Fluctuations of $β$-Jacobi Product Processes

Abstract

We study Markov chains formed by squared singular values of products of truncated orthogonal, unitary, symplectic matrices (corresponding to the Dyson index β = 1, 2, 4 respectively) where time corresponds to the number of terms in the product. More generally, we consider the β-Jacobi product process obtained by extrapolating to arbitrary β > 0. When the time scaling is preserved, 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 β = 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 β > 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 β-generalization of the interpolating process.

Extremal singular values of random matrix products and Brownian motion on GL(N,C)

“…Prior to this work, convergence of the largest log singular values to ξ(t) was known for products of Ginibre and truncated unitary matrices [LWW18,ABK19,Ahn19]. Recall that a complex Ginibre matrix is a rectangular matrix of i.i.d.…”

“…For products of square Ginibre matrices, the convergence of the largest log singular values to ξ(t) for fixed time was shown by [ABK19, ABK20, LWW18], with generalizations to products of rectangular Ginibre matrices indicated in [LWW18]. Extensions to joint time convergence and for products of truncated unitary matrices were established in [Ahn19]. The accessibility of these examples are due to determinantal and related structures available in those cases [AKW13, AIK13, KKS16, BGS18, Ahn19].…”

“…Natural examples are products of real Ginibre matrices and truncated Haar orthogonal matrices for the former, and products of quaternionic Ginibre matrices and truncated Haar symplectic matrices for the latter. These models, and oneparameter deformations of these models in the Dyson index β in the spirit of β-ensembles, were considered in [Ahn19] where tightness results were obtained at the edge in the regime M N . While convergence results beyond β = 2 are unavailable, we expect N → ∞ limits of β-deformations of Y (N ) (t) to appear, where for β = 1 and 4 this should be the corresponding diffusion on GL(N, R) and GL(N, H).…”

Lyapunov exponents for truncated unitary and Ginibre matrices

“….. Another motivation to investigate this question came from work of the second author [22,Theorem 1.2], in which the analogues of Lyapunov exponents for p-adic Date: September 16, 2021. 1 Related existence results on limits of matrix entries were obtained earlier by Bellman [5] and Furstenberg-Kesten [10]. random matrices were found to converge to a similar limit-a geometric progression rather than an arithmetic one-as n → ∞ for the p-adic analogues of Ginibre and truncated unitary matrices.…”

“…The regime where T /n converges to a constant was also studied in [3,4,18]. Though the Lyapunov exponents do not directly appear in this regime, it is known from the work of the first author [1] that the asymptotic behavior of the largest singular values for products of Ginibre matrices coincides with that of products of corners of Haar unitary matrices. This made it natural to expect that the sequence of Lyapunov exponents of both Ginibre and truncated unitary corners should converge to the same limit 0, −1, −2, .…”