Airy-kernel determinant on two large intervals

Krasovsky, Igor; Maroudas, Theo-Harris

doi:10.48550/arxiv.2108.04495

“…These oscillations were then substantially simplified by Deift, Its and Zhou in [24], who expressed them in terms of the Riemann theta function. Since then, there has been other works of this vein, see [13] for β-ensembles, [20] for partition functions of random matrix models, [34] for the sine process, [9,10,51] for the Airy process, and [11] for the Bessel process. In all these works, the Riemann theta function describes the fluctuations in the asymptotics, thereby suggesting that this function is a universal object related to the multi-cut regime of one-dimensional point processes.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Large gap asymptotics on annuli in the random normal matrix model

Charlier¹

2021

Preprint

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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 formwhere n is the number of points of the process. We determine the constants C1, . . . , C6 explicitly, as well as the oscillatory term Fn 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: when the hole region is a disk, only the constants C1, . . . , C4 were previously known, and when the hole region is an annulus, only C1 was previously known. For general values of our parameters, even C1 is new. A main discovery of this work is that Fn 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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“…These oscillations were then substantially simplified by Deift, Its and Zhou in [26], who expressed them in terms of the Riemann theta function. Since then, there has been other works of this vein, see [16] for β-ensembles, [21] for partition functions of random matrix models, [35] for the sine process, [12,13,51] for the Airy process, and [14] for the Bessel process. In all these works, the Riemann theta function describes the fluctuations in the asymptotics, thereby suggesting that this function is a universal object related to the multi-cut regime of one-dimensional point processes.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Large gap asymptotics on annuli in the random normal matrix model

Charlier

¹

2023

View full text Add to dashboard Cite

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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“…As noted above Corollary 1.8, the case 𝑔 = 1 is somewhat simpler in several respects; in particular, for 𝑔 = 1 every flow is both ergodic and has "good Diophantine properties". Hence there is no need to invoke Birkhoff's ergodic theorem when 𝑔 = 1, see [4,5,35,45]. For these reasons, we feel that Theorem 1.5 is an important novel contribution of the current paper.…”

Section: Related Workmentioning

confidence: 89%

“…Recently, for the case of two intervals (this corresponds to a genus 1 situation), Fahs and Krasovsky in [35] showed that this coefficient is identically equal to −1∕2. Large gap asymptotics for the Airy process in certain genus 1 situations were recently computed in [4,5,45], and there too simple expressions for the log coefficients appearing in the large gap asymptotics were obtained. We also mention that the multiplicative constants in the asymptotics were explicitly computed in [35,45].…”

Section: Related Workmentioning

confidence: 99%

See 1 more Smart Citation

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

Blackstone

¹

,

Charlier

²

,

Lenells

³

2023

Comm Pure Appl Math

1

0

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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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Airy-kernel determinant on two large intervals

Abstract: We find the probability of two gaps of the form (sc, sb) ∪ (sa, +∞), c < b < a < 0, for large s > 0, in the edge scaling limit of the Gaussian Unitary Ensemble of random matrices, including the multiplicative constant in the asymptotics.

Cited by 4 publications

References 22 publications

Large gap asymptotics on annuli in the random normal matrix model

Large gap asymptotics on annuli in the random normal matrix model

Large gap asymptotics on annuli in the random normal matrix model

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

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