Breakdown of Self-Averaging at Quantum Hall Transitions.

We study the full distribution of the zero-temperature Hall conductivity in a lattice model of the integer quantum Hall effect across disorder realizations. Near the plateau transition, the distributions develop heavy power-law tails with exponent α≈2.2-2.5, implying a finite mean but divergent variance. The tail persists across system sizes, correlation lengths of the disorder potential, and fillings indicating that Hall conductivity is not self-averaging at criticality. The exponent is qualitatively compatible with predictions from random-matrix models of Berry curvature fluctuations, hinting at a universal statistical structure of the quantum Hall critical regime.