半空间约束Riesz气体的限域相变
Confinement transitions in half-space constrained Riesz gases
AI总结：
该研究针对半空间约束的Riesz气体，证明其平衡测度随相互作用参数呈现二分性，在强长程 regime下存在限域相变，推广了库仑气体的相关结果。
AI中文摘要：
我们研究空间维度为任意d、相互作用参数$s \in (d-3,d)$的Riesz气体，其被硬壁约束在半空间中。我们证明平衡测度会根据相互作用范围呈现二分性：当$s \in (d-2,d)$（对应弱长程 regime）时，平衡测度始终保留非平凡的体分量，因此绝不会完全限域在壁上；相反，当$s \in (d-3,d-2]$（对应强长程 regime）时，会发生限域相变：存在一个临界壁位置，超过该位置后平衡测度完全支撑在壁上。我们的定理推广了针对库仑气体（对应$s=d-2$的情况）的最新结果。
英文摘要：
We study a Riesz gas with interaction parameter $s \in (d-3,d)$ in an arbitrary spatial dimension $d$, confined to a half-space by a hard wall. We prove that the equilibrium measure exhibits a dichotomy according to the interaction range. For $s \in (d-2,d)$, corresponding to the weakly long-ranged regime, the equilibrium measure always retains a non-trivial bulk component and thus is never completely confined to the wall. In contrast, for $s \in (d-3,d-2]$, corresponding to the strongly long-ranged regime, a confinement transition occurs: there exists a critical wall position beyond which the equilibrium measure is supported entirely on the wall. Our theorem generalises recent results established for the Coulomb gas, corresponding to the case $s=d-2$.
