Sharp Phase Transition for the Formation of Infinite Tubes

Classical bond percolation theory studies the conditions for a given point in a random graph to be connected to infinity, or "escape" to infinity, via a sequence of random edges. In this work, we present a higher-dimensional generalization of this question, asking whether a fixed loop (or, more generally, a topological sphere) can escape to infinity via a tube formed by random plaquettes. We refer to this phenomenon as tube percolation. We first compare tube percolation with previously studied higher-dimensional percolation phenomena, including face and cycle percolation. For tubes of codimension one, we further relate the critical probability for tube percolation to those for percolation of finite clusters and shielded percolation in the dual bond percolation model. Next, we introduce a tubular analogue of the classical one-arm event, the tubular one-arm event, and prove that it exhibits a sharp threshold at criticality: below criticality, its probability decays exponentially in scale, whereas above criticality, it admits a mean-field-type lower bound. The proof relies on the O'Donnell-Saks-Schramm-Servedio (OSSS) inequality together with an exploration algorithm adapted to the topology of tubes. Finally, we study the tubular box-crossing property. Unlike ordinary path connectedness, "tube connectedness" is not transitive, and thus there is no natural notion of clusters. Nevertheless, we establish an analogue of the uniqueness of the infinite cluster from classical bond percolation. Combining this result with the sharp threshold for the tubular one-arm event, we prove that the existence of a box-crossing tube also exhibits a sharp threshold.