论文标题
确定性矩阵和IID GUE矩阵中多项式中平滑函数的渐近扩展
Asymptotic expansion of smooth functions in polynomials in deterministic matrices and iid GUE matrices
论文作者
论文摘要
令$ x^n $为$ n \ times n $独立gue随机矩阵的家族，$ z^n $一个确定性矩阵的家族，$ p $ a sexchaint nodaverative dynoverative dolynomial，即任何$ n $，$ n $，$ p（x^n）$ is sexAdeAdeAdeAdeAdeAdeAdeAdeAdexoint，$ f $ f $ a smoplece a a specame a a ploock a a plose Function。我们证明，对于任何$ k $，如果$ f $足够光滑，则存在确定性常数$α_i^p（f，z^n）$，以至于$ \ mathbb {e} \ left [\ frac {1} {n} {n} {n} {n} {n} {trext {tr} {tr} \ sum_ {i = 0}^k \ frac {α_i^p（f，z^n）} {n^{2i}} \ +\ +\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ banters。特别是，如果$ x $是一个免费的半圆形系统，那么当$ f $的支持和$ p（x，z^n）$的频谱是不连接时，对于所有$ i $，$ i $，$α_i^p（f，z^n）= 0 $。作为推论，我们证明给定$α<1/2 $，对于$ n $足够大，每个特征值$ p（x^n，z^n）$ is $ n^{ - α} $ - 从$ p（x，z^n）$的频谱近距离接近。
Let $X^N$ be a family of $N\times N$ independent GUE random matrices, $Z^N$ a family of deterministic matrices, $P$ a self-adjoint non-commutative polynomial, that is for any $N$, $P(X^N)$ is self-adjoint, $f$ a smooth function. We prove that for any $k$, if $f$ is smooth enough, there exist deterministic constants $α_i^P(f,Z^N)$ such that $$ \mathbb{E}\left[\frac{1}{N}\text{Tr}\left( f(P(X^N,Z^N)) \right)\right]\ =\ \sum_{i=0}^k \frac{α_i^P(f,Z^N)}{N^{2i}}\ +\ \mathcal{O}(N^{-2k-2}) .$$ Besides the constants $α_i^P(f,Z^N)$ are built explicitly with the help of free probability. In particular, if $x$ is a free semicircular system, then when the support of $f$ and the spectrum of $P(x,Z^N)$ are disjoint, for all $i$, $α_i^P(f,Z^N)=0$. As a corollary, we prove that given $α<1/2$, for $N$ large enough, every eigenvalue of $P(X^N,Z^N)$ is $N^{-α}$-close from the spectrum of $P(x,Z^N)$.
