Wick polynomials in non-commutative probability

多图与时间有序的伊瑟里斯-威克公式

Multigraphs and Time Ordered Isserlis-Wick formulae

本文通过代数和图示两种方法，研究高斯过程与多项式变量诱导路径有序指数的计算。

Two-parameter Non-commutative Central Limit Theorem

The non-commutative Central Limit Theorem (CLT) introduced by Speicher in 1992 states that given almost any sequence of non-commutative random variables that commute or anti-commute pair-wise, the *-moments of the normalized partial sum S_N=(b_1+...+ b_N)/\sqrt{N} are given by a Wick-type formula refined to count the number of crossings in the underlying pair-partitions. When coupled with explicit matrix models, the theorem yields random matrix models for creation and annihilation operators on the q-Fock space of Bozejko and Speicher. In this paper, we derive a non-commutative CLT when the pai…

Non-commutative Polynomials of Independent Gaussian Random Matrices. The Real and Symplectic Cases

In their paper, "A new application of random matrices: Ext(C*_red(F_2)) is not a group", Haagerup and Thorbjornsen prove an extension of Voiculescu's random matrix model for independent complex self-adjoint Gaussian random matrices. We generalize their result to random matrices with real or symplectic entries (the GOE- and the GSE-ensembles) and random matrix ensembles related to these.

A pedestrian approach to the invariant Gibbs measures for the 2-d defocusing nonlinear Schrödinger equations

We consider the defocusing nonlinear Schrödinger equations on the two-dimensional compact Riemannian manifold without boundary or a bounded domain in $\R^2$. Our aim is to give a pedagogic and self-contained presentation on the Wick renormalization in terms of the Hermite polynomials and the Laguerre polynomials and construct the Gibbs measures corresponding to the Wick ordered Hamiltonian. Then, we construct global-in-time solutions with initial data distributed according to the Gibbs measure and show that the law of the random solutions, at any time, is again given by the Gibbs measure.

非线性随机矩阵的测度集中及其在神经网络和非交换多项式中的应用

Concentration of measure for non-linear random matrices with applications to neural networks and non-commutative polynomials

本文证明了多种非线性随机矩阵模型的集中不等式，并以此为推论获得了神经网络共轭核的线性谱统计及随机矩阵中非交换多项式的估计。

On the operator norm of non-commutative polynomials in deterministic matrices and iid Haar unitary matrices

Let $U^N = (U_1^N,\dots, U^N_p)$ be a d-tuple of $N\times N$ independent Haar unitary matrices and $Z^{NM}$ be any family of deterministic matrices in $\mathbb{M}_N(\mathbb{C})\otimes \mathbb{M}_M(\mathbb{C})$. Let $P$ be a self-adjoint non-commutative polynomial. In 1998, Voiculescu showed that the empirical measure of the eigenvalues of this polynomial evaluated in Haar unitary matrices and deterministic matrices converges towards a deterministic measure defined thanks to free probability theory. Let now $f$ be a smooth function, the main technical result of this paper is a precise bound of …

Meixner class of orthogonal polynomials of a non-commutative monotone Levy noise

Let $(X_t)_{t\ge0}$ denote a non-commutative monotone Lévy process. Let $ω=(ω(t))_{t\ge0}$ denote the corresponding monotone Lévy noise.. A continuous polynomial of $ω$ is an element of the corresponding non-commutative $L^2$-space $L^2(τ)$ that has the form $\sum_{i=0}^n\langle ω^{\otimes i},f^{(i)}\rangle$, where $f^{(i)}\in C_0(\mathbb R_+^i)$. We denote by $\mathbf{CP}$ the space of all continuous polynomials of $ω$. For $f^{(n)}\in C_0(\mathbb R_+^n)$, the orthogonal polynomial $\langle P^{(n)}(ω),f^{(n)}\rangle$ is defined as the orthogonal projection of the monomial $\langleω^{\otimes n…

通过树配对证明AMP的普遍性

Universality of AMP via Tree Pairings

本文通过树配对代数证明，多项式非线性AMP在特定迭代次数内与状态演化预测一致，适用于非高斯矩阵及扩展模型。