论文标题
关于确定性矩阵和IID HAAR统一矩阵的非交通性多项式的运算符规范
On the operator norm of non-commutative polynomials in deterministic matrices and iid Haar unitary matrices
论文作者
论文摘要
令$ u^n =（u_1^n，\ dots，u^n_p）$为$ n \ times n $ n $ haar统一矩阵和$ z^{nm} $是任何确定性矩阵中的任何确定性矩阵中\ Mathbb {M} _M（\ Mathbb {C}）$。令$ p $为自动偶发性多项式。 1998年，Voiculescu表明，通过自由概率理论，在HAAR单一矩阵和确定性矩阵中评估的该多项式的特征值的经验度量趋于确定性措施。现在让$ f $成为一个平稳的功能，本文的主要技术结果是$$ \ frac {1} {1} {mn} {mn} \ text {tr} \ left的期望之间的差异的精确限制。如果$ f $是7倍，我们表明它是由$ m^2 \ left \ vert f \ right \ vert _ {\ mathcal {c}^7} n^{ - 2} $界定的。作为推论，我们获得了柯林斯和男性结果的定量界限的新证明，这为在HAAR统一矩阵和确定性矩阵中评估的多项式的操作员规范提供了足够的条件，几乎可以肯定地倾斜其自由极限。 Actually we show that if $U^N$ and $Y^{M_N}$ are independent and $M_N = o(N^{1/3})$, then almost surely, the norm of any polynomial in $(U^N\otimes I_{M_N}, I_N\otimes Y^{M_N})$ converges almost surely towards its free limit.
Let $U^N = (U_1^N,\dots, U^N_p)$ be a d-tuple of $N\times N$ independent Haar unitary matrices and $Z^{NM}$ be any family of deterministic matrices in $\mathbb{M}_N(\mathbb{C})\otimes \mathbb{M}_M(\mathbb{C})$. Let $P$ be a self-adjoint non-commutative polynomial. In 1998, Voiculescu showed that the empirical measure of the eigenvalues of this polynomial evaluated in Haar unitary matrices and deterministic matrices converges towards a deterministic measure defined thanks to free probability theory. Let now $f$ be a smooth function, the main technical result of this paper is a precise bound of the difference between the expectation of $$ \frac{1}{MN} \text{Tr}\left( f(P(U^N\otimes I_M,Z^{NM})) \right) , $$ and its limit when $N$ goes to infinity. If $f$ is seven times differentiable, we show that it is bounded by $M^2 \left\Vert f\right\Vert_{\mathcal{C}^7} N^{-2}$. As a corollary we obtain a new proof with quantitative bounds of a result of Collins and Male which gives sufficient conditions for the operator norm of a polynomial evaluated in Haar unitary matrices and deterministic matrices to converge almost surely towards its free limit. Actually we show that if $U^N$ and $Y^{M_N}$ are independent and $M_N = o(N^{1/3})$, then almost surely, the norm of any polynomial in $(U^N\otimes I_{M_N}, I_N\otimes Y^{M_N})$ converges almost surely towards its free limit.
