Index distribution of Cauchy random matrices
Index distribution of Cauchy random matrices
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柯西随机矩阵的指数分布
DOI：
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发表时间：
2013
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通讯作者：
P. Vivo
中科院分区：
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作者：
R. Marino;S. Majumdar;G. Schehr;P. Vivo
使用库仑气体技术，我们分析计算一个大的 N × N 柯西随机矩阵具有 N+ 个正特征值的概率 Pβ(C)(N+,N)?>，其中 N+ 称为系综的索引。我们证明，对于大 N，该概率尺度为 Pβ(C)(N+,N)≈exp[−βN2ψC(N+/N)]?>，其中 β 是系综的戴森指数。速率函数 ψC(κ) 根据单积分计算，易于进行数值计算并适合渐近分析。我们发现速率函数在 κ = 1/2 处的最小值附近具有由对数奇点调制的二次行为。因此，大 N 的指数方差为 Var(N+) ∼ σCln N，其中 σC = 2/(βπ2) 是高斯和 Wishart 情况下相应前因子的两倍。分析结果通过数值模拟并对照精确的有限 N 公式进行检查，对于 β = 2，可以使用正交多项式导出该公式。
Using a Coulomb gas technique, we compute analytically the probability Pβ(C)(N+,N)?> that a large N × N Cauchy random matrix has N+ positive eigenvalues, where N+ is called the index of the ensemble. We show that this probability scales for large N as Pβ(C)(N+,N)≈exp[−βN2ψC(N+/N)]?>, where β is the Dyson index of the ensemble. The rate function ψC(κ) is computed in terms of single integrals that are easily evaluated numerically and amenable to an asymptotic analysis. We find that the rate function, around its minimum at κ = 1/2, has a quadratic behavior modulated by a logarithmic singularity. As a consequence, the variance of the index scales for large N as Var(N+) ∼ σCln N, where σC = 2/(βπ2) is twice as large as the corresponding prefactor in the Gaussian and Wishart cases. The analytical results are checked by numerical simulations and against an exact finite N formula which, for β = 2, can be derived using orthogonal polynomials.