受限于避开球的向量高斯自由场的宏观冻结
Macroscopic freezing of a vector Gaussian free field conditioned to avoid a ball
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中文总结 AI 辅助
研究二维离散高斯自由场在避开球条件下的宏观冻结，证明场除以log n收敛于平衡势两倍乘随机单位向量，并确定容量代价与对齐性质。
中文摘要 AI 辅助
我们考虑在边长为$n$的盒子上的固定有限个独立的二维离散高斯自由场，边界值为零，并受限于向量值场在宏观内部区域中保持在固定球外。我们证明，该场除以$\log n$后在$L^2$中依分布收敛于该区域的平衡势的两倍乘以一个均匀随机单位向量。我们还确定了条件事件的精确容量代价，并在密度为一的确定性站点集合上获得对齐，且对其相互距离没有限制。证明结合了Korzhenkova和Sepúlveda的范数排斥定理、高斯大偏差以及容量变分问题中的等式情形。
英文摘要
We consider a fixed finite number of independent two-dimensional discrete Gaussian free fields on a box of side $n$, with zero boundary values, conditioned so that the vector-valued field stays outside a fixed ball on a macroscopic interior domain. We prove that the field, divided by $\log n$, converges in distribution in $L^2$ to twice the equilibrium potential of the domain multiplied by a uniform random unit vector. We also identify the exact capacity cost of the conditioning event and obtain alignment on a deterministic density-one set of sites, with no restriction on their mutual distance. The proof combines the norm repulsion theorem of Korzhenkova and Sepúlveda with Gaussian large deviations and the equality case in the capacity variational problem.
