平均场库仑气体的谱隙
Spectral gaps for mean-field Coulomb gases
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中文总结 AI 辅助
该研究针对d≥3维排斥单组分库仑气体，在弱耦合条件下证明了粒子数一致的庞加莱不等式，得到过阻尼朗之万动力学的指数弛豫，通过单点不等式、移动极点估计等方法给出谱隙的匹配上下界。
中文摘要 AI 辅助
我们证明了维度$d\ge3$的排斥性（单组分）库仑气体存在粒子数一致的庞加莱不等式，其逆温度$β_N$可依赖于$N$。我们假设了一个以单个粒子所受有效相互作用强度$Θ_N$表述的弱耦合条件。这一结果给出了关联过阻尼朗之万动力学的指数弛豫性质。证明的核心要素是关于库仑极点（含重合极点）的数量与位置均一致的单点庞加莱不等式。通过移动极点估计和Dobrushin-Wu张量化方法得到多粒子下界，而质心坐标给出匹配的上界。
英文摘要
We prove a Poincaré inequality, uniform in the particle number, for repulsive (one-component) Coulomb gases in dimensions $d\ge3$, with inverse temperature $β_N$ that is allowed to depend on $N$. We assume a weak-coupling condition expressed in terms of the effective interaction strength $Θ_N$ seen by a single particle. This yields exponential relaxation for the associated overdamped Langevin dynamics. The main ingredient is a one-site Poincaré inequality uniform in the number and positions of the Coulomb poles, including coincident poles. A moving-pole estimate and Dobrushin-Wu tensorization yield the many-particle lower bound, while the center-of-mass coordinate gives the matching upper bound.
