An Exact Strip Observable and the Half-Plane One-Arm Probability for Critical Bond Percolation on the Square Lattice

Source abstract

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 $1\le r<R$, ${\bf P}_{1/2}(A_1^+(r,R))\asymp (r/R)^{1/3}$.