Free Energy and Fluctuations in the Random Normal Matrix Model with Spectral Gaps
(2025) In Constructive Approximation- Abstract
We study large n expansions for the partition function of a Coulomb gas (Formula presented.) where Q is a radially symmetric confining potential on the complex plane C. The droplet is not assumed to be connected, but may consist of a number of disjoint annuli and possibly a central disk. The boundary condition is “soft edge”, i.e., Q is smooth in a C-neighbourhood of the droplet. We include the following possibilities: (i) existence of “outposts”, i.e., components of the coincidence set which fall outside of the droplet, (ii) a Fisher-Hartwig singularity at the origin, (iii) perturbations Q-hn where h is a smooth radially symmetric test-function. In each case, the free energy logZn admits a large n expansion of the form (Formula... (More)
We study large n expansions for the partition function of a Coulomb gas (Formula presented.) where Q is a radially symmetric confining potential on the complex plane C. The droplet is not assumed to be connected, but may consist of a number of disjoint annuli and possibly a central disk. The boundary condition is “soft edge”, i.e., Q is smooth in a C-neighbourhood of the droplet. We include the following possibilities: (i) existence of “outposts”, i.e., components of the coincidence set which fall outside of the droplet, (ii) a Fisher-Hartwig singularity at the origin, (iii) perturbations Q-hn where h is a smooth radially symmetric test-function. In each case, the free energy logZn admits a large n expansion of the form (Formula presented.) where C1,…,C5 are certain geometric functionals. The n-dependent term Gn is bounded as n→∞; it arises in the presence of spectral gaps. We use the free energy expansions to study the distribution of fluctuations of linear statistics. We prove that the fluctuations are well approximated by the sum of a Gaussian and certain independent terms which provide the displacement of particles from one component to another. This displacement depends on n and is expressed in terms of the Heine distribution. We also prove (under suitable assumptions) that the number of particles which fall near a spectral outpost converges to a Heine distribution.
(Less)
- author
- Ameur, Yacin LU ; Charlier, Christophe LU and Cronvall, Joakim LU
- organization
- publishing date
- 2025
- type
- Contribution to journal
- publication status
- epub
- subject
- keywords
- Coulomb gas, Disconnected droplet, Fisher-Hartwig singularity, Fluctuation, Heine distribution, Orthogonal polynomials, Partition function, Spectral outpost
- in
- Constructive Approximation
- publisher
- Springer
- external identifiers
-
- scopus:105016520070
- ISSN
- 0176-4276
- DOI
- 10.1007/s00365-025-09720-9
- language
- English
- LU publication?
- yes
- id
- c1bc7a62-9234-458a-9306-562dabeb48ce
- date added to LUP
- 2026-01-09 08:24:44
- date last changed
- 2026-01-09 08:25:47
@article{c1bc7a62-9234-458a-9306-562dabeb48ce,
abstract = {{<p>We study large n expansions for the partition function of a Coulomb gas (Formula presented.) where Q is a radially symmetric confining potential on the complex plane C. The droplet is not assumed to be connected, but may consist of a number of disjoint annuli and possibly a central disk. The boundary condition is “soft edge”, i.e., Q is smooth in a C-neighbourhood of the droplet. We include the following possibilities: (i) existence of “outposts”, i.e., components of the coincidence set which fall outside of the droplet, (ii) a Fisher-Hartwig singularity at the origin, (iii) perturbations Q-hn where h is a smooth radially symmetric test-function. In each case, the free energy logZn admits a large n expansion of the form (Formula presented.) where C1,…,C5 are certain geometric functionals. The n-dependent term Gn is bounded as n→∞; it arises in the presence of spectral gaps. We use the free energy expansions to study the distribution of fluctuations of linear statistics. We prove that the fluctuations are well approximated by the sum of a Gaussian and certain independent terms which provide the displacement of particles from one component to another. This displacement depends on n and is expressed in terms of the Heine distribution. We also prove (under suitable assumptions) that the number of particles which fall near a spectral outpost converges to a Heine distribution.</p>}},
author = {{Ameur, Yacin and Charlier, Christophe and Cronvall, Joakim}},
issn = {{0176-4276}},
keywords = {{Coulomb gas; Disconnected droplet; Fisher-Hartwig singularity; Fluctuation; Heine distribution; Orthogonal polynomials; Partition function; Spectral outpost}},
language = {{eng}},
publisher = {{Springer}},
series = {{Constructive Approximation}},
title = {{Free Energy and Fluctuations in the Random Normal Matrix Model with Spectral Gaps}},
url = {{http://dx.doi.org/10.1007/s00365-025-09720-9}},
doi = {{10.1007/s00365-025-09720-9}},
year = {{2025}},
}