Fluctuations of $$\beta $$-Jacobi product processes

Limits and fluctuations of p-adic random matrix products

“…The same is true on the soft edge, namely there is another limit (also introduced in [6] and also treated in the mathematical literature in [61]) which interpolates between the Airy process and independent Gaussian statistics; we mention also work of Berezin-Strahov [10], which computed the gap probabilities of the limit. This soft edge limit was further shown by Ahn for truncated unitary matrices [2] and later for a broad class of invariant ensembles [1]. In this regime, again, the soft edge is universal, and the bulk is expected to be so.…”

“…However, Borodin-Gorin-Strahov [23] found that singular values of complex matrix products are also a degeneration of the Macdonald processes introduced by Borodin-Corwin [18]. Many techniques for analyzing such processes exist, which were applied to complex matrix products in subsequent work of Ahn [2], Ahn-Strahov [4], Ahn and the author [5], and Gorin-Sun [44]. The work [76] found analogous relations between padic matrix products and Macdonald processes.…”

“…for the set of integer signatures of length k, where we allow k = ∞. The limit object in our theorems is a random element L t,χ = (L (1) , L (2) , . .…”

“…Theorem 6.1. For any t ∈ (0, 1), χ ∈ R >0 , there is a unique Sig ∞ -valued random variable L t,χ = (L (1) , L (2) , . .…”

“…(1.2) 1 In fact, ai|ai−1. 2 It is worth clarifying that in this literature, 'local' typically refers to local fields such as Qp as opposed to global fields such as Q, and 'local statistics' in [69] refers to those which only depend on the localization of the cokernel at a finite collection of primes. Our usage of 'local' in this work, which instead follows that of the real/complex random matrix theory literature, is unrelated.…”