Edge Behavior of Two-Dimensional Coulomb Gases Near a Hard Wall

Seo, Seong-Mi

doi:10.1007/s00023-021-01126-0

Cited by 11 publications

(15 citation statements)

References 38 publications

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“…For comparison, "pointwise" root-type singularities typically produce other kinds of ingredients in the asymptotics, such as Barnes' G-function (as was discovered by Basor [6] in dimension one and by Webb and Wong [61] in dimension two), and "circular" jump-type singularities involve the error function. We also mention that ensembles with "circular" root-type singularities have been studied in [64,55], and ensembles with "elliptic" root-type singularities in [52]. In [64,52,55], the singularities are located at the hard edge and the focus was on the leading order behavior of the kernel; in particular, the (associated) Hermite polynomials do not show up in these works.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

“…We also mention that ensembles with "circular" root-type singularities have been studied in [64,55], and ensembles with "elliptic" root-type singularities in [52]. In [64,52,55], the singularities are located at the hard edge and the focus was on the leading order behavior of the kernel; in particular, the (associated) Hermite polynomials do not show up in these works.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

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On the characteristic polynomial of the eigenvalue moduli of random normal matrices

Byun¹,

Charlier²

2022

Preprint

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We study the characteristic polynomial pn(x) = n j=1 (|zj| − x) where the zj are drawn from the Mittag-Leffler ensemble, i.e. a two-dimensional determinantal point process which generalizes the Ginibre point process. We obtain precise large n asymptotics for the moment generating function E[e u π Im ln pn(r) e a Re ln pn (r) ], in the case where r is in the bulk, u ∈ R and a ∈ N. This expectation involves an n × n determinant whose weight is supported on the whole complex plane, is rotation-invariant, and has both jump-and root-type singularities along the circle centered at 0 of radius r. This "circular" root-type singularity differs from earlier works on Fisher-Hartwig singularities, and surprisingly yields a new kind of ingredient in the asymptotics, the so-called associated Hermite polynomials.

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Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

On the characteristic polynomial of the eigenvalue moduli of random normal matrices

Byun¹,

Charlier²

2022

Preprint

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“…where Bal µ| S\C , ∂C is the balayage of µ| S\C onto the boundary ∂C. The formula (1.5) expresses the fact that the portion µ| S\C is swept onto the boundary ∂C according to the balayage operation, which preserves (up to a constant) the exterior logarithmic potential in the exterior of the droplet S. See [69,Section II.4] as well as [35,71,53,14] for more details about the balayage. The balayage part of (1.5) is a density on the curve ∂C, so this part is singular with respect to the two-dimensional Lebesgue measure.…”

Section: Hard Wall Constraints In Random Matrix Theorymentioning

confidence: 99%

“…The equilibrium measure µ h in potential (1.10) can be easily computed using standard balayage techniques [69] (see also [35,Section 4.1] or [71] for details) and is given by…”

Section: Mittag-leffler Ensembles With a Hard Wall Constraintmentioning

confidence: 99%

“…Coulomb gas ensembles in the presence of a hard wall have previously been considered in the literature, but so far the focus has been on large gap probabilities (or partition functions) [50,47,53,4,5,3,1,48,29] and on the local statistics [78,64,71]. We refer to [11,70,12,23,51,59] for studies of local droplets and local statistics near soft/hard edges.…”

Section: Mittag-leffler Ensembles With a Hard Wall Constraintmentioning

confidence: 99%

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Exponential moments for disk counting statistics at the hard edge of random normal matrices

Ameur¹,

Charlier²,

Joakim³

et al. 2022

Preprint

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We consider the multivariate moment generating function of the disk counting statistics of a model Mittag-Leffler ensemble in the presence of a hard wall. Let n be the number of points. We focus on two regimes: (a) the "hard edge regime" where all disk boundaries are a distance of order 1 n away from the hard wall, and (b) the "semi-hard edge regime" where all disk boundaries are a distance of order 1 √ n away from the hard wall. As n → +∞, we prove that the moment generating function enjoys asymptotics of the form

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Large gap asymptotics on annuli in the random normal matrix model

Charlier

2023

Math. Ann.

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We consider a two-dimensional determinantal point process arising in the random normal matrix model and which is a two-parameter generalization of the complex Ginibre point process. In this paper, we prove that the probability that no points lie on any number of annuli centered at 0 satisfies large n asymptotics of the form $$\begin{aligned} \exp \Bigg ( C_{1} n^{2} + C_{2} n \log n + C_{3} n + C_{4} \sqrt{n} + C_{5}\log n + C_{6} + {\mathcal {F}}_{n} + \mathcal {O}\Big ( n^{-\frac{1}{12}}\Big )\Bigg ), \end{aligned}$$ exp ( C 1 n 2 + C 2 n log n + C 3 n + C 4 n + C 5 log n + C 6 + F n + O ( n - 1 12 ) ) , where n is the number of points of the process. We determine the constants $$C_{1},\ldots ,C_{6}$$ C 1 , … , C 6 explicitly, as well as the oscillatory term $${\mathcal {F}}_{n}$$ F n which is of order 1. We also allow one annulus to be a disk, and one annulus to be unbounded. For the complex Ginibre point process, we improve on the best known results: (i) when the hole region is a disk, only $$C_{1},\ldots ,C_{4}$$ C 1 , … , C 4 were previously known, (ii) when the hole region is an unbounded annulus, only $$C_{1},C_{2},C_{3}$$ C 1 , C 2 , C 3 were previously known, and (iii) when the hole region is a regular annulus in the bulk, only $$C_{1}$$ C 1 was previously known. For general values of our parameters, even $$C_{1}$$ C 1 is new. A main discovery of this work is that $${\mathcal {F}}_{n}$$ F n is given in terms of the Jacobi theta function. As far as we know this is the first time this function appears in a large gap problem of a two-dimensional point process.

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Edge Behavior of Two-Dimensional Coulomb Gases Near a Hard Wall

Cited by 11 publications

References 38 publications

On the characteristic polynomial of the eigenvalue moduli of random normal matrices

On the characteristic polynomial of the eigenvalue moduli of random normal matrices

Exponential moments for disk counting statistics at the hard edge of random normal matrices

Large gap asymptotics on annuli in the random normal matrix model

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