{'fraction_non_alphanumeric': 0.05667953187250996, 'fraction_numerical': 0.020885209163346612, 'mean_word_length': 4.027225786261931, 'pattern_counts': {'":': 0, '<': 13, '<?xml version=': 0, '>': 5, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 2, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Density Correlations in the Two-Dimensional Coulomb Gas

We consider a two-dimensional Coulomb gas of positive and negative pointlike unit charges interacting via a logarithmic potential. The density (rather than the charge) correlation functions are studied. In the bulk, the form-factor theory of an equivalent sine-Gordon model is used to determine the density correlation length. At the surface of a rectilinear plain wall, the universality of the asymptotic behavior of the density correlations is suggested. A scaling analysis implies a local form of the compressibility sum rule near a hard wall. A symmetry of the Coulomb system with respect to the M\"obius conformal transformation, which induces a gravitational source acting on the particle density, is established. Among the consequences, a universal term of the finite-size expansion of the grand potential is derived exactly for a disk geometry of the confining domain.


Introduction and summary
In this paper, we consider classical Coulomb systems in thermodynamic equilibrium. For the sake of simplicity, we will restrict ourselves to the case of a symmetric two-component plasma (TCP), i.e., a neutral system of two species of particles of opposite unit charges σ = ±1, living in a ν-dimensional space and interacting through the Coulomb interaction σ i σ j v(|r i − r j |). In dimension ν, the Coulomb potential v at a spatial position r ∈ where, for ν = 2, the length scale r 0 will be set to unity without any loss of generality. For the case of pointlike particles, the singularity of v(r) at the origin prevents the thermodynamic stability against the collapse of positive-negative pairs of charges: in two dimensions for small enough temperatures, in three dimensions for any temperature.
In two dimensions and for the case of pointlike charged particles, the particle density n is an "irrelevant" variable which only scales the distance. The complete description of the bulk thermodynamics is available in the whole range of inverse temperatures β < 2 where the plasma is stable against the collapse of positive-negative pairs of charges. The exact equation of state where p is the pressure and n the total particle density, has been known for a very long time [2,3]. The evaluation of other thermodynamic quantities (free energy, internal energy, specific heat, etc.) can be based on an explicit density-fugacity (n − z) relationship. This relationship was obtained only recently [4] via a mapping onto a classical two-dimensional sine-Gordon theory. Surface thermodynamics (surface tension) of the two-dimensional TCP in contact with a rectilinear dielectric wall was derived for specific boundary conditions in refs. [5,6]. The large-distance behavior of the bulk charge-charge correlation function was given in ref. [7] by exploring the form-factor method for the equivalent sine-Gordon theory. Exact formulae for pair correlations for an arbitrary interparticle distance are available just at the collapse point β = 2 [8,9], which corresponds to the free-fermion point of an equivalent Thirring model.
The long-range tail of the Coulomb force causes screening, and thus give rise to exact constraints for the charge correlations for any value of β (for a review, see ref. [10]).
In the bulk, the charge structure function S ρ (r, r ′ ) = S ρ (|r − r ′ |) obeys the Stillinger-Lovett sum rules [11], namely the zeroth-moment condition dr S ρ (r) = 0 (1.12) and the second-moment condition Hereinafter, without any loss of generality, we put the dielectric constant ǫ of a medium in which the plasma lives equal to unity.
Let us now introduce a semi-infinite TCP which occupies the half space x > 0, and denote by y the set of (ν − 1) coordinates normal to x. The plane at x = 0 is a hard wall impenetrable to the particles. The half-space x < 0 is assumed to be filled with a material of dielectric constant ǫ W : a particle of unit charge at the point r = (x > 0, y) has an electric image of charge (1−ǫ W )/(1+ǫ W ) at the point r * = (−x, y) [12]. Due to invariance with respect to translations along the wall and rotations around the x direction, 14) The electroneutrality condition (1.12) takes the form The Carnie and Chan [13] generalization of the second-moment condition (1.13) results in the dipole sum rule [14,15] ∞ The asymmetry of the screening cloud of a charged particle sitting near the wall induces a long-range tail in the charge correlation along the wall [16,17], except of the cases ǫ W = 0 (ideal dielectric wall) and ǫ W = ∞ (ideal conductor wall) which will be excluded from our considerations. One expects an asymptotic power-law decay where f ρ (x, x ′ ) obeys the sum rule [18,19] Recently [20], it was proven that for any value of When both sides of (1.19) are integrated over x from 0 to ∞, it is clear that the sum rule (1.18) for f ρ is a direct consequence of the dipole sum rule (1.16), and vice versa.
As concerns the density correlation function, according to the general theory of fluids [21], the zeroth moment of the bulk density structure function S n (r, r ′ ) = S n (|r − r ′ |) is related to the isothermal compressibility In two dimensions and for the case of pointlike charged particles, with the use of the exact equation of state (1.11), one gets explicitly [22] Recently, using the technique of a renormalized Mayer expansion [23], and in particular a remarkable "cancelation property" of specific families of renormalized diagrams [24], the second moment of S n was shown to have a simple value [25] The universal finite-size properties of two-dimensional critical systems with short-range interactions among constituents are well understood within the principle of conformal invariance [26] - [30]. For a finite system of characteristic size R, at a critical point, the dimensionless grand potential βΩ = − ln Ξ (Ξ is the grand partition function) has a large-R expansion of the form The coefficients A and B of the bulk and surface parts are non-universal. The coefficient of the ln R-term is universal, dependent only on the conformal anomaly number c of the critical theory and on the Euler number χ of the manifold on which the system is confined. In general, where h is the number of handles and b the number of boundaries of the manifold (χ = 2 for a sphere, χ = 1 for a disk, χ = 0 for an anulus or a torus). The grand potential of the two-dimensional TCP is supposed to exhibit a universal finite-size correction of type (1.24) at any temperature of the conducting regime.
Plausible, but not always rigorously justified, arguments for a critical-like behavior were first given for Coulomb gases with periodic boundary conditions [31], then for Coulomb systems confined to a domain by plain hard walls [32], by ideal-conductor walls [33] and finally by ideal-dielectric boundaries [34,35]. The explicit checks were done at the exactly solvable β = 2 inverse temperature for various geometries of confining domains. Only very recently [36], a direct derivation of the universal finite-size correction term was done for the specific case of the TCP living on the surface of a sphere of radius R. By combining the method of stereographic projection of the sphere onto an infinite plane with the linear response theory, the prefactor to the universal ln R correction term was related to the bulk second moment of the density structure function S n , eq. (1.23). The obtained result confirms the prediction (1.24) for a Coulomb system, as if we had c = −1, in full agreement with heuristic approaches and exact results at β = 2 [31]- [35].
In general, due to screening phenomena, the sum rules for the charge structure function Density correlations near a rectilinear hard wall are analysed in section 3. In subsection 3.1., we present the results of the Debye-Hückel β → 0 limit. Subsection 3.2. deals with the exactly solvable β = 2 case for the plain hard wall with ǫ W = 1. In comparison with formula (1.17) for S ρ taken at dimension ν = 2, a more rapid, but still power-law, asymptotic decay of S n along the boundary is observed, In subsection 3.3, at least for the considered plain hard wall, the universal form of the Here, we also suggest explicit forms of f ρ (x, x ′ ) and f n (x, x ′ ) for any value of β.
In section 4, we explore the scaling properties of the two-dimensional TCP confined to a disk of radius R, for the sake of simplicity by an uncharged plain hard wall. We present some important formulae which are used in the subsequent sections. The rectilinear wall is obtained as the limiting R → ∞ case of the disk. As a by-product of the formalism, we derive a local form of the compressibility (zeroth-moment) sum rule (1.22) for S n near the rectilinear hard wall.
In section 5, by using a specific (namely Möbius) conformal transformation of particle coordinates we show how the two-dimensional TCP can be mapped onto the one under the action of a gravitational source, acting in the same way on both positively and negatively charged particles. The mapping implies a new sum rule for S n in the disk geometry. In this section, the new sum rule is used to derive the counterpart of the bulk second-moment sum rule for S n (1.23) near a rectilinear wall.
Section 6 deals with the universal finite-size correction ln R-term for the disk geometry.
Using the new sum rule for S n derived in section 5, the universal term is confirmed at any value of β. This term has already been calculated at β = 2 in ref. [32].
A brief recapitulation and some concluding remarks are given in section 7.

Bulk density correlations 2.1 Weak coupling
In this subsection, we derive the asymptotic form of the density correlation function h n (r) in the bulk, at the lowest order in β. It should be noted that β-expansions of correlation functions must be taken for a fixed value of the inverse Debye length κ = (2πβn) 1/2 , since κ only fixes the length scale.
In the present case of a charge-symmetrical two-component plasma, the ordinary Ornstein-Zernike (OZ) equation split into two independent relations for the charge-charge and density-density functions [25] h respectively, where * denotes a convolution product. In the renormalized Mayer expansion [4,25], at the lowest order in β, the charge direct correlation function is given by with v being the Coulomb potential, and the density direct correlation function is given by the Meeron diagram The wavy line denotes the renormalized bond K (sum of chains). In the bulk, K(r) = −βK 0 (κr), where K 0 is a modified Bessel function.
Inserting (2.2) into (2.1a), one gets the definition of K. This is why since K 0 (x) has the asymptotic form [π/(2x)] 1/2 exp(−x). In the OZ relation (2.1b), where c n (2.3) is of order β 2 , the convolution term is easily seen to be of higher order β 3 . Thus, at lowest order in β, h n (r) = c n (r), The same analysis can be done in dimension ν = 3, where where now κ = (4πβn) 1/2 [37]. In this case, the large-r behavior is not touched by an inevitable short-distance regularization of the Coulomb potential.
However, the Ursell functions U ρ = n 2 h ρ and U n = n 2 h n are exactly known [8,9] as and where m = 2πz (z is the fugacity). Replacing the modified Bessel functions by their asymptotic expansions gives the asymtotic form of U ρ (r) as and the asymtotic form of U n (r) as

Form-factor analysis
We now determine the large-distance behavior of h n (r) in the whole stability interval of inverse temperatures 0 < β < 2 via a form-factor analysis.
The grand partition function of the TCP can be turned into (see, e.g., [38]) is the Euclidean action of the classical sine-Gordon model. In the sine-Gordon representation, the density of particles of one sign σ = ± is where · · · denotes the averaging over the sine-Gordon action (2.13), two-body densities (1.5) are expressible as follows etc. The parameter z = z + = z − , which is the fugacity renormalized by a (diverging) self-energy term, gets a precise meaning when one fixes the normalization of the cos-field.
For the TCP [4], this normalization is given by the short-distance behavior dominated by the Boltzmann factor of the Coulomb potential. The corresponding formula in the sine-Gordon picture is known in quantum field theory as the conformal normalization.
Their number depends on the inverse of the parameter The mass of the B j -breather is given by and the breather disappears from the spectrum just when m j = 2M. Under the conformal normalization (2.17), the relationship between the soliton mass M and the parameter z was established in [40], where Γ stands for the Gamma function. Using the Thermodynamic Bethe ansatz, the specific quantity was found in ref. [41]. As a thermodynamic result [4], Note that as β approaches the collapse value 2, for a fixed z, M is finite and n → ∞ as it should be.
For the underlying sine-Gordon theory, the two-point truncated correlation functions of local operators O a (a is a free parameter) can be formally written as an infinite convergent series over multi-particle intermediate states, where ǫ indexes the particles [ǫ = +(−) for a soliton (antisoliton) and ǫ = j for a breather B j (j = 1, 2, . . .)] and θ is the particle rapidity. The form factors are the matrix elements of the operator at the origin, between an N-particle in-state (as a linear superposition of free one-particle states |Z ǫ (θ) ) and the vacuum.
In the limit |r − r ′ | → ∞, the dominant contribution to the truncated correlation function in (2.23) comes from a multi-particle state with the minimum value of the total particle mass N j=1 m ǫ j , at the point of vanishing rapidities θ j → 0. Due to topological reasons, solitons and antisolitons coexist in pairs, the total mass of the pair being 2M.
The breathers B j with masses given by relation (2.19) are lighter and therefore, when they exist and their form-factor contributions do not vanish, they are the best candidates for governing the asymptotic behavior of the two-point correlation function.
The form factors of an exponential operator O a = exp(iaφ(r)) for various combinations of particles were calculated in refs. [42] - [44]. The one-breather form factors, which do not depend on the rapidity, posses the following general structure where the full dependence on σ = ±1 is presented. These form factors are nonzero only if j = odd integer. Due to the invariance of the sine-Gordon action (2.13) with respect to the transformation, it holds e ibφ = e −ibφ . With regard to (2.15), the total contribution of a given breather B j (j odd) to the charge h ρ and density h n correlation functions (1.10) is proportional to where κ denotes as usually the inverse Debye length. At asymptotically large r, (2.23) gives in agreement with the β → 0 limit (2.4). At the free-fermion point β = 2, the B 1 -breather disappears, and the soliton-antisoliton pair with mass 2M determines the correlation length: At small β, the large-distance behavior of h n is determined by the two-B 1 -breather state. Indeed, the corresponding form factor [7,43] (where the full dependence on σ = ±1 is given) has the necessary σ → −σ symmetry and thus contributes to h n . It follows from (2.19) that the mass of two B 1 -breathers, 2m 1 , is smaller than the one of the soliton-antisoliton pair, 2M, in the region β < 1. Consequently, at large distance r, in agreement with the β → 0 limit (2.5). It stands to reason that subsequently the soliton-antisoliton pair determines the large-distance asymptotic of h n , where the β-dependence of M can be deduced from eq. (2.22). At β = 2, the result (2.11) is recovered. We omit a tedious calculation of the inverse-power law dependence of prefactors on distance in (2.31) and (2.32).
One concludes that, for a given β < 2, the large-distance exponential decay of h n is faster than the one of h ρ . Although the correlation lengths depend continuously on β replaced by its bulk value n. This K has been implicitly considered in refs. [20,45], where it was used as the charge correlation function, h ρ = K. Except in the cases ǫ W = ∞ (ideal conductor wall) and ǫ W = 0 (ideal dielectric wall), K has an algebraic decay, for large The Meeron diagram (2.3) gives for the asymptotic form of the density Ursell function These formulae can be generalized to ν = 2, 3 dimensions:

β=2
When β=2 and ǫ W = 1, the charge and density Ursell functions can be expressed in terms of auxiliary functions g σ+ , where σ = ±, as [9] where m = 2πz (z is the fugacity). It will now be shown that, near the wall, each g function has a slow algebraic decay for large |y − y ′ |.
Each g is a sum of two terms The first term in the rhs of (3.6) is the same as in the bulk. It has a fast (exponential) decay and does not contribute to the asymptotic form, which is entirely due to the second term. This second term is determined by its Fourier transform as These Fourier transforms arẽ where k = (m 2 + l 2 ) 1/2 . Thus,g wall σ+ (x, x ′ ; l) is a function of l singular at l = 0. This singularity generates in g wall σ+ (x, x ′ ; y − y ′ ) an algebraic decay at large |y − y ′ |. The corresponding asymptotic expansion is obtained by splitting the integral in (3.7) into the l < 0 and l > 0 contributions, and evaluating each contribution by successive integration per partes. For instance, calling for simplicity N(l) the functiong wall σ+ (x, x ′ ; l) when l < 0, one obtains the asymptotic expansion A similar expansion holds for P (l), the functiong wall σ+ (x, x ′ ; l) when l > 0. Using these expansions finally gives The dominant term in eqs. (3.10), of order 1/(y − y ′ ) 2 , determines the asymptotic form of the charge Ursell function (3.5a) [20], But this dominant term cancels out in the expression (3.5b) of the density Ursell function, the asymptotic form of which is governed by the subdominant term in eqs. (3.10), of order (3.12)

Conjectures
The functions U ρ,n (r) and S ρ,n (r) differ from one another only by a term containing δ(r), which has no effect on their identical large-distance behavior. With regard to the definitions of the asymptotic characteristics f ρ (1.17) and f n (1.25), the results of the two previous subsections can be summarized, for ν = 2 dimensions and ǫ W = 1, by The explicit forms of f ρ and f n in the two exactly solvable cases has an appealing feature: both charge and density functions factorize in x and x ′ particle coordinates. This can be intuitively explained by the fact that one studies the leading asymptotic y → ∞ limit of pair correlations along the wall, in which probably there is no correlation between x and x ′ coordinates of particles. In both β → 0 and β = 2 cases, the decay of factorized functions into the bulk has the exponential form obtained for the corresponding bulk functions in subsections 2.1. and 2.2. With regard to the general form-factor analysis in subsection 2.3., it is therefore tempting to write down with the mass m 1 of the elementary B 1 -breather given by formula (2.27). This formula reproduces correctly the two solvable cases (3.13) and reflects the dominance of the lightest particle in the sine-Gordon spectrum in the large-distance behavior of charge correlation functions. The value of the prefactor in (3.15) is fixed by the sum rule (1.18) with ν = 2 and ǫ W = 1. As concerns the density asymptotic charasteristic f n (x, x ′ ) (3.14), the prefactor of the exponential acquires the same value in the β → 0 limit as well as at β = 2, what motivates us to suggest, taking into account large-distance asymptotics (2.31) and (2.32), that In particular, at the boundary x = x ′ = 0, f n (0, 0) is supposed to have a universal value 1/(2π 2 ) independent of β. We were not able to give some general argument for such a result.
We emphasize that formulae (3.15) and (3.16) are only conjectures which must be verified, for example, within a systematic weak-coupling expansion in the presence of a dielectric wall. Such calculations for the correlation functions are tedious and far from simple, and will not be presented here.

Scaling analysis in a disk
Under the neutrality constraint N + = N − , the grand partition function of the twodimensional symmetric TCP in a domain D, bounded by an impermeable hard wall (for simplicity, uncharged and with no image forces, ǫ W = 1) is written as where Q N,N is the configuration integral and denotes the interaction Boltzmann weight of N particles of charge +1 with coordinates . From the explicit representations etc., one readily gets the important relations etc.
The domain of interest in this section is the disk of radius R, D = {|r| ≤ R}. When one rescales the particle coordinates as p i = Rp ′ i and n i = Rn ′ i , it becomes evident that Ξ(z, R) depends only on the dimensionless combination z 2 R 4−β . As a consequence, If one accepts the expected value of c = −1 in the universal ln R-term of the large-R expansion (1.24) (χ = 1 for the disk), ln Ξ = −βΩ takes the form where p is the bulk pressure, and γ is the surface tension. The bulk particle density n is given by the equation which gives the equation of state (1.11).
The particle density n(r, R) depends on both the radius R and the distance 0 ≤ r ≤ R from the centre of the disk. By using the above mentioned scaling transformation of particle coordinates, one has n(r, R) = z 1/(1−β/4) g z 1/(2−β/2) r, z 1/(2−β/2) R (4.10) with an unknown function g. The origin can be moved to the boundary via the coordinate transformation x = R − r. In order to distinguish between the two different functions, we will use the obvious notation Similarly as in (4.10), with an unknown functionḡ different from g. The definition (4.11) implies The transition from the disk of radius R to a rectilinear hard wall can be understood as the limiting R → ∞ procedure, with lim R→∞ n R (x) = n(x) being the particle density at a finite distance x ≥ 0 from the plain hard wall.
It is evident that the derivative of R where R is a position vector of any point at the disk boundary. Hereinafter, the integration range from 0 to R formally means that 0 ≤ {|p i |, |n i |} ≤ R. As a result, ∂ ∂R ln Ξ = 2πRn R (0) (4.14) With regard to eqs. (4.6) and (4.9), the finite-size correction of the particle density at the boundary reads The well-known contact theorem n(0) = [1 − (β/4)]n [46,47] results as the R → ∞ limit of eq. (4.15).
By using the same procedure for Ξ n(r, R) one arrives at where, in polar coordinates, R = (R, ϕ). Using eq. (4.13b) and after some simple algebra, one can pass from (4.16) to In the limit R → ∞ and for a fixed x, n R (x) = n(x)+O(1/R), so lim R→∞ ∂n R (x)/∂R → 0.
When one introduces the y-coordinate as follows y = Rϕ, eq. (4.17) takes the form and we recover the two-dimensional version of the WLMB equation [48,49].
We are now ready to take advantage of the exact grand-canonical relation (4.4b). With regard to the scaling form of n R (x), eq. (4.12), it holds In the limit R → ∞, the last term on the rhs of eq. (4.19) vanishes. Inserting the resulting z∂n(x)/∂z into (4.4b), one finally obtains This is the local form of the compressibility sum rule in the presence of a plain hard wall.
Indeed, in the limit x → ∞, n(x) → n and ∂n(x)/∂x → 0 faster than any power-law due to screening. Consequently, in the bulk, (4.20) reproduces (1.22). It is straightforward to prove that eq. (4.20) is valid for any value of ǫ W .

Möbius invariance and sum rules
We now consider the TCP confined to a two-dimensional domain D, and study the action of the Möbius conformal transformation (with free complex parameters ad − bc = 0) of complex particle coordinates (z,z) on the configuration integral Q N,N (4.1b). Under the conformal transformation (5.1), the domain D is mapped onto the one denoted as D ′ , the surface element dzdz is written as the square of the distance between two particles takes the form so that the Boltzmann factor (4.2) reads Finally, putting c =c = 1 and a = r 0 ,ā =r 0 , one finds Taking for the domain a disk with its center at the origin, D = {|r| ≤ R}, let us set b = R 2 and d =r 0 (r 0r0 = R 2 ), besides the already chosen c = 1 and a = r 0 , so that the Möbius conformal transformation (5.1) takes the form It is straightforward to show that under this transformation the disk zz ≤ R 2 maps onto a "dual" domain D ′ defined by the inequality If the gravitational point is outside of the disk, i.e. r 0r0 > R 2 , the transformation (5.7) maps the disk onto itself. Relation (5.5) results into If r 0r0 < R 2 , the disk is mapped onto its complement in the two-dimensional space and eq. (5.5) gives In both cases, the gravitational source lies outside of the dual domain, as is required by the condition of stability (see the comment in the above paragraph). There exists a special Möbius transformation which maps the disk directly onto a half-plane, but we will not discuss this case.
Although the present paragraph will not be used in the following, let us remark that the self-duality of the disk system (5.9) produces a relation between the grand partition functions of type (5.6), with D ′ = D = disk andz = z(r 0r0 − R 2 ) 2−(β/2) . Since Ξ(z, β, R) depends on β and the combination z 2 R 4−β , Ξ(z, β, R|r 0 ) will depend on β and the combinationz 2 R 4−β /(r 0r0 − R 2 ) 4−β . Let x = |r 0 | − R be the distance of the gravitational point from the surface of the disk. The consequent ratio R/(2xR + x 2 ) diverges in the limit R → ∞, x → 0. In this case, the asymptotic formula (4.6) can be applied. When x is finite, the asymptotic formula (4.6) is applicable only when one multiplies z by an appropriate R-dependent constant via changing the zero reference energy of the gravitational potential.
We will now use the self-dual relation (5.9) to derive a new sum rule (5.14) and its more detailed version (5.17) for the density structure function S n in the case of the disk geometry. First, we rewrite the rhs of (5.9) as follows Then, since R 2 /(r 0r0 ) < 1, |p i /r 0 | < 1 and |n i /r 0 | < 1, we perform the large-|r 0 | expansion for each z = {p i , n i }. The term of order (r 0r0 ) 0 in (5.11) exactly reproduces the lhs of (5.9), and the coefficients of higher-order terms in inverse powers of r 0 andr 0 must be identically equal to zero. The terms 1/r 0 and 1/r 0 trivially vanish. The 1/(r 0r0 ) term vanishes when Within the grand-canonical formalism (4.1)-(4.3), the equality (5.13) is equivalent to Everything that has been done in the case of the configuration integral can be adapted to the particle density. Let us "rotate" the Möbius transformation (5.7) around the origin by the multiplication factor (r 0 /r 0 ) in order to obtain the identity z ′ = z in the limit |r 0 | → ∞, and denote by a the ratio R/r 0 whose absolute value is smaller than 1, For a given point r = (r,r) in the interior of the disk, the previously developed formalism When one expands the rhs of (5.16) in powers of a = a x + ia y , |a| < 1, the requirement of the nullity of the coefficients attached to a x and a y leads to the relation The new sum rule (5.14) results from the more detailed eq. (5.17) when one integrates R 0 d 2 r both sides of the latter, then uses an integration per partes and finally applies the equality (4.4a).
This new sum rule (5.17) has a consequence for the density near the boundary. Let us divide eq. (5.17) by |r|, and move the origin to the boundary via the coordinate transformation r = R − x described in section 4. Then, using eq. (4.19), and in the R → ∞ limit, we find after some simple algebra that This formula determines the leading correction of the density n R (x), at a fixed distance x from the wall, with respect to its asymptotic R → ∞ value n( Multiplying eq. (4.18) by x, integrating then over x from 0 to ∞ and performing an integration per partes, one finds On the other hand, according to the relation (4.4a), it holds for the disk Assuming the large-R expansion (4.6) and passing to the boundary-distance variable In the limit R → ∞, this equation implies the obvious boundary relation We will suppose that the half-infinite Coulomb system has good screening properties into the bulk, i.e., [n(x) − n] decays faster than any inverse-power law as x → ∞ and all Inserting (5.24) into (5.25) and after some algebra we arrive at Actually, it can be shown that the boundary sum rule (5.26) can be derived directly from the second-moment sum rule in the bulk (1.23) by using a simple assumption. Indeed, S n (x, x ′ ; y) can be decomposed as the sum of the bulk structure factor plus a surface term: 6 Universal finite-size correction for the disk In most considerations of the previous two sections, we have assumed the validity of the large-R expansion of ln Ξ (4.6) for the disk, in particular the value −1/6 of the coefficient of the universal ln R-term. Here, we prove this assumption.
Using the formula |r − r ′ | 2 = |r| 2 + |r ′ | 2 − 2r · r ′ , we write As concerns the rhs of (6.4), the unknown integral over the disk area can be represented In the limit R → ∞, due to good screening properties of the plasma, only the term survives. Based on the scaling analysis of section 3, since n(x) = nḡ( √ nx), this integral is dimensionless, and therefore its derivative with respect to z vanishes. Consequently, without the integral on the rhs of eq. (6.4), the solution of that equation combined with (6.5) reads f = − 1 6 ln Rz 1/(2−β/2) + const + · · · (6.7) in full agreement with the expected universal large-R behavior (4.6).

Concluding remarks
The The TCP in contact with a rectilinear hard wall of dielectric constant ǫ W was mapped onto an integrable boundary sine-Gordon model in two cases: ǫ W → ∞ (ideal metal wall, Dirichlet boundary condition) [5] and ǫ W = 0 (ideal dielectric wall, Neumann boundary condition) [6]. Interestingly, just in these two cases the charge and density correlation functions decay exponentially also along the boundary. The next potentially integrable case corresponds to the plain hard wall with ǫ W = 1: at least, the explicit solution is available at the free-fermion point [9]. For this case, we have suggested an explicit form of the asymptotic characteristics along the wall, f ρ (x, x ′ ) (3.15) and f n (x, x ′ ) (3.16), as a natural interpolation between the exact results in the β → 0 limit and at β = 2. The universal value of the prefactor = 1/(2π 2 ) in f n deserves attention. To check the validity of our conjectures, one should go beyond the Debye-Hückel limit.
The scaling analysis of section 4 leads to a new local version of the compressibility sum rule in the presence of a hard wall (4.20).
In the crucial section 5, a symmetry of the TCP confined to a disk with respect to the Möbius conformal transformation of particle coordinates, which induces a gravitational source point interacting with the particles, is established. This symmetry is of interest by itself: it produces a path between the pure Coulomb system and the one under the action of a gravitational point. The gravitational point acts in the same way on positive and negative charges, so that the total particle density is relevant. As a result, the new sum rule (5.14) for the density correlation function and its local version (5.17) are derived for the disk geometry of the Coulomb system. The finite-size formula (5.18) for the particle density at a fixed distance from the wall and the boundary version (5.26) of the bulk second moment of S n (1.23) also deserve attention.
The new density sum rule (5.14) is used in section 6 to derive the universal finite-size correction term of the grand potential for the disk geometry. We would like to stress a fundamental difference between the derivation of the universal finite-size correction term for a system which is finite in each direction (a disk or a sphere, a density sum rule is relevant) and for a system which is infinite at least in one direction (a strip or a ν-dimensional slab, a charge sum rule is relevant [33,34]).
In most cases, we have restricted ourselves to the symmetrical-charge TCP and ǫ W = 1 in order to maintain the clarity of presentation. The extension of the formalism to the asymmetric TCP and arbitrary ǫ W is usually straightforward.