Skip to main content

Advertisement

Log in

Exact and Asymptotic Features of the Edge Density Profile for the One Component Plasma in Two Dimensions

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

There is a well known analogy between the Laughlin trial wave function for the fractional quantum Hall effect, and the Boltzmann factor for the two-dimensional one-component plasma. The latter requires continuation beyond the finite geometry used in its derivation. We consider both disk and cylinder geometry, and focus attention on the exact and asymptotic features of the edge density. At the special coupling \(\Gamma := q^2/k_BT=2\) the system is exactly solvable. In particular the \(k\)-point correlation can be written as a \(k \times k\) determinant, allowing the edge density to be computed to first order in \(\Gamma - 2\). A double layer structure is found, which in turn implies an overshoot of the density as the edge of the leading support is approached from the interior. Asymptotic analysis shows that the deviation from the leading order (step function) value is different for the interior and exterior directions. For general \(\Gamma \), a Gaussian fluctuation formula is used to study the large deviation form of the density for \(N\) large but finite. This asymptotic form involves thermodynamic quantities which we independently study, and moreover an appropriate scaling gives the asymptotic decay of the limiting edge density outside of the plasma.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Alastuey, A., Jancovici, B.: On the two-dimensional one-component Coulomb plasma. J. Phys. 42, 1–12 (1981)

    Article  MathSciNet  Google Scholar 

  2. Ameur, Y., Hedenmalm, H., Makarov, N.: Fluctuations of eigenvalues of random matrices. Duke Math. J. 159, 31–81 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bernevig, B.A., Haldane, F.D.M.: Model fractional quantum Hall states and Jack polynomials. Phys. Rev. Lett. 100, 246802 (2008)

    Article  ADS  Google Scholar 

  4. Bernevig, B.A., Regnault, N.: The anatomy of Abelian and non-Abelian fractional quantum Hall states. Phys. Rev. Lett. 103, 206801 (2009)

    Article  ADS  Google Scholar 

  5. Can, T., Forrester, P.J., Téllez, G., Wiegmann, P.: Singular behaviour at the edge of Laughlin states. Phys. Rev. B 89, 235137 (2014)

    Article  ADS  Google Scholar 

  6. Choquard, Ph, Forrester, P.J., Smith, E.R.: The two-dimensional one-component plasma at \(\Gamma = 2\): the semiperiodic strip. J. Stat. Phys. 33, 13–22 (1983)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  7. Ciftja, O., Wexler, C.: Monte Carlo simulation method for Laughlin-like states in a disk geometry. Phys. Rev. B 67, 075304 (2003)

    Article  ADS  Google Scholar 

  8. Datta, N., Morf, R., Ferrari, R.: Edge of the Laughlin droplet. Phys. Rev. B 53, 10906–10915 (1996)

    Article  ADS  Google Scholar 

  9. Forrester, P.J.: Finite size corrections to the free energy of Coulomb systems with a periodic boundary condition. J. Stat. Phys. 63, 491–504 (1991)

    Article  ADS  MathSciNet  Google Scholar 

  10. Forrester, P.J.: Fluctuation formula for complex random matrices. J. Phys. A 32, L159–L163 (1999)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  11. Forrester, P.J.: Log-Gases and Random Matrices. Princeton University Press, Princeton (2010)

    Book  MATH  Google Scholar 

  12. Forrester, P.J.: Spectral density asymptotics for Gaussian and Laguerre \(\beta \)-ensembles in the exponentially small region. J. Phys. A 45, 075206 (2012)

    Article  ADS  MathSciNet  Google Scholar 

  13. Forrester, P.J.: Large deviation eigenvalue density for the soft edge Laguerre and Jacobi \(\beta \)-ensembles. J. Phys. A 45, 145201 (2012)

    Article  ADS  MathSciNet  Google Scholar 

  14. Forrester, P.J., Mays, A.: A method to calculate correlation functions for \(\beta = 1\) random matrices of odd size. J. Stat. Phys. 134, 443–462 (2009)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  15. Ginibre, J.: Statistical ensembles of complex, quaternion, and real matrices. J. Math. Phys. 6, 440–440 (1965)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  16. Jancovici, B.: Exact results for the two-dimensional one-component plasma. Phys. Rev. Lett. 46, 386–388 (1981)

    Article  ADS  MathSciNet  Google Scholar 

  17. Jancovici, B., Manificat, G., Pisani, C.: Coulomb systems seen as critical systems: finite-size effects in two dimensions. J. Stat. Phys. 76, 307–330 (1994)

    Article  ADS  MATH  Google Scholar 

  18. Laughlin, R.B.: Anomalous quantum Hall effect: an incompressible quantum fluid with fractionally charge excitations. Phys. Rev. Lett. 50, 1395–1398 (1983)

    Article  ADS  Google Scholar 

  19. Morf, R., Halperin, B.I.: Monte Carlo evaluation of trial wave functions for the fractional quantized Hall effect: disk geometry. Phys. Rev. B 33, 2221–2246 (1986)

    Article  ADS  Google Scholar 

  20. Rider, B., Virág, B.: The noise in the circular law and the Gaussian free field, IMRN 2007 (2007), rnm006

  21. Saff, E.B., Totik, V.: Logarithmic Potentials with External Fields. Springer, Berlin (1997)

    Book  MATH  Google Scholar 

  22. Šamaj, L., Wagner, J., Kalinay, P.: Translation symmetry breaking in the one-component plasma on the cylinder. J. Stat. Phys. 117, 159–178 (2004)

    Article  ADS  MATH  Google Scholar 

  23. Sari, R.R., Merlini, D.: On the \(\nu \)-dimensional one-component classical plasma: the thermodynamic limit revisited. J. Stat. Phys. 76, 91–100 (1976)

    Article  ADS  MathSciNet  Google Scholar 

  24. Shakirov, S.: Exact solution for mean energy of 2d Dyson gas at \(\beta = 1\). Phys. Lett. A 375, 984–989 (2011)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  25. Téllez, G.: Exactly solvable models in statistical mechanics of Coulomb systems. Rev. Acad. Colomb. Cienc. 37, 61–74 (2013)

    Google Scholar 

  26. Téllez, G., Forrester, P.J.: Finite size study of the 2dOCP at \(\Gamma =4\) and \(\Gamma =6\). J. Stat. Phys. 97, 489–521 (1999)

    Article  ADS  MATH  Google Scholar 

  27. Téllez, G., Forrester, P.J.: Expanded Vandermonde powers and sum rules for the two-dimensional one-component plasma. J. Stat. Phys. 148, 824–855 (2012)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  28. Thouless, D.J.: Theory of the quantised Hall effect. Surf. Sci. 142, 147–154 (1984)

    Article  ADS  Google Scholar 

  29. Wiegmann, P.: Nonlinear hydrodynamics and fractionally quantized solitons at the fractional quantum Hall edge. Phys. Rev. Lett. 108, 206810 (2012)

    Article  ADS  Google Scholar 

  30. Zabrodin, A., Wiegmann, P.: Large-\(N\) expansion for the 2D Dyson gas. J. Phys. A 39, 8933 (2006)

    Article  ADS  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

The work of P.W. and T.C. was supported by NSF DMS-1156636 and DMS-1206648. The work of P.F. was supported by the Australian Research Council through the DP ‘Characteristic polynomials in random matrix theory’. G.T. acknowledges financial support from Facultad de Ciencias, Uniandes.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to P. J. Forrester.

Additional information

Dedicated to the memory of Bernard Jancovici (1930–2013) and his work on sum rules and exact solutions for Coulomb systems.

Appendices

Appendix 1

The purpose of this appendix is to derive (3.10).

According to the definitions, for general \(\Gamma \) in soft cylinder geometry

$$\begin{aligned} \langle U_1 \rangle ^\mathrm{c}&= - {1 \over 2} \int _0^W dx_1 \int _0^W dx_2 \int _{-\infty }^\infty dy_1 \int _{-\infty }^\infty dy_2 \, \log 2 \Big | \sin {\pi ((x_1 - x_2) + i (y_1 - y_2)) \over W} \Big | \nonumber \\&\quad \times \rho _{(2)}(\vec {r}_1,\vec {r}_2), \end{aligned}$$
(5.45)

where \(\vec {r}_j = (x_j,y_j)\). Generalizing (2.11), we know that for \(\Gamma = 2\) [6]

$$\begin{aligned} \rho _{(2)}(\vec {r}_1,\vec {r}_2) = \rho _{(1)}(\vec {r}_1) \rho _{(1)}(\vec {r}_2) + \rho _{(2)}^T(\vec {r}_1,\vec {r}_2) \end{aligned}$$

where \( \rho _{(1)}(\vec {r})\) is given by (2.11) and

$$\begin{aligned} \rho _{(2)}^T(\vec {r}_1,\vec {r}_2)&\!=\! - {2 \rho _b \over W^2} e^{- \pi (y_1 \!-\! y_2)^2} \sum _{q_1 = 0}^{N-1} \exp \Big \{ \!- 2 \pi \rho _b \Big ( {y_1 \!+\! y_2 \over 2} - {q_1 \!+\! 1/2 \over W \rho _b} \Big )^2 \!+\! 2 \pi i q_1 {(x_1 \!-\! x_2) \over W} \Big \} \nonumber \\&\quad \times \sum _{q_2 = 0}^{N-1} \exp \Big \{ - 2 \pi \rho _b \Big ( {y_1 + y_2 \over 2} - {q_2 + 1/2 \over W \rho _b} \Big )^2 - 2 \pi i q_2 {(x_1 - x_2) \over W} \Big \} \end{aligned}$$

(the case \(l=2\) of (4.13)). Our task then is to compute some explicit multiple integrals.

Making use of the Fourier expansion (5.39), elementary calculations show

$$\begin{aligned}&- {1 \over 2} \int _0^W dx_1 \int _0^W dx_2 \int _{-\infty }^\infty dy_1 \int _{-\infty }^\infty dy_2 \, \log 2 \Big | \sin {\pi ((x_1 - x_2) + i (y_1 - y_2)) \over W} \Big | \rho _{(2)}^T(\vec {r}_1,\vec {r}_2) \nonumber \\&\quad = - {1 \over \pi } \sum _{l=1}^{N-1} {N - l \over l} \int _{-\infty }^\infty dy_1 \, e^{-y_1^2} \int _{y_1 + \sqrt{2 \pi /\rho _b} l / W}^\infty dy_2 \, e^{-y_2^2} + {N \over W \sqrt{\rho _b}} \end{aligned}$$
(5.46)

and

$$\begin{aligned}&- {1 \over 2} \int _0^W dx_1 \int _0^W dx_2 \int _{-\infty }^\infty dy_1 \int _{-\infty }^\infty dy_2 \, \log 2 \Big | \sin {\pi ((x_1 - x_2) + i (y_1 - y_2)) \over W} \Big |\nonumber \\&\qquad \times \rho _{(1)}(\vec {r}_1) \rho _{(1)}(\vec {r}_2) \nonumber \\&\quad = - {\pi \over W^2 \rho _b} \sum _{l=1}^{N-1} (N-l) l + {2 \over W} \sum _{l=1}^{N-1} (N-l) {l \over W \rho _b} \int _{-\infty }^\infty dy_1 \, e^{-y_1^2} \int _{y_1 + \sqrt{2 \pi /\rho _b} l / W}^\infty dy_2 \, e^{-y_2^2} \nonumber \\&\qquad - {1 \over W \sqrt{\rho _b}} \sum _{l=1}^{N-1} (N-l) e^{-\pi l^2/\rho _b W^2}. \end{aligned}$$
(5.47)

This reduces our task to analyzing certain one-dimensional sums in the large \(N\) limit.

The first sum in (5.47) is elementary, and we have

$$\begin{aligned} \sum _{l=1}^{N-1} (N-l) l = N(N^2 - 1)/6. \end{aligned}$$
(5.48)

For the remaining sums, the leading and first order correction for large \(N\) can be obtained by making use of the trapezoidal rule

$$\begin{aligned} \sum _{k=1}^N f(k h) = {1 \over h} \int _0^{Nh} f(x) \, dx - \Big ( {f(0) - f(Nh) \over 2} \Big ) + \mathcal O( h^2). \end{aligned}$$
(5.49)

In this regards, the portion of the first summation in (5.46),

$$\begin{aligned} - {N \over \pi } \sum _{l=1}^{N-1} {1\over l} \int _{-\infty }^\infty dy_1 \, e^{-y_1^2} \int _{y_1 + \sqrt{2 \pi /\rho _b} l / W}^\infty dy_2 \, e^{-y_2^2} \end{aligned}$$

requires preliminary manipulation, since a literal application of (5.49) is not possible. This is due to the corresponding \(f(x)\) not being integrable about \(x=0\). Thus we write

$$\begin{aligned}&\sum _{l=1}^{N-1} {1 \over l} \int _{-\infty }^\infty dy_1 \, e^{-y_1^2} \int _{y_1 + \sqrt{2 \pi /\rho _b} l / W}^\infty dy_2 \, e^{-y_2^2} \nonumber \\&\quad = \sum _{l=1}^K {1 \over l} \Big \{ \int _{-\infty }^\infty dy_1 \, e^{-y_1^2} \int _{y_1 + \sqrt{2 \pi /\rho _b} l / W}^\infty dy_2 \, e^{-y_2^2} - \int _{-\infty }^\infty dy_1 \, e^{-y_1^2} \int _{y_1}^\infty \, e^{-y_2^2} \Big \} \nonumber \\&\qquad + \Big ( \int _{-\infty }^\infty dy_1 \, e^{-y_1^2} \int _{y_1}^\infty \, e^{-y_2^2} \Big ) \sum _{l=1}^K {1 \over l} \nonumber \\&\qquad + \sum _{l=K+1}^N {1 \over l} \int _{-\infty }^\infty dy_1 \, e^{-y_1^2} \int _{y_1 + \sqrt{2 \pi /\rho _b} l / W}^\infty dy_2 \, e^{-y_2^2} , \end{aligned}$$
(5.50)

where \(K = \Big [ W \sqrt{2 \pi \over \rho _b} \Big ]\).

With \(H_K\) denoting the harmonic numbers, it is a standard result that

$$\begin{aligned} \sum _{l=1}^K {1 \over l} =: H_K = \log K + \mathbf{C} + {1 \over 2K} + \mathcal O \Big ( {1 \over K^2} \Big ). \end{aligned}$$
(5.51)

The remaining sums in (5.50) can all be analyzed using (5.49). Doing this and combining with (5.51) shows

$$\begin{aligned}&- {1 \over \pi } \sum _{l=1}^{N-1} {1 \over l} \int _{-\infty }^\infty dy_1 \, e^{-y_1^2} \int _{y_1 + \sqrt{2 \pi /\rho _b} l / W}^\infty dy_2 \, e^{-y_2^2} \nonumber \\&\quad = - {N \over 2} \log \Big ( \sqrt{\rho _b \over 2} {W \over 2} \Big ) - {N \mathbf{C} \over 4} - {N \over 2 W \sqrt{\rho _b}} + {\sqrt{\rho _b} \over 2 \pi } W + \mathcal O(1) \end{aligned}$$
(5.52)

and

$$\begin{aligned}&{2 \over W} \sum _{l=1}^{N-1} (N\!-l) {l \over W \rho _b} \int _{-\infty }^\infty dy_1 \, e^{-y_1^2} \int _{y_1 \!+\! \sqrt{2 \pi /\rho _b} l / W}^\infty dy_2 \, e^{-y_2^2} \!-\! {1 \over W \sqrt{\rho _b}} \sum _{l=1}^{N-1} (N\!-l) e^{-\pi l^2/\rho _b W^2} \nonumber \\&\quad = \Big ( {N \over 4} - {1 \over 3 \pi } W \sqrt{\rho _b} \Big ) + \Big ( - {N \over 2} - {N \over 2 W \sqrt{\rho _b}} + {\sqrt{\rho _b} W \over 2 \pi } \Big ) + \mathcal O(1). \end{aligned}$$
(5.53)

Substituting (5.52) in (5.46), (5.53) and (5.48) in (5.47), and using these results to evaluate the RHS of (5.45) gives (3.10).

Appendix 2

Consider the soft cylinder with leading order density profile in the \(y\)-direction \(\tilde{\rho _b} = \rho _b \chi _{0 < y < W} \). For large \(W\), \(n \in \mathbb {Z}^+\), we see that

$$\begin{aligned}&\int _{-\infty }^\infty (y - W/2)^{2n} (\rho _{(1)}^{N,\mathrm{c}}(y) - \tilde{\rho _b}) dy \sim \frac{2n (2n-1)}{2} \Big ( \frac{W}{2} \Big )^{2n-2} \tilde{M_2}, \quad \nonumber \\&\quad \tilde{M_2} := \int _{-\infty }^\infty y^2 (\rho _{(1)}^{N,\mathrm{c}}(y) - \tilde{\rho _b}) dy. \end{aligned}$$

A readily verifiable consequence is that to leading order

$$\begin{aligned} \rho _{(1)}(y) - \tilde{\rho _b} = \frac{\tilde{M_2}}{4} \big (\delta '' (W-y) + \delta '' (y) \big ). \end{aligned}$$
(5.54)

We observe that the RHS of (5.54), multiplied by the measure \(dy\), is independent of \(W\) if we scale \(y \mapsto Wy\), \(x \mapsto Wx\), \(\frac{1}{W} \tilde{M_2} \mapsto M_2\), where

$$\begin{aligned} {M_2} := \int _{-\infty }^\infty y^2 (\rho _{(1)}^{N,\mathrm{c}}(y) |_{L=W=1}- N \chi _{0<y<1}) dy \end{aligned}$$
(5.55)

Thus (5.38) follows, provided we can show that \(M_2\) has the evaluation (5.33).

For this latter task we observe from the explicit formula for the partition function implied by (2.8) that

$$\begin{aligned} W \frac{\partial }{\partial W} \log Z_{N, \Gamma } (W,L) = \frac{-\pi \Gamma \rho _b W^2}{3}N + \Gamma \pi \rho _b \bigg \langle \sum _{l=1}^N y_l^2 \bigg \rangle _{\widehat{IQ}_{N, \Gamma }(W,L)}. \end{aligned}$$

Changing variables \(x_l \mapsto x_l / L\), \(y_l \mapsto y_l / L\) and setting \(W=L\) this reads

$$\begin{aligned} W \frac{\partial }{\partial W} \log Z_{N, \Gamma } (W,L)|_{W=L}&= \frac{-\pi \Gamma N^2}{3} + \Gamma \pi N \bigg \langle \sum _{l=1}^N y_l^2 \bigg \rangle _{\widehat{IQ}_{N, \Gamma }(1,1)} \nonumber \\&= \int _{-\infty }^\infty y^2 (\rho _{(1)}^{N,\mathrm{c}} (y) |_{L=W=1} - N \chi _{0<y<1})dy =: M_2. \end{aligned}$$
(5.56)

Thus we seek an independent computation of the LHS of (5.56).

To provide such a computation, we first observe

$$\begin{aligned} W \frac{\partial }{\partial W} = -\rho _b \frac{\partial }{\partial \rho _b}. \end{aligned}$$
(5.57)

Next we note that scaling in disk geometry together with the expected universality of the leading large \(N\) behaviour of the partitions in disk and cylinder geometries implies that for large \(N\)

$$\begin{aligned} Z_{N, \Gamma } (W,L) |_{\rho _b = N/{WL}} \sim e^{N(\Gamma / 4 - 1)\log \rho _b + Ng(\Gamma ) + \mathcal {O}(\sqrt{N})} \end{aligned}$$
(5.58)

for some \(g(\Gamma )\). Substituting (5.57) and (5.58) in the LHS of (5.56) gives (5.33)

Appendix 3

In this appendix, we study the behavior of the density in the cylinder when \(y\rightarrow -\infty \) for finite \(N\) and \(W\), when \(\Gamma /2\) is an integer. We will consider first \(N\) and \(W\) as independent variables. Let \(\tilde{W}=\rho _b W^2\) and \(\tilde{y}=\rho _b W y\) be the rescaled lengths by the characteristic length \(1/(\rho _b W)\). Considerations leading to the configuration integral (3.19) can be extended to obtain the density profile [22]

$$\begin{aligned} \rho _{(1)}^{N,c}(y)= \rho _b \sqrt{\frac{\Gamma }{\tilde{W}}} \sum _{l=0}^{(N-1)\Gamma /2} a_l^\mathrm{c} \exp \left[ -\frac{2\pi \Gamma }{\tilde{W}} \left( \tilde{y}-N+\frac{1}{2}+\frac{2l}{\Gamma } \right) ^2 \right] \end{aligned}$$
(5.59)

with

$$\begin{aligned} a_l^\mathrm{c}=\frac{1}{Q_{N,\Gamma }^\mathrm{c *}} \sum _{\mu \, |\, l\in \mu } {(c_\mu ^{(N)}(\Gamma /2))^2 \over \prod _i m_i!} e^{\pi \Gamma \sum _{j=1}^N (2\mu _j/\Gamma + 1/2)^2/\tilde{W}}, \end{aligned}$$
(5.60)

where the sum runs over all partitions which include \(l\). If \(\tilde{y}\rightarrow -\infty \), then

$$\begin{aligned} \rho _{(1)}^{N,c}(y) \underset{y\rightarrow -\infty }{\sim } \rho _b \sqrt{\frac{\Gamma }{\tilde{W}}}\, e^{-\pi \Gamma (\tilde{y}-1/2)^2/\tilde{W}} a_{(N-1)\Gamma /2}^\mathrm{c} . \end{aligned}$$
(5.61)

To compute \(a_{(N-1)\Gamma /2}^\mathrm{c}\), one needs to consider in (5.60) all the partitions \(\mu \) with \(c_{\mu }^{(N)}(\Gamma /2)\ne 0\) and \(\mu _1=(N-1)\Gamma /2\). The partition \(\tilde{\mu }=(\mu _2,\mu _3,\ldots ,\mu _{N})\) is a partition of \(\Gamma (N-1)(N-2)/4\) with \(\Gamma (N-2)/2 \le \mu _2 \le \cdots \le {\mu }_N\), and due to a factorization property satisfied by the coefficients of the partitions [4], one has

$$\begin{aligned} c_{((N-1)\Gamma /2,\tilde{\mu })}^{(N)}(\Gamma /2) = c_{\tilde{\mu }}^{(N-1)}(\Gamma /2) \,. \end{aligned}$$
(5.62)

Therefore \(\tilde{\mu }\) corresponds to a partition for a system with \(N-1\) particles (this is not surprising as taking \(y \rightarrow \infty \) effectively removes that particle; see a similar argument in [14]). Then

$$\begin{aligned} a_{(N-1)\Gamma /2}^\mathrm{c}= \frac{Q_{N-1,\Gamma }^\mathrm{c *}(\tilde{W})}{Q_{N,\Gamma }^\mathrm{c *}(\tilde{W})} \, e^{\pi \Gamma (N-1/2)^2}, \end{aligned}$$
(5.63)

and using (3.20), this leads to

$$\begin{aligned} \rho _{(1)}^{N,c}(y)&\underset{y\rightarrow -\infty }{\sim } \rho _b \left( \frac{2\pi }{\sqrt{\tilde{W}}}\right) ^{\Gamma /2} e^{-\pi \Gamma (\tilde{y}^2-\tilde{y})/\tilde{W}} \nonumber \\&\quad \times \exp \left[ \beta [F^{c}_{N,\Gamma }(\tilde{W})-F^{c}_{N-1,\Gamma }(\tilde{W})] -\left( 1-\frac{\Gamma }{4}\right) \log \rho _b -\frac{\pi \Gamma }{3\tilde{W}}\right] . \qquad \qquad \end{aligned}$$
(5.64)

Now, consider the limit \(N\rightarrow \infty \), and \(\tilde{W}\rightarrow \infty \), but with \(N\) and \(\tilde{W}\) independent. Using the universal properties of the free energy (3.22), we have

$$\begin{aligned} \rho _{(1)}^{N,c}(y)&\underset{y\rightarrow -\infty }{\sim } \rho _b \left( \frac{2\pi }{\sqrt{\tilde{W}}}\right) ^{\Gamma /2} e^{-\pi \Gamma (\tilde{y}^2-\tilde{y})/\tilde{W}} \nonumber \\&\quad \times \exp \left[ \beta f(\Gamma ,1) + (1-2\Gamma ) \frac{\pi }{6\tilde{W}}+o(1/N)+o(1/\tilde{W}) \right] . \end{aligned}$$
(5.65)

Notice that in the difference \(F^{c}_{N,\Gamma }(\tilde{W})-F^{c}_{N-1,\Gamma }(\tilde{W})\), as \(\tilde{W}\) is kept fixed, the surface tension terms in (3.22) cancel out, leading to a next order correction of order \(\mathcal{O}(1/N)\) instead of a naively expected \(\mathcal{O}(1/\sqrt{N})\). In the scaled edge \(\tilde{W}=N\rightarrow \infty \) and \(\tilde{y}\mapsto N y\) this can be compared to (5.42). Indeed if one takes \(y\rightarrow -\infty \) in (5.42), then (5.65) is recovered. The \(o(1)\) term in (5.42) for \(y \rightarrow - \infty \) should be (5.43).

As an illustration of the results, for \(\Gamma =4\), Fig. 3 shows a plot of the numerically computed

$$\begin{aligned} \log (\rho _{(1)}^{N,c}(\sqrt{N}y)/\tilde{\rho }_{(1)}^{N,c}(\sqrt{N}y)) +\beta f(\Gamma ,1), \end{aligned}$$

for various values of \(N=\tilde{W}\) confirming the expected behavior as \(y\rightarrow -\infty \). In the plot, \(\tilde{\rho }_{(1)}^{N,c}\) denotes the right hand side of (5.42).

Fig. 3
figure 3

Exact numerically computed density profile in the soft cylinder compared to the scaled form (5.42). From bottom to top, \(W^2=N=2,3,4,5,6,7,8,9,10,11,12,13,14\)

In Fig. 4, the value of the limit of \(\log (\rho _{(1)}^{N,c}(y)/\tilde{\rho }_{(1)}^{N,c}(\sqrt{N}y))+\beta f(\Gamma ,1)\) as \(y\rightarrow -\infty \) is plotted against \(1/N\), showing indeed a linear behavior as expected

$$\begin{aligned} \lim _{y\rightarrow -\infty } \log \frac{\rho _{(1)}^{N,c}(y)}{\tilde{\rho }_{(1)}^{N,c}(\sqrt{N}y)e^{-\beta f(\Gamma ,1)}} = \beta f(\Gamma ,1) +\frac{\pi }{6}(1-2\Gamma ) \frac{1}{N} +o(1/N)\,. \end{aligned}$$
(5.66)
Fig. 4
figure 4

Numerical value of the LHS of (5.66) as a function of \(N\) (red dots) and a linear regression done with values of \(N>7\) (blue dashed line) (Color figure online)

Very similar figures are obtained for \(\Gamma =6\) and 8 (not shown). Doing a numerical regression of Fig. 4 provides an alternative way to obtain numerically \(g(\Gamma )=\beta f(\Gamma ,1)\), and verify the \(1/N\) finite size correction. Table 3 shows the values obtained for \(g(\Gamma )\) and the \(1/N\) correction for \(\Gamma =4,\) 6, 8, and compares them to the estimations of free energy per particle on the sphere [26] and the expected value \(\pi (1-2\Gamma )/6\) of the \(1/N\) correction. As this method for estimating the free energy per particle relies on fitting an expression with \(1/N\) corrections, it seems as equally reliable as the one used in [26] for the 2dOCP on the sphere when the universal \(\log N\) correction is subtracted to the free energy.

Table 3 Estimation of the free energy \(g(\Gamma )=\beta f(\Gamma ,1)\) per particle obtained from (5.66)

Similar considerations can be done for the soft disk. The density profile is [26]

$$\begin{aligned} \rho _{(1)}^\mathrm{d}(r)=(\Gamma /2) \rho _b e^{-\pi \Gamma \rho _b r^2/2} \sum _{l=0}^{(N-1)\Gamma /2} a_l^\mathrm{d} \, (\Gamma \pi \rho _b r^2/2)^{l} \,, \end{aligned}$$
(5.67)

with

$$\begin{aligned} a_l^\mathrm{d}=\frac{N!\pi ^{N}}{Q_{N,\Gamma }^\mathrm{d}(\rho _b)\, l!} \sum _{\mu \,|\,l\in \mu } \frac{(c_\mu ^{(N)}(\Gamma /2))^2}{\prod _{i}m_i!} \prod _{j=1}^{N-1} \mu _j!\,. \end{aligned}$$
(5.68)

The leading behavior of the density as \(r\rightarrow \infty \) is given by

$$\begin{aligned} \rho _{(1)}^\mathrm{d}(r)\underset{y\rightarrow -\infty }{\sim } (\Gamma /2)\,\rho _b\, e^{-\pi \Gamma \rho _b r^2/2}\, a_{(N-1)\Gamma /2}^\mathrm{d} (\pi \rho _b\Gamma r^2/2)^{(N-1)\Gamma /2} \,. \end{aligned}$$
(5.69)

Again, the coefficient \(a_{(N-1)\Gamma /2}^\mathrm{d}\) is related to the ratio of two partition functions with \(N\) and \(N-1\) particles

$$\begin{aligned} a_{(N-1)\Gamma /2}^\mathrm{d}= N \pi \frac{Q_{N-1,\Gamma }^\mathrm{d}(\rho _b)}{Q_{N,\Gamma }^\mathrm{d}(\rho _b)} \,. \end{aligned}$$
(5.70)

Using (3.1), we find

$$\begin{aligned}&\rho _{(1)}^\mathrm{d}(r)\underset{y\rightarrow -\infty }{\sim } \frac{\pi \rho _b}{N^{\Gamma /4}} \exp \left[ -\frac{N\Gamma }{2}\left( \frac{\pi \rho _b r^2}{N}-1\right) \right] \left( \frac{\pi \rho _b r^2}{N} \right) ^{(N-1)\Gamma /2}\nonumber \\&\quad \times \exp \left[ \beta f(\Gamma ,\rho _b)-\left( 1-\frac{\Gamma }{4}\right) \log (\pi \rho _b) +\frac{\beta \mu (\Gamma ,\rho _b)\sqrt{\pi }}{\sqrt{\rho _b N}} +(1-\Gamma )\frac{1}{12N}+o(1/N) \right] \nonumber \\ \end{aligned}$$
(5.71)

In the scaled edge, with \(r\mapsto \sqrt{N}r\) and \(\rho _b =1/\pi \), taking \(r\rightarrow \infty \) in (5.29) reproduces (5.71), but here the \(o(1)\) has non zero \(\mathcal{O}(1/\sqrt{N})\) corrections — except for \(\Gamma = 2\) when \(\beta \mu (\Gamma ,\rho _b)\) vanishes — as opposed to the soft cylinder geometry.

Appendix 4

In this appendix we present a detailed derivation of the exterior asymptotes of \(A(y)\) as in (4.10). From Proposition 1, \(A(y)\) can be written as a sum of four terms, each of which is analyzed separately below.

Lemma 9

The asymptotic expansion of \(A_{1}(y)\) outside the droplet is

$$\begin{aligned} A_{1}(y) \mathop {\sim }_{y \rightarrow -\infty } - \frac{1}{8} \frac{e^{- 2y^{2}}}{\sqrt{2\pi } |y|} + \frac{1}{16\pi \sqrt{2}} \frac{e^{- 2y^{2}}}{ y^{2}} +\frac{1}{32 \sqrt{2\pi } |y|^{3}} e^{- 2y^{2}} + \mathcal {O}(y^{-4}e^{- 2y^{2}}). \end{aligned}$$

Proof

This asymptotic expansion can be obtained by differentiating \(A_{1}(- |y|)\) with respect to \(|y|\), and integrating from \(|y|\) to \(\infty \), with the result

$$\begin{aligned} A_{1}(- |y|)&= - \frac{1}{8} \mathrm{erfc}(\sqrt{2}|y|) + \frac{1}{2 \sqrt{6\pi }} \int _{|y|}^{\infty } dt \, e^{- 2t^{2}/3} \mathrm{erfc}(2 t/\sqrt{3}), \\&= - \frac{1}{8} \mathrm{erfc}(\sqrt{2}|y|) + \frac{1}{16\pi \sqrt{2}} \frac{e^{- 2y^{2}}}{ y^{2}} + \mathcal {O}(y^{-4}e^{- 2y^{2}}). \end{aligned}$$

The second line is obtained by expanding the complementary error function in the integrand for large \(t\), and integrating by parts. The result follows by keeping the next to leading order term in the large \(|y|\) expansion of the first term. \(\square \)

This can be used to show the following.

Lemma 10

The leading order asymptote of \(A_{2}(y)\) outside the droplet is

$$\begin{aligned} A_{2}(y) \mathop {\sim }_{y \rightarrow -\infty } \mathcal {O}\left( y^{-2} e^{- 2y^{2}}\right) . \end{aligned}$$

Proof

Applying a sequence of integration by parts, we can rewrite \(A_{2}(y)\) in terms of \(A_{1}(y)\) as

$$\begin{aligned} A_{2}(y)&= -\frac{1}{4 \sqrt{2\pi }} y e^{- 2y^{2}} + \frac{1}{4 \sqrt{2}\pi } e^{- 2y^{2}} + \frac{3}{4 \sqrt{6\pi }} y e^{- 2y^{2}/3} \mathrm{erfc}(-2 y/\sqrt{3})\nonumber \\&\quad + \left( \frac{1}{2} +2 y^{2}\right) A_{1}(y). \end{aligned}$$

Using the asymptotic expansion for \(A_{1}(-|y|)\) above, only the term with a pre-exponential factor of \(\mathcal {O}(y^{-2})\) remains. \(\square \)

Next, we consider the leading asymptote of \(A_{3}(y)\). This follows by a straightforward expansion for large \(-y \gg 1\).

Lemma 11

The leading asymptote of \(A_{3}(y)\) outside the droplet is

$$\begin{aligned} A_{3}(y) \mathop {\sim }_{y \rightarrow -\infty } \frac{1}{2 \sqrt{2\pi }} |y| e^{- 2y^{2}} + \mathcal {O} \left( y^{- 3} e^{- 2y^{2}}\right) . \end{aligned}$$

Proof

After replacing the error functions appearing in \(A_{3}(y)\) with their large \(|y|\) asymptotic expansions, this result follows by straightforward algebra.

Lemma 12

The asymptote of \(A_{4}(y)\) outside the droplet is

$$\begin{aligned} A_{4}(y) \mathop {\sim }_{y \rightarrow -\infty } \frac{1}{4 \sqrt{2\pi }} \frac{\log |y|}{|y|} e^{- 2y^{2}} + \frac{1}{4\sqrt{2\pi }} \left( \frac{\mathbf{C}}{2}+ \log 2\right) \frac{e^{- 2y^{2}}}{ |y|} + \mathcal {O} \left( y^{- 2} e^{- 2y^{2}}\right) \!. \end{aligned}$$

Proof

Using the fact that the integrand is symmetric in its arguments \(t_{1}\) and \(t_{2}\), we can rewrite \(A_{4}(y)\) for \(y < 0\) as

$$\begin{aligned} A_4(-|y|) =&-\frac{1}{\sqrt{2\pi }}\int _0^\infty dt_1 \int _0^{\infty }dt_2 \frac{e^{-2(t_1+|y|)^2}}{t_1-t_2} \bigg ( \mathrm{erf} (t_1-t_2) + \mathrm{erf} \big (\sqrt{2}(t_2+|y|) \big )\bigg ). \end{aligned}$$

After a change of variables,

$$\begin{aligned} A_4(-|y|)&= -\frac{1}{\sqrt{2\pi }}\int _{|y|}^\infty dt_1 \int _{|y|}^{\infty }dt_2 \frac{e^{-2 t_{1}^2}}{t_1-t_2} \bigg ( \mathrm{erf} (t_1-t_2) + \mathrm{erf} \big (\sqrt{2}t_2 \big )\bigg )\\&\sim -\frac{1}{\sqrt{2\pi }}\int _{|y|}^\infty dt_1 \int _{|y|}^{\infty }dt_2 \frac{e^{-2 t_{1}^2}}{t_1-t_2} \bigg ( \mathrm{erf} (t_1-t_2) +1 - \frac{e^{- 2t _{2}^{2}}}{\sqrt{2\pi } t_{2}}\bigg ). \end{aligned}$$

Integrating over \(t_{2}\) this reads

$$\begin{aligned} A_{4}(-|y|) \sim - \frac{1}{\sqrt{2\pi }} \int _{|y|}^{\infty } dt_{1} e^{- 2 t_{1}^{2}} \bigg (\log ( t_{1} - |y|) + \frac{\mathbf{C}}{2} + \log 2 - \int _{0}^{|y| - t_{1}} \frac{\mathrm{erf}(t)}{t} dt\bigg ). \end{aligned}$$

We can expand the last integral as an asymptotic series in \((|y| - t_{1})\). The leading term is \(2 (|y| - t_{1})/\sqrt{\pi }\), which, upon integrating with respect to \(t_{1}\), becomes

$$\begin{aligned} \int _{|y|}^{\infty } dt_{1} e^{- 2 t_{1}^{2}} (|y| - t_{1}) =\mathcal {O}\left( y^{- 2} e^{- 2 y^{2}} \right) . \end{aligned}$$
(5.72)

The first term in parentheses can be similarly developed as an asymptotic series. A change of variables \(x = t_{1} - |y|\), followed by a rescaling \(x = \xi /|y|\), makes the Gaussian factor \(\exp \left( - 2(x + |y|)^{2}\right) = e^{- 2 y^{2}} \exp \left( - \frac{2\xi ^{2}}{y^{2}} - 4 \xi \right) \). After a Laurent expansion in \((\xi /y)^{2}\), the integral becomes

$$\begin{aligned} \int _{|y|}^{\infty } dt_{1}e^{- 2t_{1}^{2}} \log (t_{1}-|y|)&= \frac{e^{- 2y^{2}}}{y} \int _{0}^{\infty }d\xi \log (\xi /y) e^{- 4 \xi } \left( 1 - \frac{2 \xi ^{2}}{y^{2}} + \frac{2 \xi ^{4}}{y^{4}} + \ldots \right) \nonumber \\&= - \frac{\log |y|}{4|y|}e^{- 2y^{2}} -2 \left( \frac{\mathbf{C}}{2} + \log (2)\right) \frac{1}{4 |y|}e^{- 2y^{2}} \nonumber \\&\quad \;\quad + \mathcal {O}\left( \frac{\log y}{y^{3}} e^{- 2y^{2}}\right) . \end{aligned}$$
(5.73)

The next term can be evaluated easily and its large distance asymptote reads

$$\begin{aligned} \int _{|y|}^{\infty } dt_{1} e^{- 2t_{1}^{2}} \left( \frac{\mathbf{C}}{2} + \log 2\right) \sim \left( \frac{\mathbf{C}}{2} + \log 2\right) \frac{e^{- 2y^{2}}}{4 |y|}. \end{aligned}$$
(5.74)

Combining (5.72), (5.73), and (5.74) gives the stated asymptote. \(\square \)

This exhaustive analysis demonstrates that the leading asymptote outside indeed arises from \(A_{3}(y)\), and moreover

$$\begin{aligned}&A(y) \mathop {\sim }_{y \rightarrow -\infty } \frac{1}{2 \sqrt{2\pi }} |y| e^{- 2y^{2}} + \frac{1}{4 \sqrt{2\pi }} \frac{\log |y|}{|y|} e^{- 2y^{2}} + \frac{1}{4\sqrt{2\pi }} \left( \frac{\mathbf{C}-1}{2}+ \log 2\right) \frac{e^{- 2y^{2}}}{ |y|}\nonumber \\&\quad +\mathcal {O} \left( y^{- 2} e^{- 2y^{2}}\right) . \end{aligned}$$
(5.75)

Appendix 5

In this appendix we present a more detailed proof of equation (5.14) in Lemma 5. A direct computation of the LHS for the antisymmetric parts of \(A_{1}(y)\), \(A_{2}(y)\) and \(A_{3}(y)\) gives

$$\begin{aligned} \sum _{i = 1}^{3} \left( A_{i,a}''(y) + 4 y A_{i,a}'(y)\right) = \left( 2y^{2}+\frac{1}{2}\right) \left( \mathrm{erf}(\sqrt{2/3}y)-\mathrm{erf}(\sqrt{2}y)\right) + \frac{ \sqrt{6}}{ \sqrt{\pi }} y e^{-2 y^{2}/3}\!. \end{aligned}$$
(5.76)

For \(A_{4}(y)\), we write the LHS as \(e^{-2y^{2}}\partial _{y}\left( e^{2 y^{2}} \partial _{y} A_{a,4}(y)\right) \), and carry out the operations in the sequence implied. First, the antisymmetric part must be written in a suitable form. Taking advantage of the symmetry of the integrand and changing variables, the double integral can be written as

$$\begin{aligned} A_4(y)&= - \frac{1}{2 \sqrt{2\pi }} \int _{-y}^{\infty } \int _{-y}^{\infty } dt_{1} dt_{2} F(t_{1},t_{2}),\\ F(t_{1},t_{2})&= \frac{1}{t_1-t_2} \bigg (e^{-2t_1^2} \Big (\mathrm{erf} (t_1-t_2) + \mathrm{erf} \big (\sqrt{2}t_2 \big ) \Big )\\&\qquad \qquad \qquad + e^{-2 t_2^2} \Big (\mathrm{erf} (t_1-t_2) - \mathrm{erf} \big (\sqrt{2} t_1 \big ) \Big ) \bigg ). \end{aligned}$$

Using the fact that \(\lim _{y \rightarrow \infty }A_{4}(y) = 0\), this can be written equivalently as

$$\begin{aligned} A_{4}(y)&= \frac{1}{2\sqrt{2\pi }} \int _{-\infty }^{\infty }dt_{1} \int _{-\infty }^{-y} \, dt_{2} \,F(t_{1},t_{2}) + \frac{1}{2\sqrt{2\pi }} \int _{-\infty }^{-y}dt_{1} \int _{-y}^{\infty }dt_{2}F(t_{1},t_{2}) \\&= \frac{1}{\sqrt{2\pi }} \int _{-\infty }^{\infty }dt_{1} \, \int _{-\infty }^{-y}dt_{2} \, F(t_{1},t_{2}) -\frac{1}{2\sqrt{2\pi }} \int _{-\infty }^{-y}dt_{1} \int _{-\infty }^{-y}dt_{2}F(t_{1},t_{2})\\&= \frac{1}{\sqrt{2\pi }} \int _{-\infty }^{\infty }dt_{1} \int _{-\infty }^{-y} \, dt_{2} \, F(t_{1},t_{2}) + A_{4}(-y), \end{aligned}$$

and thus

$$\begin{aligned} A_{a,4}(y)&= \frac{1}{2\sqrt{2\pi }} \int _{-\infty }^{\infty }dt_{1} \int _{-\infty }^{-y}dt_{2 } \,F(t_{1},t_{2}). \end{aligned}$$

From this, we apply the LHS to get

$$\begin{aligned} e^{- 2y^{2}}\partial _{y}\left( e^{2y^{2}} \partial _{y}A_{a,4}(y)\right)&= -\frac{\sqrt{2}}{ \sqrt{\pi }} \int _{-\infty }^{\infty } dx \, e^{-2(x-y)^{2} } \Big (\mathrm{erf} (x) - \mathrm{erf} \big (\sqrt{2}y \big ) \Big )\nonumber \\&= \mathrm{erf}(\sqrt{2}y)-\mathrm{erf}(\sqrt{2}y/\sqrt{3}). \end{aligned}$$

Combining this with (5.76) proves the lemma.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Can, T., Forrester, P.J., Téllez, G. et al. Exact and Asymptotic Features of the Edge Density Profile for the One Component Plasma in Two Dimensions. J Stat Phys 158, 1147–1180 (2015). https://doi.org/10.1007/s10955-014-1152-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10955-014-1152-2

Keywords

Navigation