \sum_{k=1} ^M \delta_{\mathbf{w}_j} \right)&= Con\mathrm{supp} (\mu_{\rm eq}) \nonumber\\ NJ |\mathbf{x}|^2 -2 J \log |\, . \,| \star \left(J \mu_{\rm eq}- \sum_{k=1} ^M \delta_{\mathbf{w}_j} \right)&\geq Con\mathrm{supp} (\mu_{\rm eq}) ^c \end{align}\tag{29}\] for a constant \(C\in \mathbb{R}\) (Lagrange multiplier for the mass constraint). A useful characterization [36] is that if 29 holds for some probability measure \(\mu_{\rm eq}\) and some constant \(C\), then \(\mu_{\rm eq}\) must be the unique minimizer. We thus argue that 26 satisfies this, with \(C= C_R\) as in 27 .
First observe that 25 and 17 imply that 26 indeed is a probability measure. Next it follows from Newton’s theorem (see [37] that \[\label{eq:Newton} -\frac{1}{\pi} \log |\, . \,| \star \mathbb{1}_{D(0,R)} (\mathbf{x}) = \begin{cases} - R^2 \log |\mathbf{x}|for|\mathbf{x}| \geq R\\ - \frac{|\mathbf{x}|^2}{2} + \frac{R^2}{2} -R^2 \log Rfor|\mathbf{x}| \leq R. \end{cases}\tag{30}\] Combining with 16 and observing that \[NJ r^2 -2 NJ R^2 \log r \geq C_Rforr\geq R\] we find that \[\begin{gather} NJ |\mathbf{x}|^2 -2 J \log |\, . \,| \star \left(J \mu_{\rm eq}- \sum_{k=1} ^M \delta_{\mathbf{w}_j} \right) \\= NJ \left(|\mathbf{x}|^2 - \frac{2}{\pi} \log |\, . \,| \star \mathbb{1}_{D(0,R)}\right) - 2 J \log |\, . \,| \star \left(\sum_{k=1} ^M \delta_{\mathbf{w}_j} - \frac{N}{\pi} \mathbb{1}_{H}\right) \end{gather}\] indeed satisfies the desired conditions 29 . Multiplying those by \(\mu_{\rm eq}\) and integrating we find the expression of the energy in 27 , thus concluding the proof of Item (i).
We turn to Item (ii). Let \(\mu_{\rm eq}^\mathbf{a}\) be the equilibrium measure corresponding to the pinned charges at \(\mathbf{w}_1 (\mathbf{a}) = \mathbf{w}_1 + \mathbf{a}, \dots, \mathbf{w}_M (\mathbf{a}) = \mathbf{w}_M + \mathbf{a}\). From 27 we have that \[\nabla_\mathbf{a}E^{\rm MF}(\mathbf{a}) = - \frac{N^2}{2 \pi^2} \nabla_\mathbf{a}D\left( \mathbb{1}_{D(0,R) \setminus H(\mathbf{a})}, \mathbb{1}_{D(0,R) \setminus H(\mathbf{a})}\right).\] Denote \[\mathrm{Emp}^\mathbf{a}:= \frac{\pi}{N} \sum_{j=1}^M \delta_{\mathbf{w}_j (\mathbf{a})}\] and write \[\begin{align} D\left( \mathbb{1}_{D(0,R) \setminus H(\mathbf{a})}, \mathbb{1}_{D(0,R) \setminus H(\mathbf{a})}\right) &= D\left( \mathbb{1}_{D(0,R)}, \mathbb{1}_{D(0,R)}\right) + D\left( \mathbb{1}_{H(\mathbf{a})}, \mathbb{1}_{H(\mathbf{a})}\right) - 2 D\left( \mathbb{1}_{D(0,R) }, \mathbb{1}_{H(\mathbf{a})}\right) \\ &= D\left( \mathbb{1}_{D(0,R)}, \mathbb{1}_{D(0,R)}\right) + D\left( \mathbb{1}_{H(\mathbf{a})}, \mathbb{1}_{H(\mathbf{a})}\right) - 2 D\left( \mathbb{1}_{D(0,R) \setminus H(\mathbf{a})}, \mathbb{1}_{H(\mathbf{a})}\right) \\ &- 2 D\left( \mathbb{1}_{H(\mathbf{a})}, \mathbb{1}_{H(\mathbf{a})}\right) \\ &= D\left( \mathbb{1}_{D(0,R)}, \mathbb{1}_{D(0,R)}\right) - D\left( \mathbb{1}_{H(\mathbf{a})}, \mathbb{1}_{H(\mathbf{a})}\right) - 2 D\left( \mathbb{1}_{D(0,R) \setminus H(\mathbf{a})}, \mathrm{Emp}^\mathbf{a}\right)\\ &= D\left( \mathbb{1}_{D(0,R)}, \mathbb{1}_{D(0,R)}\right) - D\left( \mathbb{1}_{H(\mathbf{a})}, \mathbb{1}_{H(\mathbf{a})}\right) + 2 D\left( \mathbb{1}_{H(\mathbf{a})}, \mathrm{Emp}^\mathbf{a}\right) \\ &- 2 D\left( \mathbb{1}_{D(0,R)}, \mathrm{Emp}^\mathbf{a}\right) \end{align}\] where we used 16 to get the third equality. It follows from Definition 1 that \(H(\mathbf{a})\) is just \(H(0)\) translated by \(\mathbf{a}\). Hence only the very last term of the right-hand side does depend on \(\mathbf{a}\). Recalling 30 we find that \[\begin{align} \nabla_\mathbf{a}D\left( \mathbb{1}_{D(0,R)}, \mathrm{Emp}^\mathbf{a}\right) &= - \frac{\pi^2}{2N} \nabla_\mathbf{a}\left( \sum_{j=1}^M |\mathbf{w}_j + \mathbf{a}|^2\right) \\ &= - \frac{\pi^2}{N} \sum_{j=1} ^M \left( \mathbf{w}_j + \mathbf{a}\right). \end{align}\] Combining with the two previous equations gives 28 .
As regards Item (iii), first note that since \(H_1 \cap H_2 = \varnothing\) we have from Definition 1 that \(H= H_1 \cup H_2\) is a screening region for the total set of points \(\mathbf{w}_k\in \mathbb{R}^2, k= 1 \dots M\). Hence 27 and 26 lead to \[\begin{align} 2 E^{\rm MF}_{12} &= C_{R_{12}}- \frac{N^2}{\pi^2} \left( D\left(\mathbb{1}_{D(0,R_{12})},\mathbb{1}_{D(0,R_{12})}\right) - 2 D\left(\mathbb{1}_{D(0,R_{12})},\mathbb{1}_{H_1} \right) - 2D\left(\mathbb{1}_{D(0,R_{12})},\mathbb{1}_{H_2} \right) \right) \\ &- \frac{N^2}{\pi^2} \left(D\left(\mathbb{1}_{H_1},\mathbb{1}_{H_1}\right) + D\left(\mathbb{1}_{H_2},\mathbb{1}_{H_2}\right) + 2 D\left(\mathbb{1}_{H_1},\mathbb{1}_{H_2}\right) \right) \end{align}\] with related expressions for \(E^{\rm MF}_{1}, E^{\rm MF}_{2}\). Hence \[\begin{align} \label{eq:MFEdiff} 2\left(E^{\rm MF}_{12} - E^{\rm MF}_{1} - E^{\rm MF}_{2}\right) &= C_{R_{12}} - C_{R_{1}} - C_{R_{2}} \nonumber\\ &- \frac{N^2}{\pi^2} \left( D\left(\mathbb{1}_{D(0,R_{12})},\mathbb{1}_{D(0,R_{12})}\right) - D\left(\mathbb{1}_{D(0,R_{1})},\mathbb{1}_{D(0,R_{1})}\right) - D\left(\mathbb{1}_{D(0,R_{2})},\mathbb{1}_{D(0,R_{2})}\right)\right) \nonumber\\ &+2 \frac{N^2}{\pi^2} D\left(\mathbb{1}_{D(0,R_{12})\setminus D(0,R_{1})},\mathbb{1}_{H_1}\right) + 2 \frac{N^2}{\pi^2} D\left(\mathbb{1}_{D(0,R_{12})\setminus D(0,R_{2})},\mathbb{1}_{H_2}\right) \nonumber\\ &- 2 \frac{N^2}{\pi^2}D\left(\mathbb{1}_{H_1},\mathbb{1}_{H_2}\right) \end{align}\tag{31}\] Returning to 30 we have \[D\left(\mathbb{1}_{D(0,R)},\mathbb{1}_{D(0,R)}\right) = \frac{\pi ^2 R^4}{4} - \pi^2 R^4 \log R.\] On the other hand, using Newton’s theorem [37] again implies that the Coulomb potential generated by \(\mathbb{1}_{D(0,R_{12})\setminus D(0,R_{1})}\) is constant inside \(D(0,R_{1})\), wherein \(H_1\) is included. Hence, using 16 , \[\begin{align} D\left(\mathbb{1}_{D(0,R_{12})\setminus D(0,R_{1})},\mathbb{1}_{H_1}\right) &= - \frac{\pi}{N} \sum_{j=1} ^{M_1} \log |\, . \,| \star \mathbb{1}_{D(0,R_{12})\setminus D(0,R_{1})} (\mathbf{w}_{1,j}) \\ &= - \pi \frac{M_1}{N} \log |\, . \,| \star \mathbb{1}_{D(0,R_{12})\setminus D(0,R_{1})} (0)\\ &= \pi ^2 \frac{M_1}{N} \left(R_{1}^2 \log R_{1} - \frac{R_{1}^2}{2} - R_{12}^2 \log R_{12} + \frac{R_{12}^2}{2}\right) \end{align}\] and a similar expression with \(R_1,H_1\) replaced by \(R_2,H_2\). Since \(H_1\cap H_2 = \varnothing\) it also follows from 16 that \[D\left(\mathbb{1}_{H_1},\mathbb{1}_{H_2}\right) = - \frac{\pi^2}{N^2} \sum_{j=1}^{M_1} \sum_{k=1}^{M_2} \log |\mathbf{w}_{1,j} - \mathbf{w}_{2,k}|.\] Combining the above calculations and inserting them in 31 leads to ?? . ◻
Our general strategy for proving Theorem 1 is as follows:
\(\bullet\) Since we are dealing with \(M\) distinct charges distributed around \(\mathbf{a}\), we can apply a simple exact formula for the corresponding partition function, originating in [29], [30] and used extensively in [31].
\(\bullet\) The formula gives ?? up to the log of a determinant based on the finite \(N\) Ginibre correlation kernel. Replacing the latter with the infinite area, translation invariant, correlation kernel, and controling the error thus made, ?? follows suit.
Let then \[\label{eq:ref32points} \left(\mathbf{w}_1,\ldots,\mathbf{w}_{M}\right)\in \mathbb{R}^{2M}\tag{32}\] be a reference cloud of distinct points. We assume ?? and ?? .
We identify the vectors \(\mathbf{w}_1,\ldots,\mathbf{w}_M\) with complex numbers \(w_1,\ldots,w_M\) and \(\mathbf{a}\) with the complex number \(a\). Define \[\label{eq:partdiltran} \mathcal{Z}_{N}(\mathbf{a}) := \int_{\mathbb{R}^{2N}}\prod_{1\leq j < k } |z_j - z_k| ^2 e^{-N\sum_{j=1}^N |z_j|^2} \prod_{j=1}^N \prod_{k= 1}^M \left|z_j- (w_k + a)\right|^{2} dz_1\ldots dz_N.\tag{33}\] We shall use the a priori information that 8 is already known rigorously up to \(o_N (N)\): \[\label{eq:a32priori} F_N (\mathbf{a}) = -\frac{1}{2} \log \mathcal{Z}_{N}(\mathbf{a}) = E^{\rm MF}(N,N,M) - \frac{N}{4} \log N + \frac{N}{2} \beta f_2 (\beta) + o_N (N)\tag{34}\] where \(E^{\rm MF}(N,N,M)\) is the mean-field energy from section 3 at \(J= N\) and \(\beta f_2 (\beta)\) is the infinite area Jellium free-energy density, at inverse temperature \(\beta = 2\), as defined in [5]. We use the above at \(\beta = 2\) where estimates for the Ginibre ensemble imply \[2 f_2 (2) = 2 \left( \frac{\log 2\pi}{2} - 1 \right).\]
We note that the validity of 34 is usually investigated for a smooth, fixed external potential, not that generated by point charges that we consider. However, since the singularities generated by the point charges are outside of the droplet, a careful inspection of the known proofs shows that they carry over to our case. In fact, our arguments below only require the direction \(\leq\) of 34 which, as per 3 , is the “easy” direction of the variational principle. Constructing a good trial state is sufficient for our needs.
We start our investigation of the remainder term in 34 by recalling an exact formula:
Lemma 1 (Exact expression for partition functions with pinned unit charges).
With the notation above \[\begin{gather}
\label{eq:exact32part}
\mathcal{Z}_{N}(\mathbf{a}) = \mathcal{Z}_{N+M} (0) \frac{N!}{(N+M)!}\\
\det_{M\times M} \left[\mathrm{K}_{N+M}(w_i + a,w_j +a) \right] \frac{\prod_{j=1}^M e^{N|w_j +a|^2}}{\prod_{1\leq i <j \leq M} |w_i-w_j|^2}
\end{gather}\qquad{(9)}\] where \[\label{eq:KN}
K_{J} (z,w) = e ^{-\frac{N}{2}|z|^2 -\frac{N}{2}|w|^2} \sum_{j= 0}^{J} \frac{N^{j+1}}{\pi j!} z^j \overline{w}^j\qquad{(10)}\] with the appropriate normalization is the Ginibre correlation kernel for \(J\)
particles in a background charge density \(-4N\).
Proof. This originates in [29], [30], see for example [31] for a proof of ?? . We used that for \(M\) distinct points \(w_1,\dots,w_M\) \[\begin{align} \label{eq:Ginibre32marginal} \frac{1}{M!}&\det_{M\times M} \left[\mathrm{K}_{N+M}(w_i,w_j) \right] = \rho ^{(M)}_{N+M} (w_1,\ldots,w_M) := \frac{{N+M \choose M}}{\mathcal{Z}_{N+M}(0,0)}\nonumber\\ &\int_{\mathbb{R}^N}\prod_{1\leq j < k \leq N+M} |w_j - w_k| ^2 e^{-N\sum_{j=1}^{N+M} |w_j|^2} dw_{N+1} \ldots dw_{N+M} \end{align}\tag{35}\] the \(M\)-particles reduced density of a Ginibre ensemble with \(N+M\) particles and correlation kernel \(K_{N+M}\) as in ?? . ◻
We will need some accurate estimates on the determinant appearing in ?? . This is to ensure that the errors we will later make by replacing it with the \(N+M\to \infty\) version will indeed be negligible compared with its main contribution.
Lemma 2 (Lower bound on the determinant).
Under the previously stated assumptions, for a fixed positive constant \(C>0\) \[\label{eq:bound32det321} \det_{M\times M} \frac{\pi}{N}
\left[\mathrm{K}_{N+M}(w_i + a,w_j +a) \right] \geq \exp (- C \left( c - c\log c \right) N)\qquad{(11)}\] where \(c = M/N.\)
Proof. Starting from ?? and recalling the notation ?? we find \[\begin{gather} \label{eq:pouet} -\log \mathcal{Z}_{N}(\mathbf{a}) = -\log \mathcal{Z}_{N+M} (0) - 2 \mathcal{H}_N (\mathbf{w}_1,\ldots,\mathbf{w}_M) + \log \frac{(N+M)!}{N!} - M \log \frac{N}{\pi} \\- \log \det_{M\times M} \frac{\pi}{N} \left[\mathrm{K}_{N+M}(w_i + a,w_j +a) \right]. \end{gather}\tag{36}\] From Stirling’s formula we get \[\begin{align} \label{eq:AMN} A(M,N) &= \log \frac{(M+N)!}{N!} - M \log \frac{N}{\pi} \nonumber \\&= \frac{1}{2}\log \frac{N+M}{N}+(N+M) \log\frac{N+M}{N} + M(\log \pi - 1) +o_N (1)\nonumber \\&=(1+c) N \log (1+c) - cN \left( 1-\log\pi\right) + \frac{1}{2} \log (1+c) + o_N (1) \end{align}\tag{37}\] whereas asymptotics for the Ginibre ensemble recalled in 59 lead to (\(\beta = 2\)) \[\begin{gather} \label{eq:useGin} -\log \mathcal{Z}_{N+M} (0) + 2 A (M,N) = \frac{3}{4} (1+c)^2 N^2 - \frac{(1+c)^2}{2} N^2 \log (1+c) \\ - (1+c)\frac{N}{2} \log \left( N \right)+ (N+M)\beta f_2 (\beta) + c O(N)\\ = 2E^{\rm MF}(N+M,N,0) - \frac{1+c}{2} \log N + (N+M)\beta f_2 (\beta) + c O(N) \end{gather}\tag{38}\] where \(E^{\rm MF}(N+M,N,0)\) is the mean-field energy from Section 3 with \(J=N+M,M= 0\) and \(O(N)\) is a linear function of \(N\).
Combining with 34 with 36 and 38 and then in inserting ?? we find \[\begin{gather} \label{eq:quasi32det32bound} \log \det_{M\times M} \frac{\pi}{N} \left[\mathrm{K}_{N+M}(w_i + a,w_j +a) \right] \\ = 2E^{\rm MF}(N+M,N,0) - 2E^{\rm MF}(N,N,M) - \frac{N^2}{\pi} \int_{H} |\mathbf{x}|^2 d\mathbf{x}+ \frac{N ^2}{\pi ^2} D \left(\mathbb{1}_{H},\mathbb{1}_{H}\right) \\ + cN \beta f_2 (\beta) + cN \log c -cN \left( 1- \log \pi\right) + c O (N) \end{gather}\tag{39}\] where \(H\) is the screening region of the pinned charges. There now remains to observe that the terms on the second line cancel to conclude the proof.
Indeed, with \[M = cNandR= \sqrt{1+c}\] it follows from 26 27 that \[\begin{align} 2E^{\rm MF}(N+M,N,0) &= N(N+M)(1+c) - N (N+M)(1+c) \log (1+c) \\ &- \frac{N^2}{\pi^2} D\left(\mathbb{1}_{D(0,R)},\mathbb{1}_{D(0,R)}\right)\\ 2E^{\rm MF}(N,N,M)&= N^2 (1+c) - N^2 (1+c) \log (1+c) \\&- \frac{N^2}{\pi^2} D\left(\mathbb{1}_{D(0,R)},\mathbb{1}_{D(0,R)}\right) - \frac{N^2}{\pi^2} D\left(\mathbb{1}_{H},\mathbb{1}_{H}\right) + 2 \frac{N^2}{\pi^2} D\left(\mathbb{1}_{D(0,R)},\mathbb{1}_{H}\right) \end{align}\] and hence \[\begin{gather} 2E^{\rm MF}(N+M,N,0) - 2E^{\rm MF}(N,N,M) - \frac{N^2}{\pi} \int_{H} |\mathbf{x}|^2 d\mathbf{x}+ \frac{N ^2}{\pi ^2} D \left(\mathbb{1}_{H},\mathbb{1}_{H}\right) \\ = NM (1+c) - NM (1+c) \log (1+c) - 2 \frac{N^2}{\pi^2} D\left(\mathbb{1}_{D(0,R)},\mathbb{1}_{H}\right) - \frac{N^2}{\pi} \int_{H} |\mathbf{x}|^2 d\mathbf{x}= 0 \end{gather}\] where we used 30 and 17 to compute \(D\left(\mathbb{1}_{D(0,R)},\mathbb{1}_{H}\right)\) in the last step.
Inserting in 39 and exponentiating the resulting expression concludes the proof. ◻
We now use the exact formula from Lemma 1 to investigate the effect of a joint translation of the pinned charges. To this effect we first replace the correlation kernel \(K_{N+M}\) by the corresponding, infinite area, kernel \(K_\infty\). The error thus made is controled thanks to Lemma 2.
Lemma 3 (Inserting the translation-invariant kernel).
Let \[\begin{align}
\label{eq:Kinfty}
K_{\infty} (z,w) &= e ^{-\frac{N}{2}|z|^2 -\frac{N}{2}|w|^2} \sum_{j= 0}^{\infty} \frac{N^{j+1}}{\pi j!} z^j \overline{w}^j\nonumber \\
&=\frac{N}{\pi} e^{-\frac{N}{2}\left(|z|^2 + |w|^2 - 2 z \overline{w} \right)}\nonumber \\
&= \frac{N}{\pi} e^{-\frac{N}{2}\left(|z-w|^2 - \mathrm{i}(\mathbf{z}-\mathbf{w}) \cdot (\mathbf{z}+ \mathbf{w})^{\perp} \right)}
\end{align}\qquad{(12)}\] and \[\begin{gather}
\label{eq:Zinfty}
\mathcal{Z}_{N}^\infty (\mathbf{a}) := \mathcal{Z}_{N+M} (0) \frac{N!}{(N+M)!}\\
\det_{M\times M} \left[\mathrm{K}_{\infty}(w_i + a,w_j +a) \right] \frac{\prod_{j=1}^M e^{N|w_j +a|^2}}{\prod_{1\leq i <j \leq M} |w_i-w_j|^2}
\end{gather}\qquad{(13)}\] we have that, for \(|\mathbf{a}|,c\) small enough, \[\label{eq:Zinfty2}
-\log \mathcal{Z}_{N}(\mathbf{a}) = -\log \mathcal{Z}_{N}^\infty (\mathbf{a}) + o_N(1)\qquad{(14)}\] in the limit \(N\to \infty\).
Proof. Comparing ?? with ?? , we need to prove that \[\begin{gather} \left|\det_{M\times M}\left[\frac{\pi}{N}\mathrm{K}_{N+M}(w_i + a,w_j +a) \right]-\det_{M\times M}\left[\frac{\pi}{N}\mathrm{K}_{\infty}(w_i + a,w_j +a) \right]\right|\\ \ll \det_{M\times M}\left[\frac{\pi}{N}\mathrm{K}_{N+M}(w_i + a,w_j +a) \right] \end{gather}\] in the limit \(N\to \infty\). In view of Lemma 2 it suffices to prove that \[\label{eq:bound32det322} \left|\det_{M\times M}\left[\frac{\pi}{N}\mathrm{K}_{N+M}(w_i + a,w_j +a) \right]-\det_{M\times M}\left[\frac{\pi}{N}\mathrm{K}_{\infty}(w_i + a,w_j +a) \right]\right| \leq e^{-D N}\tag{40}\] for some fixed \(D>0\), and then use the fact that \(c=M/N\) is assumed small enough.
We prove 40 by arguing as in [38]. Let \(H_k\) be the matrix
whose \(k-1\) first columns are the vectors \[v_{kj} := \left(\frac{\pi}{N} K_{\infty}(w_i + a,w_j +a)\right)_{i=1,\ldots,M}\] for \(j=1 \ldots k-1\)
whose \(k\)-th column is the vector \[v_{kk} := \left(\frac{\pi}{N} K_{N+M}(w_i + a,w_k +a) - \frac{\pi}{N}\mathrm{K}_{\infty}(w_i + a,w_j +a) \right)_{i=1,\ldots,M}\]
whose \(M-k\) last columns are the vectors \[v_{kj} := \left(\frac{\pi}{N} K_{N+M}(w_i + a,w_j +a)\right)_{i=1,\ldots,M}\] for \(j=k+1 \ldots M\).
By linearity of the determinant with respect to columns we have \[\label{eq:lin32det} \det_{M\times M}\left[\frac{\pi}{N}\mathrm{K}_{N+M}(w_i + a,w_j +a) \right]-\det_{M\times M}\left[\frac{\pi}{N}\mathrm{K}_{\infty}(w_i + a,w_j +a) \right] = \sum_{k=1} ^M \det_{M\times M} H_k\tag{41}\] and by Hadamard’s inequality \[\label{eq:Hadamard} \left| \det_{M\times M} H_k\right| \leq \prod_{j=1} ^M \left(\sum_{i=1} ^M |v_{kj}^i|^2 \right)^{1/2}\tag{42}\] with \(v_{kj}^i\) the \(i\)-th element of the vector \(v_{kj}\). We will bound the above terms using the estimates on correlation kernels recalled in Appendix 6. To this end, note that ?? with \(r_2\) small enough and a choice of \(|\mathbf{a}|\) small enough imply that \[|w_j+a| \leq \sqrt{1 +c} - \delta\] for some \(\delta >0\), so that we may in particular use 64 to obtain \[\label{eq:Kdiff32main} \left| \mathrm{K}_{N+M}(w_i + a,w_j +a)-\mathrm{K}_{\infty}(w_i + a,w_j +a)\right| \leq C e^{-C\delta N}\tag{43}\] for all \(i,j\).
Hence, using 60 and 64 we have, for \(j\neq k\) \[\label{eq:use32Hadamard} \sum_{i=1} ^M |v_{kj}^i|^2 \leq C \sum_{i=1} ^M \left( e^{-N|w_i - w_j| ^2} + C e^{-CN}\right).\tag{44}\] But, in view of our choice of configuration \(\mathbf{w}_1,\ldots,\mathbf{w}_M\), in particular ?? , the points can be sorted into clusters whose distance to a given \(w_j\) is between \(LN^{-1/2}\) and \((L+1)N^{-1/2}\), for integers \(L\). The number of points in the \(L\)-th cluster cannot exceed \(CL\) for some fixed constant \(C\), and drops to \(0\) for \(L\geq C N^{1/2}\). Hence for \(j\neq k\) \[\label{eq:use32Hadamard322} \sum_{i=1} ^M |v_{kj}^i|^2 \leq C \sum_{L=0} ^{C\sqrt{N}} CL \left( e^{-C L ^2} + C e^{-CN}\right) \leq C.\tag{45}\] On the other hand 43 gives, for \(j=k\) \[\label{eq:main32gain} \sum_{i=1} ^M |v_{kk}^i|^2 \leq M e^{-C \delta N}\tag{46}\] Hence, combining 41 and 1 with 45 and 46 we obtain a bound for the left-hand side of 40 of the order \(M^{3/2} C^M e^{-C\delta N}\). Recalling that \(M= cN\) and that \(\delta\) can be bounded below by a fixed positive constant for \(c,|\mathbf{a}|\) small enough yields the desired 40 . ◻
We now use translation-invariance of the Ginibre process (whose correlation kernel is \(K_\infty\)) to compute the gradient of the modified partition function ?? :
Lemma 4 (Translation of the pinned charges).
With \(\mathcal{Z}_{N}^\infty (\mathbf{a})\) as in ?? we have that \[\label{eq:moveonehole2}
\nabla_\mathbf{a}\log \mathcal{Z}_{N}^\infty (\mathbf{a})= 2 c \mathbf{a}N^2 + 2 N \sum_{j=1} ^M \mathbf{w}_j\qquad{(15)}\]
Proof. We use that the \(\log\) of ?? is the sum of several terms, only two of which do depend on \(\mathbf{a}\). In particular, the van der Monde determinant in the denominator gives no contribution.
We have \[\sum_{j=1}^M |\mathbf{w}_j + \mathbf{a}| ^2 = M |\mathbf{a}|^2 + 2 \mathbf{a}\cdot \sum_{j=1}^M \mathbf{w}_j + \sum_{j=1}^M |\mathbf{w}_j|^2\] and hence, recalling 15 , \[\nabla_\mathbf{a}\log \mathcal{Z}_{N}^\infty (\mathbf{a}) = 2c N^2 \mathbf{a}+ 2 N \sum_{j=1} ^M \mathbf{w}_j- \nabla_\mathbf{a}\log \det_{M\times M} \left[\mathrm{K}_{\infty}(w_i +a,w_j +a) \right]\] and there remains to observe that \(\det_{M\times M} \left[\mathrm{K}_{\infty}(w_i +a,w_j +a) \right]\) does not depend on \(\mathbf{a}\) either. Indeed, according to 35 and ?? , it is proportional to the \(M-\)particles density of a translation-invariant point process (the Ginibre point process on the full plane). More precisely, using the third formula in ?? \[\nabla_\mathbf{a}\mathrm{K}_{\infty}(w_i +a,w_j +a) = -\mathrm{i}N \mathbf{a}^\perp \cdot (\mathbf{w}_i - \mathbf{w}_j) \mathrm{K}_{\infty}(w_i +a,w_j +a)\] and hence, expanding the determinant, \[\begin{gather} \nabla_\mathbf{a}\det_{M\times M} \left[\mathrm{K}_{\infty}(w_i +a,w_j +a)\right] \\ = -\mathrm{i}N \mathbf{a}^\perp \cdot \sum_{\sigma \in \mathfrak{S}_M} \mathrm{sgn}(\sigma) \sum_{j=1} ^M (\mathbf{w}_j - \mathbf{w}_{\sigma(j)})\prod_{i=1}^M \mathrm{K}_{\infty}\big(w_i +a,w_{\sigma(i)}+a\big) = 0 \end{gather}\] because certainly \[\sum_{j=1} ^M \mathbf{w}_j - \sum_{j=1} ^M\mathbf{w}_{\sigma(j)} = 0\] for any permutation. This concludes the proof. ◻
We may now conclude the proof of Theorem 1. The argument is similar to ideas of [23].
Proof of Theorem 1. Starting from Lemma 3, we have that, under the stated assumptions \[F^{\rm Corr}_N (\mathbf{a}) = - \frac{1}{2} \mathcal{Z}_{N}^\infty (\mathbf{a}) - N^2 E^{\rm MF}(\mathbf{a}) + o_N (1)\] and \[F^{\rm Corr}_N (\mathbf{a}) - F^{\rm Corr}(0) = - \frac{1}{2} \mathcal{Z}_{N}^\infty (\mathbf{a}) - N^2 E^{\rm MF}(\mathbf{a}) + \frac{1}{2} \mathcal{Z}_{N}^\infty (0) + N^2 E^{\rm MF}(0) + o_N (1).\] But, combining 28 and ?? we conclude that the map \[\mathbf{a}\mapsto - \frac{1}{2} \mathcal{Z}_{N}^\infty (\mathbf{a}) - N^2 E^{\rm MF}(\mathbf{a})\] is constant, and thus complete the proof. ◻
The main technical input in the proof of Theorem 3 is a decoupling lemma for the determinant obtained by applying Lemma 1 to \(\mathcal{Z}_N(\mathbf{a}_1,\ldots,\mathbf{a}_n)\). We show that the main contribution is the product of the determinants obtained from applying the lemma to \(\mathcal{Z}_N(\mathbf{a}_j)\) for \(j=1,\ldots,\mathbf{a}_n)\). This is certainly intuitive: the multiple-holes-configuration’s total determinant is made of diagonal blocks corresponding to each of the one-hole determinants, complemented with off-diagonal blocks whose fast decay can be controled via the estimates recalled in Appendix 6. This is a clustering property for correlation functions of a Ginibre ensemble when their arguments are sufficiently separated in space.
The rest of the proof follows by inserting the exact formulae for Ginibre partition functions that we recall in Appendix 6 and comparing with the properties of the mean-field problem discussed in Section 3.
We state the decoupling lemma directly for the infinite Ginibre ensemble, replacing finite-\(N\) correlation kernels by \(K_\infty\).
Lemma 5 (Decoupling the multiple-holes determinant).
We concatenate the \(n\) lists of points \(\left( \mathbf{w}_{j,k}\right)_{k=1 \ldots M_j}\) (with \(j=1\ldots n\)) into a single list \(\mathbf{W}= (\mathbf{w}_1,\ldots,\mathbf{w}_M)\) of cardinal \(M\) to define the \(M\times M\) matrix \[\label{eq:big32mat}
\mathcal{K}^M := \left( \frac{\pi}{N} K_{\infty} (w_j,w_k)\right)_{1\leq j, k \leq M}.\qquad{(16)}\] Under Assumptions 1 and 2 we have that \[\label{eq:decouple32fin}
\log \det_{M\times M} \mathcal{K}^M = \sum_{j=1} ^M \log \det_{M_j\times M_j} \left(\frac{\pi}{N} K_{\infty} (\mathbf{w}_{j,k},\mathbf{w}_{j,\ell})_{1\leq k,\ell \leq M_j} \right) +o_N (1)\qquad{(17)}\] where, by contrast with ?? we use the
labeling of points into several different groups.
Proof. We define \[\label{eq:big32mat32bis} \mathcal{K}^{M,j} := \left( \frac{\pi}{N} K_{\infty} (w_j,w_k)\right)_{1\leq j, k \leq \sum_{k=1} ^j M_k}\tag{47}\] similary to \(\mathcal{K}_M\), but concatenating only the first \(j\) groups of points. That way in particular \(\mathcal{K}^M = \mathcal{K}^{M,n}\). It suffices to prove that \[\label{eq:decouple} \log \det \mathcal{K}^{M,j} = \log \det_{M_j\times M_j} \left(\frac{\pi}{N} K_{\infty} (\mathbf{w}_{j,k},\mathbf{w}_{j,\ell})_{1\leq k,\ell \leq M_j} \right) + \log \det \mathcal{K}_{M,j-1} +o_N (1)\tag{48}\] for all \(j=2,\ldots,M\) and iterate this relation. We next fix \(j\geq 2\) and prove 48 . Proceeding by induction we are free to assume \[\label{eq:decouple32int} \log \det \mathcal{K}_{M,j-1} = \sum_{k=1} ^{j-1} \log \det_{M_k\times M_k} \left(\frac{\pi}{N} K_{\infty} (\mathbf{w}_{k,\ell},\mathbf{w}_{k,m})_{1\leq \ell,m \leq M_k} \right) +o_N (1)\tag{49}\] We split the points entering in the definition of \(\mathcal{K}^{M,j}\) into two groups: the \(A\) group consisting of the points \[\mathbf{w}^A_{1}, \ldots, \mathbf{w}^A_{M_j} = \mathbf{w}_{j,1},\ldots,\mathbf{w}_{j,M_j}\] and the \(B\) group consisting of the other points, \[\mathbf{w}^B_{1}, \ldots, \mathbf{w}^B_{N_{j-1}} = \mathbf{w}_{k,\ell}, \quad k= 1 \ldots j-1, \ell = 1, \ldots, M_k\] with \[N_j = \sum_{k=1} ^j M_j.\] We then expand the determinant \[\det \mathcal{K}_{M,j} = \sum_{\sigma \in \Sigma_{N_j}} \operatorname{sgn}(\sigma) \prod_{k=1} ^{N_j} \mathcal{K}^{M,j}_{k,\sigma(k)}\] where the sum is over the permutation group of \(N_j\) elements. For clarity of notation we assume that \(M_j \leq N_{j-1}\), with simple modifications to the sequel in case the relation is reversed.
Next we split the previous sum according to the number \(m\) of \(A\) elements that the permutation \(\sigma\) sends to \(B\) elements. We will denote \[I_m = (i_1,\ldots,i_m), \quad J_m = (j_1,\ldots,j_m)\] generic \(m\)-elements subsets of \(\{1,\ldots,M_j\}\) and \(\{1,\ldots,N_{j-1}\}\) respectively, and use them to label these inter-groups permutations. Then \[\begin{gather} \det \mathcal{K}_{M,j} = \sum_{m=0} ^{M_j} (-1)^m \sum_{I_m} \sum_{J_m} \sum_{\sigma \in \Sigma_{M_j-m}} \sum_{\sigma' \in \Sigma_{N_{j-1}-m}} \operatorname{sgn}(\sigma) \operatorname{sgn}(\sigma') \\ \prod_{k=1} ^m \widetilde{K}\left(\mathbf{w}_{i_k}^A,\mathbf{w}_{j_k}^B\right) \prod_{h\in I_m^c} \prod_{f\in J_m^c} \widetilde{K}\left(\mathbf{w}_h^A,\mathbf{w}_{\sigma(h)} ^A\right) \widetilde{K}\left(\mathbf{w}_f^B,\mathbf{w}_{\sigma'(f)} ^B\right) \end{gather}\] with \[\widetilde{K}:= \frac{\pi}{N} K_\infty\] and where the sums over permutations \(\sigma,\sigma'\) are (with an abuse of notation) over the indices of \[I_m^c := \left\{1,\ldots,M_j\right\}\setminus I_m\] and \[J_m^c :=\left\{1,\ldots,N_{j-1}\right\}\setminus J_m\] respectively. Grouping some terms we reduce the above to \[\begin{gather} \label{eq:combinatoire} \det \mathcal{K}_{M,j} = \sum_{m=0} ^{M_j} (-1)^m \sum_{I_m} \sum_{J_m} \prod_{k=1} ^m \widetilde{K}\left(\mathbf{w}_{i_k}^A,\mathbf{w}_{j_k}^B\right) \\ \det_{(M_j - m)\times (M_j - m)} \left( \widetilde{K}(\mathbf{w}_h^A,\mathbf{w}_{h'} ^A)\right)_{h,h'\in I_m ^c} \, \times \, \det_{(N_{j-1} - m)\times (N_{j-1} - m)} \left( \widetilde{K}(\mathbf{w}_f^B,\mathbf{w}_{h'} ^B)\right)_{f,f'\in J_m ^c}\\ =: \det_{M_j\times M_j} \left(\frac{\pi}{N} K_\infty (\mathbf{w}_{j,k},\mathbf{w}_{j,\ell})_{1\leq k,\ell \leq M_j} \right) \times \det \mathcal{K}_{M,j-1} +I_{m\geq 1} \end{gather}\tag{50}\] where we have isolated the \(m=0\) term in the last equality. Taking the \(\log\) and using \[\log (x+y) = \log (x) + \log \left(1+\frac{x}{y}\right)\] yields the desired terms from the right-hand side of 48 , with an error suitably small if we prove that \[\label{eq:decouple10} I_{m \geq 1} \ll \det_{M_j\times M_j} \left(\frac{\pi}{N} K_{\infty} (\mathbf{w}_{j,k},\mathbf{w}_{j,\ell})_{1\leq k,\ell \leq M_j} \right) \times \det \mathcal{K}_{M,j-1}\tag{51}\] for large \(N\), where \(I_{m\geq 1}\) is sum from 50 , minus the \(m=0\) term.
Under our assumptions, Lemma 2 applies to the two determinants above and gives the lower bound \[\det_{M_j\times M_j} \left(\frac{\pi}{N} K_{\infty} (\mathbf{w}_{j,k},\mathbf{w}_{j,\ell})_{1\leq k,\ell \leq M_j} \right) \times \det \mathcal{K}_{M,j-1} \geq e^{-C (c - c\log c)N}.\] Hence, for sufficiently small \(c\), it suffices to prove that \[\label{eq:decouple11} \left|I_{m \geq 1}\right| \leq e^{-CN}\tag{52}\] for a fixed constant \(C>0\). This will imply 51 , and inserting in 50 will conclude the proof.
We now turn to the proof of 52 . Recall that the points from groups \(A\) and \(B\) are by definition separated by a minimal, finite distance. As per 60 and Assumption 2 we find that, for any set of indices \(I_m,J_m\), \[\prod_{k=1} ^m \widetilde{K}\left(\mathbf{w}_{i_k}^A,\mathbf{w}_{j_k}^B\right) \leq e^{-C d^2 m N}\] where \(d\) is the minimal distance between points of the \(A\) and \(B\) groups. On the other hand, arguing as in the proof of Lemma 3, Hadamard’s inequality gives, with an argument similar to 44 , \[\left|\det_{(M_j - m)\times (M_j - m)} \left( \widetilde{K}(\mathbf{w}_h^A,\mathbf{w}_{h'} ^A)\right)_{h,h'\in I_m ^c} \right| \leq C^{M_j - m} \leq C^{cN}\] and \[\left|\det_{(N_{j-1} - m)\times (N_{j-1} - m)} \left( \widetilde{K}(\mathbf{w}_f^B,\mathbf{w}_{h'} ^B)\right)_{f,f'\in J_m ^c}\right| \leq C^{N_{j-1} - m}\leq C^{cN}\] for all such terms appearing in 50 . We have used that by definition \(M_j,N_{J-1}\leq M = cN\). Inserting these bounds in 50 and counting terms with \(m\) links from group \(A\) to group \(B\) we find \[\begin{align} |I_{m\geq 1}| &\leq \sum_{m = 1} ^{M_j} \frac{M_j ! N_{j-1}!}{(M_j-m)!(N_{j-1}-m)!} C^{2cN} e^{-Cd^2 mN} \\&\leq \sum_{m = 1} ^{M_j} e^{m\log (M_j)} e^{m\log N_{j-1} } e^{-C mN} \\&\leq \sum_{m = 1} ^{M_j} e ^{2 m\log M} e^{-C mN} \leq e^{-CN} \end{align}\] if the constant \(r_1\) in Assumption 2 is small enough. Indeed, this assumption implies \(c\leq r_1 d^2\). This concludes the proof. ◻
Lemma 5 will allow to compare \(F_N (\mathbf{a}_1,\ldots,\mathbf{a}_M)\) to \(\sum_{j=1}^M F_N (\mathbf{a}_j)\). Subtracting the appropriate mean-field energies and using results from Section 3 will then conclude the proof of Theorem 3. Let us first give the direct comparison between free energies. We denote \[\label{eq:int32ener} \mathcal{I}_{\mathrm{Int}}^{jk} := \sum_{\ell = 1} ^{M_j} \sum_{m=1} ^{M_k} -\log |\mathbf{w}_{j,\ell} - \mathbf{w}_{k,m}|.\tag{53}\]
Proposition 5 (Comparison of multiple-holes and single-holes free energies).
Under Assumptions 1 and 2 we have that \[\begin{gather}
\label{eq:freeeners}
F_N (\mathbf{a}_1,\ldots,\mathbf{a}_n) = \sum_{j=1} ^n F_N (\mathbf{a}_j) - \sum_{1\leq j < k \leq n} \mathcal{I}_{\mathrm{Int}}^{jk} \\
+ \frac{3N^2}{8} \left( (1-n) + c^2 - \sum_{j=1} ^n c_j^2 - \frac{2}{3}(1+c)^2 \log (1+c) + \frac{2}{3}\sum_{j=1} ^n (1+c_j)^2 \log (1+c_j)\right)\\
+\frac{n-1}{4} \frac{\log N}{N} + \frac{(n-1)}{2}\left(\frac{\log 2\pi}{2} - 1\right)N + \frac{5(n-1)}{24}\log N\\
+\frac{(n-1)}{2}\left(\zeta' (-1) + \frac{\log 2\pi}{2}\right) + \frac{1}{24}\left(\log (1+c) - \sum_{j=1} ^n \log (1+c_j) \right) + o_N (1).
\end{gather}\qquad{(18)}\]
Proof. Reproducing the proof of Lemma 1 to compute \(\mathcal{Z}_N (\mathbf{a}_1,\ldots,\mathbf{a}_n)\) we obtain \[\begin{gather} 2 F_N (\mathbf{a}_1,\ldots,\mathbf{a}_n) = -\log \mathcal{Z}_N (\mathbf{a}_1,\ldots,\mathbf{a}_n) \\ = -\log \mathcal{Z}_{N+M} (0) - \sum_{j=1}^M |\mathbf{w}_j|^2 + 2 \sum_{1\leq j<k \leq M} \log |\mathbf{w}_j - \mathbf{w}_k| + A(M,N) + \log \det_{M\times M} \left( \frac{\pi}{N} K_{N+M} (w_j,w_k) \right) \end{gather}\] where we have for now concatenated all points in a single list, as in proofs of the preceding subsection, and \(A(M,N)\) is as in 37 Reorganizing terms and arguing as in the proof of Lemma 3 we find \[\begin{gather} 2 F_N (\mathbf{a}_1,\ldots,\mathbf{a}_n) = -\log \mathcal{Z}_{N+M} (0) + A(M,N) - 2 \sum_{j=1}^M \mathcal{H}_N \left(\mathbf{w}_{j,1}, \ldots,\mathbf{w}_{j,M_j}\right) \\ - 2 \sum_{1\leq j < k \leq n} \mathcal{I}_{\mathrm{Int}}^{jk} + \log \det_{M\times M} \left( \frac{\pi}{N} K_{\infty} (w_j,w_k) \right) +o_N (1) \end{gather}\] using the notation ?? and 53 . Next, using Lemma 5 we have \[\log \det_{M\times M} \left( \frac{\pi}{N} K_{\infty} (w_j,w_k) \right) = \sum_{j=1} ^M \log \det_{M_j\times M_j} \left(\frac{\pi}{N} K_{\infty} (\mathbf{w}_{j,k},\mathbf{w}_{j,\ell})_{1\leq k,\ell \leq M_j} \right) +o_N (1).\] Using Lemma 3 once more thus leads to \[\begin{gather} 2 F_N (\mathbf{a}_1,\ldots,\mathbf{a}_n) = -\log \mathcal{Z}_{N+M} (0) + A(M,N) - 2 \sum_{j=1}^M \mathcal{H}_N \left(\mathbf{w}_{j,1}, \ldots,\mathbf{w}_{j,M_j}\right) \\ - 2 \sum_{1\leq j < k \leq n} \mathcal{I}_{\mathrm{Int}}^{jk} + \sum_{j=1} ^M \log \det_{M_j\times M_j} \left(\frac{\pi}{N} K_{\infty} (\mathbf{w}_{j,k},\mathbf{w}_{j,\ell})_{1\leq k,\ell \leq M_j} \right) +o_N (1). \end{gather}\] We next use Lemma 1 “backwards” to deduce \[\begin{gather} \label{eq:pre32final} 2 F_N (\mathbf{a}_1,\ldots,\mathbf{a}_n) = 2 \sum_{j=1} ^n F_N (\mathbf{a}_j) - 2 \sum_{1\leq j < k \leq n} \mathcal{I}_{\mathrm{Int}}^{jk} -\log \mathcal{Z}_{N+M} (0) + A(M,N) \\ + \sum_{j=1} ^n \left( \log \mathcal{Z}_{N+M_j} - A (M_j,N)\right) +o_N (1). \end{gather}\tag{54}\] Combining 37 and 59 we obtain, for any \(J\propto N\), \[\begin{gather} -\log \mathcal{Z}_{N+J} (0;0) + A(J,N) = \frac{3}{4} (N+J) ^2 - \frac{(N+J)^2}{2} \log \frac{N+J}{N} \\ - \frac{1}{2}(M+J) \log N - \left(\frac{\log 2\pi}{2} - 1\right)(N+J) \\ - \frac{5}{12} \log N - \frac{5}{12} \log \frac{N+J}{N} - \zeta'(-1) - \frac{\log 2\pi}{2} + o_N(1). \end{gather}\] Using the above for \(J=M\) and \(J=M_j\), \(j=1 \ldots n\), recalling that \(M=cN, M_j = c_j N\) with \(\sum_j M_j = M\) leads to \[\begin{gather} -\log \mathcal{Z}_{N+M} (0) + A(M,N) + \sum_{j=1} ^n \left( \log \mathcal{Z}_{N+M_j} - A (M_j,N)\right) = \\ \frac{3N^2}{4} \left( (1-n) + c^2 - \sum_{j=1} ^n c_j^2 - \frac{2}{3}(1+c)^2 \log (1+c) + \frac{2}{3}\sum_{j=1} ^n (1+c_j)^2 \log (1+c_j)\right)\\ +\frac{n-1}{2} \frac{\log N}{N} + (n-1)\left(\frac{\log 2\pi}{2} - 1\right)N + \frac{5(n-1)}{12}\log N\\ +(n-1)\left(\zeta' (-1) + \frac{\log 2\pi}{2}\right) + \frac{1}{12}\left(\log (1+c) - \sum_{j=1} ^n \log (1+c_j) \right) + o_N (1). \end{gather}\] Inserting in 54 we finally obtain ?? . ◻
There remains a single step to conclude the
Proof of Theorem 3. Let \(E^{\rm MF}(\mathbf{a}_1,\ldots,\mathbf{a}_N),E^{\rm MF}(\mathbf{a}_j)\) be the mean-field energies with all culsters of pinned charged present (respectively, with only the \(j\)-th one present), as defined in Section 3. Subtracting \(E^{\rm MF}(\mathbf{a}_1,\ldots,\mathbf{a}_N)\) from both sides of ?? , adding and subtracting \(\sum_{j=1}^n E^{\rm MF}(\mathbf{a}_j)\) to the right-hand side there only remains to observe that \[\begin{gather} \label{eq:split32MFE322} E^{\rm MF}(\mathbf{a}_1,\ldots,\mathbf{a}_n) - \sum_{j=1}^n E^{\rm MF}(\mathbf{a}_j) = - \sum_{1\leq j < k \leq n} \mathcal{I}_{\mathrm{Int}}^{jk} \\ +\frac{3N^2}{8} \left( (1-n) + c^2 - \sum_{j=1} ^n c_j^2 - \frac{2}{3}(1+c)^2 \log (1+c) + \frac{2}{3}\sum_{j=1} ^n (1+c_j)^2 \log (1+c_j)\right). \end{gather}\tag{55}\] This follows from inspection of ?? with an induction on \(n\). Each induction step is identical to the \(n=2\) one, modulo changing notation. Consider then two clusters of \(M_1 = c_1 N\) and \(M_2 = c_2 N\) points, corresponding radii in 25 \[R_1 ^2 = 1+c_1, \quad R_2 ^2 = 1+c_2, \quad R_{12} ^2 = 1+ c = 1+c_1 +c_2\] and constants 27 . Comparing ?? with 55 we need to show that \[\begin{align} -\frac{3}{8} &+ \frac{3}{8} c^2 - \frac{3}{8} \sum_{j=1} ^2 c_j^2 - \frac{1}{4}(1+c)^2 \log (1+c) + \frac{1}{4}\sum_{j=1} ^2 (1+c_j)^2 \log (1+c_j) \\ &= \frac{1}{2N^2}\left(C_{R_{12}} -C_{R_{1}} - C_{R_{2}}\right) \\ &-\frac{1}{2}\left(\frac{R_{12}^4}{4} -R_{12}^4 \log R_{12} - \frac{R_{1}^4}{4} + R_{1}^4 \log R_{1}- \frac{R_{2}^4}{4}+ R_{2}^4 \log R_{2}\right)\nonumber\\ &- c_1 \left(R_{12}^2 \log R_{12} - \frac{R_{12}^2}{2} - R_{1}^2 \log R_{1} + \frac{R_{1}^2}{2}\right) \nonumber\\ &- c_2 \left(R_{12}^2 \log R_{12} - \frac{R_{12}^2}{2} - R_{2}^2 \log R_{2} + \frac{R_{2}^2}{2}\right). \end{align}\] But, using 27 , the terms on the second line give altogether \[\frac{1}{2} \left((1+c_1) \log (1+c_1) + (1+c_2) \log (1+c_2) - (1+c) \log (1+c) - 1\right)\] while those on the third line amount to \[\begin{gather} \frac{1}{4} (1+c)^2 \log (1+c) - \frac{1}{4} (1+c_1)^2 \log (1+c_1) -\frac{1}{4} (1+c_2)^2 \log (1+c_2) - \frac{1}{8} \left((1+c^2) - (1+c_1^2) - (1+c_2^2)\right)\\ =\frac{1}{4} (1+c)^2 \log (1+c) - \frac{1}{4} (1+c_1)^2 \log (1+c_1) -\frac{1}{4} (1+c_1)^2 \log (1+c_1) + \frac{1}{8} \left(1 - c^2 + c_1^2 + c_2^2\right) \end{gather}\] and those on the fourth and fifth line add up to \[\frac{c}{2} (1+c) - \frac{c}{2} (1+c) \log (1+c) - \frac{c_1}{2} (1+c_1) + \frac{c_1}{2} (1+c_1) \log (1+c_1) - \frac{c_2}{2} (1+c_2) + \frac{c_2}{2} (1+c_2) \log (1+c_2),\] leading to the desired identity. ◻
Ginibre partition function for \(J\) particles \[\label{eq:Ginibre32def} \mathcal{Z}^{\mathrm{Gin}}_J := \int_{\mathbb{R}^{2J}} \prod_{1\leq j < k \leq J} |z_j - z_k| ^2 e ^{-N\sum_{j=1}^J |z_j|^2} dz_1 \ldots dz_J.\tag{56}\] We recalled in [31] that \[\label{eq:Ginibre32part320} \mathcal{Z}^{\mathrm{Gin}}_J = \frac{\pi^J \prod_{k=1}^J k!}{N^{J(J+1)/2}}\tag{57}\] For \(N=J\), we have (cf e.g. [23]) \[\begin{gather} \label{eq:Ginibre32part322} -\log \mathcal{Z}^{\mathrm{Gin}}_N = - \log \mathcal{Z}_{N}(0,0) = \frac{3}{4} N^2 -\frac{1}{2} N \log N - \left( \frac{\log 2\pi}{2} - 1 \right) N \\ - \frac{5}{12} \log N - \zeta'(-1) - \frac{\log (2\pi)}{2} + o_N (1) \end{gather}\tag{58}\] and since \[\mathcal{Z}^{\mathrm{Gin}}_J = \frac{\pi^J \prod_{k=1}^J k!}{J^{J(J+1)/2}} \left(\frac{J}{N}\right) ^{J(J+1)/2}\] we deduce that, in the general case of a mismatch between particle number \(J\) and background charge density \(N\), \[\begin{gather} \label{eq:Ginibre32part323} -\log \mathcal{Z}^{\mathrm{Gin}}_J = - \log \mathcal{Z}_J (0,0) \\ = - \frac{J(J+1)}{2} \log \frac{J}{N} + \frac{3}{4} J^2 -\frac{1}{2} J \log J - \left( \frac{\log 2\pi}{2} - 1 \right) J \\ - \frac{5}{12} \log J - \zeta'(-1) - \frac{\log (2\pi)}{2} + o_J (1) \end{gather}\tag{59}\]
We collect some bounds on the Ginibre correlation kernel(s) that can be found, inter alias, in [31]. First we have [31] \[\label{eq:Kmodule} | \mathrm{K}_\infty(z,w) | = \frac{N}{\pi} e^{-N|z-w|^2/2}.\tag{60}\] Also, from [31], for all \(M\geq 0\) \[\label{eq:Kmodule2} |\mathrm{K}_{N+M}(z,w)| \leq \frac{N}{\pi} e^{-N( |z|-|w|)^2 /2} .\tag{61}\] If \(|z|,|w|\leq 1 - \delta\), starting from [31] we get \[\label{eq:Kdiff32pre} \left|\mathrm{K}_{N+M}(z,w)- \mathrm{K}_{\infty}(z,w)\right| \leq C N ^{1/2} e^{-\frac{N}{2} \left(1-|z|+1-|w|\right)} .\tag{62}\] because the function \(\varphi(x)\) used therein is decreasing and convex, so that \(\varphi (x) \geq \varphi (1) + \varphi'(1) (x-1)\). It follows that, for \(|z|,|w|\leq 1 - \delta\) \[\label{eq:Kdiff32pre322} \left|\mathrm{K}_{N+M}(z,w)- \mathrm{K}_{\infty}(z,w)\right| \leq C N ^{1/2} e^{-CN \left(||z|-1| + ||w|-1|\right)} .\tag{63}\] Not that, in the proof of [31], \(n\) was assumed fixed in the limit \(N\to \infty\) so that the radius of the droplet for \(N+n\) Ginibre particles in a background density \(-4N\) was \(\sim 1\). For \(M\propto N\), adapting the estimates therein we find that \[\label{eq:Kdiff} \left|\mathrm{K}_{N+M}(z,w)- \mathrm{K}_{\infty}(z,w)\right| \leq C N ^{1/2} e^{-CN \left(\left||z|-\sqrt{1+c}\right| + \left||w|-{1+c}\right|\right)}.\tag{64}\] if \(|z|,|w|\leq \sqrt{1+c} - \delta\). Indeed \(\sqrt{1+c}\) is the radius of a Ginibre droplet for \(N+M\) particles, \(M= cN\).
Always understood as the one-component plasma, hereafter.↩︎
In the convention we follow, the leading MF term is of order \(N^2\), the LDA term of order \(N\) being often considered the leading one when dealing with a neutral homogeneous system [13].↩︎
For systems on surfaces, the number of handles is also involved.↩︎
“One can hear the number of holes in a drum”, see e.g. [27] and references therein↩︎