Asymptotic Expansion of Smooth Functions in Polynomials in Deterministic Matrices and iid GUE Matrices
Asymptotic Expansion of Smooth Functions in Polynomials in Deterministic Matrices and iid GUE Matrices
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确定性矩阵和独立同分布 GUE 矩阵中多项式平滑函数的渐近展开
DOI：
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发表时间：
2020
影响因子：
2.4
通讯作者：
Félix Parraud
中科院分区：
文献类型：
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作者：
Félix Parraud
设$$X^N$$X N是一族N×N×N独立的GUE随机矩阵，$$Z^N$$Z N是一族确定性矩阵，P是一个自伴的非交换多项式，即对任意N，$P(X^N，Z^N)$P(X N，Z N)是自伴的，f是光滑函数.证明了对于任意k，如果f足够光滑，则存在确定性常数$α_i^P(f，Z^N)$$αi P(f，Z N)使得$$\Begin{Align}\Mathbb{E}\Left[\FRAC{1}{N}\Tr}\Left(f(P(X^N，Z^N))\Right]\=\\Sum_{i=0}^k\FRAC{\Alpha_I^P(f，Z^N)}{N^{2i}}数学{O}(N^{-2k-2})。E1NTrf(P(XN，ZN))=∑i=0kαIP(f，ZN)N2I+O(N-2k-2).此外，借助自由概率显式地建立了常数$αIPP(f，ZN)。特别地，如果x是自由半圆系统，则当f的支集与$$P(x，Z^N)$$P(x，ZN)的谱不交时，$α_i^P(f，Z^N)=0$$αi P(f，Z^N)=0对所有$$i in{N}$$i∈N.作为推论，我们证明了对于足够大的N，给定$$α<；1/2$$α<；1/2，$$P(X^N，Z^N)$$P(XN，ZN)的每个本征值都是$$N^-$N-α-接近$$P(x，Z^N)$P(x，ZN)的谱。
Let $$X^N$$ X N be a family of $$N\times N$$ N × N independent GUE random matrices, $$Z^N$$ Z N a family of deterministic matrices, P a self-adjoint noncommutative polynomial, that is for any N , $$P(X^N,Z^N)$$ P ( X N , Z N ) is self-adjoint, f a smooth function. We prove that for any k , if f is smooth enough, there exist deterministic constants $$\alpha _i^P(f,Z^N)$$ α i P ( f , Z N ) such that $$\begin{aligned} \mathbb {E}\left[ \frac{1}{N}\text {Tr}\left( f(P(X^N,Z^N)) \right) \right] \ =\ \sum _{i=0}^k \frac{\alpha _i^P(f,Z^N)}{N^{2i}}\ +\ \mathcal {O}(N^{-2k-2}) . \end{aligned}$$ E 1 N Tr f ( P ( X N , Z N ) ) = ∑ i = 0 k α i P ( f , Z N ) N 2 i + O ( N - 2 k - 2 ) . Besides, the constants $$\alpha _i^P(f,Z^N)$$ α i P ( f , Z N ) are built explicitly with the help of free probability. In particular, if x is a free semicircular system, then when the support of f and the spectrum of $$P(x,Z^N)$$ P ( x , Z N ) are disjoint, $$\alpha _i^P(f,Z^N)=0$$ α i P ( f , Z N ) = 0 for all $$i\in \mathbb {N}$$ i ∈ N . As a corollary, we prove that given $$\alpha <1/2$$ α < 1 / 2 , for N large enough, every eigenvalue of $$P(X^N,Z^N)$$ P ( X N , Z N ) is $$N^{-\alpha }$$ N - α -close to the spectrum of $$P(x,Z^N)$$ P ( x , Z N ) .