论文标题
随机矩阵的对数能量的单调性
Monotonicity of the logarithmic energy for random matrices
论文作者
论文摘要
众所周知，半圆形定律（这是Wigner定理中的限制分布）是第二刻的对数能量惩罚的最小化器。 Girko和Marchenko-Pastur定理也是一个非常相似的事实。在这项工作中，我们阐明了一种有趣的现象，表明该功能是按照矩阵维度沿平均经验光谱分布的单调的。这让人联想到沿Boltzmann方程的Boltzmann熵的单调性，沿着ergodic Markov过程的自由能的单调性以及熵或自由熵的Shannon单调性沿经典或自由的中央限制。虽然我们仅验证高斯单位合奏，复杂的吉尼布集合和方形的laguerre单一集合的这种单调性现象，但数值模拟实际上表明它更为普遍。我们在此过程中获得了可能具有独立关注的上述模型对数能量的明确公式。
It is well-known that the semi-circle law, which is the limiting distribution in the Wigner theorem, is the minimizer of the logarithmic energy penalized by the second moment. A very similar fact holds for the Girko and Marchenko--Pastur theorems. In this work, we shed the light on an intriguing phenomenon suggesting that this functional is monotonic along the mean empirical spectral distribution in terms of the matrix dimension. This is reminiscent of the monotonicity of the Boltzmann entropy along the Boltzmann equation, the monotonicity of the free energy along ergodic Markov processes, and the Shannon monotonicity of entropy or free entropy along the classical or free central limit theorem. While we only verify this monotonicity phenomenon for the Gaussian unitary ensemble, the complex Ginibre ensemble, and the square Laguerre unitary ensemble, numerical simulations suggest that it is actually more universal. We obtain along the way explicit formulas of the logarithmic energy of the mentioned models which can be of independent interest.
