实对称一维随机带阵的转移矩阵方法
Transfer matrix approach for the real symmetric 1D random band matrices
摘要：
本文将最近发展起来的一维带矩阵超对称转移矩阵法的严格应用推广到正交对称的情形。我们考虑$N\times N$块带矩阵组成的$W\times W$随机高斯块（参数化$j,k \inΛ=[1,n]\cap \mathbb{Z}$，$N=nW$）与一个固定的项目的方差$J_{jk}=W^{-1}(δ_{j,k}+βΔ_{j,k})$在每个块。考虑到极限$W, n\to\infty$，我们证明了这样的矩阵的特征多项式的第二关联函数的行为在大部分的频谱表现出交叉附近的阈值$W\sim \sqrt{N}$。
英文摘要：
This paper adapts the recently developed rigorous application of the supersymmetric transfer matrix approach for the 1d band matrices to the case of the orthogonal symmetry. We consider $N\times N$ block band matrices consisting of $W\times W$ random Gaussian blocks (parametrized by $j,k \inΛ=[1,n]\cap \mathbb{Z}$, $N=nW$) with a fixed entry's variance $J_{jk}=W^{-1}(δ_{j,k}+βΔ_{j,k})$ in each block. Considering the limit $W, n\to\infty$, we prove that the behavior of the second correlation function of characteristic polynomials of such matrices in the bulk of the spectrum exhibit a crossover near the threshold $W\sim \sqrt{N}$.
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