Fluctuations of β-Jacobi product processes

• 成果类型: Article
• 署名作者: Ahn, Andrew
• 署名单位: Massachusetts Institute of Technology (MIT)
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-022-01109-0
• 发表日期: 2022
• 页码: 57-123
• 关键词: root systems; RANDOM MATRICES; gaussian fluctuations; asymptotics; spectrum; unitary; models; edge

摘要

We study Markov chains formed by squared singular values of products of truncated orthogonal, unitary, symplectic matrices (corresponding to the Dyson index beta = 1, 2, 4 respectively) where time corresponds to the number of terms in the product. More generally, we consider the beta-Jacobi product process obtained by extrapolating to arbitrary beta > 0. For fixed time (i.e. number of factors is constant), we show that the global fluctuations are jointly Gaussian with explicit covariances. For time growing linearly with matrix size, we show convergence of moments after suitable rescaling. When beta = 2, our results imply that the right edge converges to a process which interpolates between the Airy point process and a deterministic configuration. This process connects a time-parametrized family of point processes appearing in the works of Akemann-Burda-Kieburg and Liu-Wang-Wang across time. In the arbitrary beta > 0 case, our results show tightness of the particles near the right edge. The limiting moment formulas correspond to expressions for the Laplace transform of a conjectural beta-generalization of the interpolating process.