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 models which can be of independent interest.

众所周知，半圆定律是维格纳定理中的极限分布，是二阶矩惩罚的对数能量的最小者。 Girko 和 Marchenko-Pastur 定理也存在非常相似的事实。在这项工作中，我们揭示了一个有趣的现象，表明该函数在矩阵维度上沿着平均经验光谱分布是单调的。这让人想起玻尔兹曼熵沿玻尔兹曼方程的单调性、自由能沿遍历马尔可夫过程的单调性，以及熵或自由熵沿经典或自由中心极限定理的香农单调性。虽然我们只验证了高斯酉系综、复吉尼布系综和方形拉盖尔酉系综的这种单调性现象，但数值模拟表明它实际上更普遍。一路上我们获得了模型对数能量的明确公式，这些公式可能是独立感兴趣的。