On boundary confinements for the Coulomb gas
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On boundary confinements for the Coulomb gas Yacin Ameur1 · Nam-Gyu Kang2 · Seong-Mi Seo2 Received: 10 May 2020 / Revised: 10 May 2020 / Accepted: 14 October 2020 / Published online: 1 November 2020 © Springer Nature Switzerland AG 2020
Abstract We introduce a family of boundary confinements for Coulomb gas ensembles, and study them in the two-dimensional determinantal case of random normal matrices. The family interpolates between the free boundary and hard edge cases, which have been well studied in various random matrix theories. The confinement can also be relaxed beyond the free boundary to produce ensembles with fuzzier boundaries, i.e., where the particles are more and more likely to be found outside of the boundary. The resulting ensembles are investigated with respect to scaling limits and distribution of the maximum modulus. In particular, we prove existence of a new point field—a limit of scaling limits to the ultraweak point when the droplet ceases to be well defined. Keywords Random normal matrices · Scaling limits · Planar orthogonal polynomials · Universality · Soft edge · Hard edge Mathematics Subject Classification 82D10 · 60G55 · 46E22 · 42C05 · 30D15
1 Introduction and main results In the theory of Coulomb gas ensembles, it is natural to consider different kinds of boundary confinements. The most well-known examples are the “free boundary”, where particles are admitted to range freely outside of the droplet, and the “hard
B
Seong-Mi Seo [email protected] Yacin Ameur [email protected] Nam-Gyu Kang [email protected]
1
Department of Mathematics, Faculty of Science, Lund University, P.O. BOX 118, 221 00 Lund, Sweden
2
School of Mathematics, Korea Institute for Advanced Study, 85 Hoegiro, Dongdaemun-gu, Seoul 02455, Republic of Korea
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edge”, where they are completely confined to it. On the other hand, notions of weakly confining potentials have attracted attention recently, where the boundary is softer than a free boundary, i.e., particles are more likely to be found outside of the boundary. In this note, we introduce a one-parameter family of edge confinements, ranging all the way between an idealized “ultraweak” edge and a hard edge. Our construction can be applied to general Coulomb gas ensembles in any dimension and for any inverse temperature β. However, we shall here be content to develop the theory only in the determinantal, two-dimensional case, i.e., we will consider ensembles of eigenvalues of random normal matrices. The study of universality in free boundary ensembles has been the focus of several recent works [4,5,18]. Notably, in the paper [18], it is shown that with free boundary confinement, the point field with intensity function R(z) = ϕ(z+¯z ) appears universally (i.e., for a “sufficiently large” class of ensembles) when rescaling about a regular boundary point, where ϕ, the “free boundary function”, is given by 1 ϕ(z) = b1 (z) := √ 2π
0
−∞
e−(z−t)
2 /2
dt =
z 1 erfc √ . 2 2
(1.1)
For the hard edge Ginibre ensemble, a dir
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