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Peer-reviewedCC BY-SA 4.0Symmetry, Integrability and Geometry: Methods and Applications

Partition Functions of Two-Dimensional Coulomb Gases with Circular Root- and Jump-Type Singularities

  • Kohei Noda1
  1. 1Université Catholique de Louvain, Belgium
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Abstract

In this paper, we study the random polynomial pn(ρ):=∏j=1n (|zj|−ρ), where the points {zj}j=1n are the eigenvalues of random normal matrices with a radially symmetric potential. We establish precise large n asymptotic expansions for the moment generating function 𝔼[e(u Imlog pn(ρ))/π eaRelog pn(ρ)], u∈ℝ, a>−1, where ρ>0 lies in the bulk of the spectral droplet. The asymptotic expansion is expressed in terms of parabolic cylinder functions, which confirms a conjecture of Byun and Charlier. This also provides the first free energy expansion of two-dimensional Coulomb gases with general circular root- and jump-type singularities. While the a=0 case has already been widely studied in the literature due to its relation to counting statistics, we also obtain new results for this special case.

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Cite

BibTeXRIS
BibTeX
@article{noda2026partition,
  title = {{Partition Functions of Two-Dimensional Coulomb Gases with Circular Root- and Jump-Type Singularities}},
  author = {Kohei Noda},
  journal = {Symmetry, Integrability and Geometry: Methods and Applications},
  year = {2026},
  month = oct,
  doi = {10.3842/sigma.2026.106},
  url = {https://sigma-journal.com/2026/106/},
  note = {CC BY-SA 4.0},
}
RIS
TY  - JOUR
TI  - Partition Functions of Two-Dimensional Coulomb Gases with Circular Root- and Jump-Type Singularities
AU  - Kohei Noda
PY  - 2026
DA  - 2026/10/30
JO  - Symmetry, Integrability and Geometry: Methods and Applications
DO  - 10.3842/sigma.2026.106
UR  - https://sigma-journal.com/2026/106/
AB  - In this paper, we study the random polynomial p_n(ρ):=∏_j=1^n (|z_j|−ρ), where the points {z_j}_j=1^n are the eigenvalues of random normal matrices with a radially symmetric potential. We establish precise large n asymptotic expansions for the moment generating function 𝔼[e^(u Imlog p_n(ρ))/π e^aRelog p_n(ρ)], u∈ℝ, a>−1, where ρ>0 lies in the bulk of the spectral droplet. The asymptotic expansion is expressed in terms of parabolic cylinder functions, which confirms a conjecture of Byun and Charlier. This also provides the first free energy expansion of two-dimensional Coulomb gases with general circular root- and jump-type singularities. While the a=0 case has already been widely studied in the literature due to its relation to counting statistics, we also obtain new results for this special case.
SN  - 1815-0659
ER  -