The exponential excursion-height rate of planar Ising disks

We determine the exponential rate at which a topology-preserving planar Ising disk reaches a large area during one excursion from a single square. For addition rate $\beta$, deletion rate $1$, and every $0<\beta<1/\kappa$, the excursion maximum $\xi$ satisfies $\lim_{T\to\infty}\Prob(\xi\ge T)^{1/T}=\kappa\beta$, where $\kappa$ is the area growth constant of square-lattice self-avoiding polygons. This proves Conjecture 7.11 of Baryshnikov and Onaran. The lower bound comes from concatenating arbitrary fixed-area disks with one connector square and distributing a unit flow over the resulting growth paths. Cumulative column areas recover each completed prefix, which bounds congestion without requiring disjoint paths within a block.