Quantum walker trapped by self-similarity of the Sierpi\'nski carpet

We study the dynamics of a single quantum particle on a finite-size square lattice with a fractal structure resembling the Sierpi\'nski carpet, and compare it to the dynamics on a uniform lattice of the same size. For a particle initially localized at a corner of the lattice, we monitor the probability of finding it near the initial and opposite corners using zone-integrated probabilities, allowing a consistent comparison across fractal orders. While on the uniform lattice the particle reaches the opposite corner ballistically, in a time proportional to the lattice size, on the Sierpi\'nski lattice it becomes increasingly confined to the vicinity of its initial position. We show that this trapping builds up self-similarly across the whole hierarchy of corner zones of the lattice.