The multiplicative constant for the Meijer-G kernel determinant

Charlier, Christophe; Lenells, Jonatan; Mauersberger, Julian

doi:10.1088/1361-6544/abd996

Cited by 5 publications

(4 citation statements)

References 46 publications

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“…The problem of determining large gap asymptotics is a notoriously difficult problem in random matrix theory with a long history [39,41,50]. There have been several methods that have proven successful to solve large gap problems of one-dimensional point processes, among which: the Deift-Zhou [25] steepest descent method for Riemann-Hilbert problems [10,18,19,22,24,[27][28][29]49], operator theoretical methods [33,34,75], the "loop equations" [15,16,56,57], and the Brownian carousel [31,64,72,73].…”

Section: Methods Of Proofmentioning

confidence: 99%

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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Section: Methods Of Proofmentioning

confidence: 99%

Large gap asymptotics on annuli in the random normal matrix model

Charlier

2023

Math. Ann.

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“…Consider the integral of 𝝎 along the closed contour composed of a large interval along the real axis and a semicircle in the upper half-plane. Then using (12) and the definition of 𝜏 in (13) we obtain in the case 𝑣 1 = −𝑣 2 that 𝑢(∞) + 𝑑 = 0 modℤ with 𝑢(𝑧) considered on the first sheet. Therefore also in the general case of 𝑣 1 , 𝑣 2 , by continuity,…”

Section: Outside Parametrix and 𝜽-Functionsmentioning

confidence: 99%

“…(Analogous results on the probability of a large gap were obtained for the Airy‐kernel determinant in [1, 16, 37], and for the Bessel‐kernel determinant in [19, 23], see [33] for an overview. For further related results on gap probabilities see [5, 9–12, 26] and references therein. )…”

Section: Introductionmentioning

confidence: 99%

Sine‐kernel determinant on two large intervals

Fahs,

Krasovsky

2023

Comm Pure Appl Math

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“…The determination of large gap asymptotics is a classical problem in random matrix theory with a long history. There exist various results on large gap asymptotics in the case of a gap on a single interval (the so‐called “one‐cut regime”), see [25, 30, 32, 33, 43, 53] for the sine process, [2, 24] for the Airy process, [27, 34] for the Bessel process, [14, 20, 21] for the Wright's generalized Bessel and Meijer‐ G point processes, [22] for the Pearcey process, [3, 8–13, 15–18, 23] for thinned‐deformations of these universal point processes, and [31, 50, 52] for the sine‐β, Airy‐β and Bessel‐β point processes. We also refer to [44] and [38] for two overviews.…”

Section: Introductionmentioning

confidence: 99%

The Bessel kernel determinant on large intervals and Birkhoff's ergodic theorem

Blackstone

Charlier

Lenells

2023

Comm Pure Appl Math

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The Bessel process models the local eigenvalue statistics near 0 of certain large positive definite matrices. In this work, we consider the probability where and is any non‐negative integer. We obtain asymptotics for this probability as the size of the intervals becomes large, up to and including the oscillations of order 1. In these asymptotics, the most intricate term is a one‐dimensional integral along a linear flow on a g‐dimensional torus, whose integrand involves ratios of Riemann θ‐functions associated to a genus g Riemann surface. We simplify this integral in two generic cases: (a) If the flow is ergodic, we compute the leading term in the asymptotics of this integral explicitly using Birkhoff's ergodic theorem. (b) If the linear flow has certain “good Diophantine properties”, we obtain improved estimates on the error term in the asymptotics of this integral. In the case when the flow is both ergodic and has “good Diophantine properties” (which is always the case for , and “almost always” the case for ), these results can be combined, yielding particularly precise and explicit large gap asymptotics.

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The multiplicative constant for the Meijer-G kernel determinant

Abstract: We compute the multiplicative constant in the large gap asymptotics of the Meijer-G point process. This point process generalizes the Bessel point process and appears at the hard edge of Cauchy–Laguerre multi-matrix models and of certain product random matrix ensembles.

Cited by 5 publications

References 46 publications

Large gap asymptotics on annuli in the random normal matrix model

Large gap asymptotics on annuli in the random normal matrix model

Sine‐kernel determinant on two large intervals

The Bessel kernel determinant on large intervals and Birkhoff's ergodic theorem

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