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 $\alpha \approx 2.2-2.5$, 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.

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• Anderson localization
• Electrical conductivity
• Integer quantum Hall effect
• Mesoscopics
• Topological phase transition
• Disordered systems
• Topological materials