Large gap asymptotics for Airy kernel determinants with discontinuities. (English) Zbl 1475.37014
The so-called Airy point process is a very important object that represents a collection of random points on the real line. This process arises as a scaling limit in different probabilistic models, and is said to be universal. It is an instance of a determinantal point process for which the kernel is specified by an Airy function.
One can also study this process after removing some of its points. This gives rise to the so-called thinned Airy process, which is exactly the model considered in this paper. If \(\overline{x}\) are some points \(x_{1},\ldots, x_{m}\) on the real line, and \(\overline{s}\) are values \(s_{1},\ldots, s_{m}\) in \((0,1)\), the function \(F(\overline{x},\overline{s})\) represents the probability having a gap on the interval \((x_{m},\infty)\) in the thinned Airy process, in which the particle in \((x_{j}, x_{j-1})\) is removed using the probability \(s_{j}\). The authors provide some asymptotic results for the function \(F(\overline{x},\overline{s})\) when the vector \(\overline{x}\) becomes very large.
The main technique used for the analysis is based on the Hilbert-Riemann problem device, a well-known tool in complex analysis. The authors also compare their results with formulas previously found in the literature, in particular when \(m=1\).
One can also study this process after removing some of its points. This gives rise to the so-called thinned Airy process, which is exactly the model considered in this paper. If \(\overline{x}\) are some points \(x_{1},\ldots, x_{m}\) on the real line, and \(\overline{s}\) are values \(s_{1},\ldots, s_{m}\) in \((0,1)\), the function \(F(\overline{x},\overline{s})\) represents the probability having a gap on the interval \((x_{m},\infty)\) in the thinned Airy process, in which the particle in \((x_{j}, x_{j-1})\) is removed using the probability \(s_{j}\). The authors provide some asymptotic results for the function \(F(\overline{x},\overline{s})\) when the vector \(\overline{x}\) becomes very large.
The main technique used for the analysis is based on the Hilbert-Riemann problem device, a well-known tool in complex analysis. The authors also compare their results with formulas previously found in the literature, in particular when \(m=1\).
MSC:
| 37A50 | Dynamical systems and their relations with probability theory and stochastic processes |
| 15B52 | Random matrices (algebraic aspects) |
| 15A15 | Determinants, permanents, traces, other special matrix functions |
| 60B20 | Random matrices (probabilistic aspects) |
| 60G55 | Point processes (e.g., Poisson, Cox, Hawkes processes) |
| 34M50 | Inverse problems (Riemann-Hilbert, inverse differential Galois, etc.) for ordinary differential equations in the complex domain |
Software:
DLMFReferences:
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