扩散温度尺度下二维库仑气体的一致对数索伯列夫不等式
Uniform logarithmic Sobolev inequalities for the 2D Coulomb gas at the diffusive temperature scale
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中文总结 AI 辅助
该研究针对扩散温度尺度下的二维库仑气体系综，证明其满足与N无关的一致对数索伯列夫不等式，还推导了加权平面高斯测度的庞加莱与对数索伯列夫不等式，常数仅依赖总方差和总指数。
中文摘要 AI 辅助
对于N≥2且β>0，考虑正则二维库仑气体系综：在ℂ^N上，dℙ_{N,β}(Z)=Z_{N,β}^{-1}e^{-β|Z|²/2}∏_{i<j}|z_i-z_j|^{β/N}dZ。对关联指数中的N⁻¹因子将该系综置于扩散（即高温）尺度的情况，我们证明：对每个固定的逆温度β>0，该吉布斯测度在无碰撞紧支光滑函数生成的闭狄利克雷型域上满足对数索伯列夫不等式（LSI），且正常数与N无关；当β在(0,∞)的紧子集上变化时，LSI常数可保持一致。第二个主要结果针对平面高斯测度，其权重为点距离正幂的有限乘积，证明了庞加莱不等式与对数索伯列夫不等式，常数仅取决于基础高斯测度的总方差和总指数，与点的数量或位置无关。为证明完整标记庞加莱不等式，我们将方差分解为置换不变部分和标签依赖部分，分别通过加权∂̄估计和带随机置换的相对坐标条件控制；此外，单位点对数索伯列夫不等式、条件熵不等式及排斥配分估计给出缺陷对数索伯列夫不等式，Rothaus紧缩法结合所得完整标记庞加莱不等式与缺陷对数索伯列夫不等式。未断言β→0或β→∞时的一致性。
英文摘要
For $N\ge2$ and $β>0$, consider the canonical 2D Coulomb gas ensemble \begin{equation} \mathrm{d}\mathbb{P}_{N,β}(Z) =\mathsf{Z}_{N,β}^{-1}e^{-β|Z|^2/2} \prod_{i<j}|z_i-z_j|^{β/N}\,\mathrm{d}Z \qquad\text{on }\mathbb{C}^N. \end{equation} The factor $N^{-1}$ in the pair exponent places the ensemble at the diffusive, or high-temperature, scale. We prove that, for every fixed inverse temperature $β>0$, this Gibbs measure satisfies a logarithmic Sobolev inequality (LSI) on the closed Dirichlet-form domain generated by collision-free compactly supported smooth functions, with a positive constant independent of $N$. The LSI constant may be chosen uniformly when $β$ ranges over a compact subset of $(0,\infty)$. A second main result establishes Poincaré and logarithmic Sobolev inequalities for planar Gaussian measures weighted by finite products of positive powers of distances to points, with constants depending only on the total variance of the underlying Gaussian measure and the total exponent, not on the number or locations of the points. To prove the full labeled Poincaré inequality, we decompose variance into permutation-invariant and label-dependent parts, controlled respectively by a weighted $\bar\partial$ estimate and by relative-coordinate conditioning with random transpositions; separately, one-site logarithmic Sobolev inequalities, a conditional entropy inequality, and a repulsive partition estimate give a defective logarithmic Sobolev inequality. Rothaus tightening combines the resulting full labeled Poincaré inequality with the defective logarithmic Sobolev inequality. No uniformity as $β\to0$ or $β\to\infty$ is asserted.
