Asymptotic Transitions Between Airy, Pearcey, and Sine Kernels in Random Matrix Theory — E8 Intelligence Research

FINDING: The search results are dominated by generic linear-algebra kernel videos (null space, kernel trick) with only one substantive mathematical hit — a 2024 arXiv paper on asymptotic transitions between Airy, Pearcey, and sine kernels in random matrix theory. | MATH: The paper (arXiv:2412.10596) provides complete asymptotic expansions of the extended Airy kernel \\(K_{\\mathrm{Ai}}(x,y)\\) and extended Pearcey kernel \\(K_{\\mathrm{Pe}}(x,y)\\) under rescalings that drive them to the sine kernel \\(K_{\\sin}(x,y) = \\frac{\\sin(x-y)}{\\pi(x-y)}\\). The Airy kernel: \\(K_{\\mathrm{Ai}}(x,y) = \\frac{\\mathrm{Ai}(x)\\mathrm{Ai}'(y) - \\mathrm{Ai}'(x)\\mathrm{Ai}(y)}{x-y}\\). The sine kernel is the bulk universal limit; Airy is the hard-edge limit; Pearcey is the cusp singularity. The transition parameter is typically a scaling variable \\(s\\) such that as \\(s \\to \\infty\\) (or \\(0\\)), one kernel asymptotically collapses to another. | CONNECTION: The sine kernel's eigenvalues are the spacings of the bulk — Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com