Local laws and rigidity for Coulomb gases at any temperature
Local laws and rigidity for Coulomb gases at any temperature
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DOI：
10.1214/20-aop1445
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发表时间：
2019-06
期刊：
影响因子：
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通讯作者：
S. Armstrong;S. Serfaty
中科院分区：
文献类型：
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作者：
S. Armstrong;S. Serfaty
我们研究库仑气体在任何维度$d \geq 2$和在一个广泛的温度范围。我们在微观尺度上证明了能量、分离和点数的局部规律。这就得到了极限点过程的存在性，它推广了任意温度和尺寸的Ginibre点过程。在均匀背景密度的情况下，局部定律与自由能的定量扩展结合在一起，具有新的显式错误率。这些估计具有明确的温度依赖性，允许处理非常大或非常小的温度，并显示出一个新的最小长度尺度的刚度取决于温度。它们也适用于能量最小化（正式零度）。该方法基于尺度上的自举，通过引入次加性和超加性近似能量，揭示了能量模表面项的可加性。
We study Coulomb gases in any dimension $d \geq 2$ and in a broad temperature regime. We prove local laws on the energy, separation and number of points down to the microscopic scale. These yield the existence of limiting point processes generalizing the Ginibre point process for arbitrary temperature and dimension. The local laws come together with a quantitative expansion of the free energy with a new explicit error rate in the case of a uniform background density. These estimates have explicit temperature dependence, allowing to treat regimes of very large or very small temperature, and exhibit a new minimal lengthscale for rigidity depending on the temperature. They apply as well to energy minimizers (formally zero temperature). The method is based on a bootstrap on scales and reveals the additivity of the energy modulo surface terms, via the introduction of subadditive and superadditive approximate energies.