Journal of Statistical Physics

, Volume 158, Issue 5, pp 1147–1180 | Cite as

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

  • T. Can
  • P. J. ForresterEmail author
  • G. Téllez
  • P. Wiegmann


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.


Two dimensional Coulomb system Fractional quantum Hall effect Edge density profile 



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.


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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • T. Can
    • 1
    • 2
  • P. J. Forrester
    • 3
    Email author
  • G. Téllez
    • 4
  • P. Wiegmann
    • 1
  1. 1.Department of PhysicsUniversity of ChicagoChicagoUSA
  2. 2.Simons Center for Geometry and PhysicsStony Brook UniversityStony BrookUSA
  3. 3.Department of Mathematics and StatisticsUniversity of MelbourneParkvilleAustralia
  4. 4.Departamento de FísicaUniversidad de Los AndesBogotáColombia

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