精确的条带可观测量和临界键逾渗在方格上的半平面单臂概率
An Exact Strip Observable and the Half-Plane One-Arm Probability for Critical Bond Percolation on the Square Lattice
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- National University of Singapore（新加坡国立大学）
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中文总结 AI 辅助
本文证明了方格上临界伯努利键逾渗的半平面单臂指数，通过修正Ikhlef-Ponsaing插值并建立精确条带公式，结合RSW定理得到单臂概率的幂律衰减。
中文摘要 AI 辅助
我们证明了方格上临界伯努利键逾渗的半平面单臂指数。Ikhlef和Ponsaing获得了沿奇数宽度条带运行的唯一无限逾渗壳层通过指定边界边的概率的精确公式。然而，他们的插值将有理函数视为多项式。我们清除完整分母，直接在$q^3=1$处证明坐标方向的Laurent次数界，并从删除关系建立精确的条带公式。在齐次点，qKZ表达式是该边界通过事件的伯努利概率。有限局部修正、RSW定理和单臂拟乘性随后给出，对于$1\le r<R$一致地，${\bf P}_{1/2}(A_1^+(r,R))\asymp (r/R)^{1/3}$。
英文摘要
We prove the half-plane one-arm exponent for critical Bernoulli bond percolation on the square lattice. Ikhlef and Ponsaing obtained an exact formula for the probability that the unique infinite percolation hull running along an odd-width strip passes through a prescribed boundary edge. Their interpolation, however, treats a rational function as a polynomial. We clear the full denominator, prove a coordinatewise Laurent-degree bound directly at $q^3=1$, and establish the exact strip formula from the deletion relations. At the homogeneous point, the qKZ expression is the Bernoulli probability of this boundary-passage event. A finite local modification, the RSW theorem, and one-arm quasi-multiplicativity then give, uniformly for $1\le r<R$, ${\bf P}_{1/2}(A_1^+(r,R))\asymp (r/R)^{1/3}$.
