Abstract:For a non-zero parameter $q$, we define Hermite trace polynomials, which are multivariate polynomials indexed by permutations. We prove several combinatorial properties for them, such as expansions and product formulas. The linear functional determined by these trace polynomials is a state for $q = \frac{1}{N}$ for $N$ a non-zero integer. For such $q$, Hermite trace polynomials of different degrees are orthogonal. The product formulas extend to the closure with respect to the state. The state can be identified with the expectation induced by the $N \times N$ Hermitian Brownian motion. Hermite trace polynomials are martingales for this Brownian motion, while the elements in the closure can be interpreted as stochastic integrals with respect to it. Using the grading on the algebra, we prove several chaos decompositions for such integrals, as well as analyze corresponding creation and annihilation operators. In the univariate, pure trace polynomial case, trace Hermite polynomials can be identified with the Hermite polynomials of matrix argument.
From: Michael Anshelevich [view email]
[v1]
Tue, 26 Jul 2022 20:52:59 UTC (33 KB)
[v2]
Fri, 11 Jul 2025 01:36:37 UTC (37 KB)
[v3]
Wed, 1 Apr 2026 21:24:05 UTC (40 KB)
[v4]
Wed, 17 Jun 2026 18:17:54 UTC (41 KB)
[v5]
Wed, 16 Sep 2026 02:01:05 UTC (41 KB)
此内容由惯性聚合(RSS阅读器)自动聚合整理，仅供阅读参考。 原文来自 — 版权归原作者所有。