At the integer quantum Hall transition, wave functions exhibit multifractal statistics — their amplitudes fluctuate with correlations that encode the critical exponents of the transition. The maximum wave-function amplitude in a finite system is a natural extreme-value observable, but extracting clean critical information from it has been difficult because the maximum couples to non-universal features of the system.
The paper shows that the maximum amplitude decomposes multiplicatively into two factors: a global gain and an intrinsic extreme component. The gain is approximately log-normal and captures non-universal effects — system-specific features like boundary conditions and coupling to external channels. The intrinsic extreme component carries the universal critical information.
Before decomposition, the statistics of the maximum are contaminated by the gain. The non-universal contribution obscures the critical scaling. After gain normalization — dividing out the log-normal factor — the remaining distribution changes qualitatively: it becomes a clean probe of the underlying correlated criticality.
The decomposition is structural, not a computational trick. The gain factor represents how efficiently the system couples to external measurements; the intrinsic extreme represents how the critical wave function distributes its weight internally. These are physically distinct processes that happen to multiply together in the maximum amplitude. Separating them requires identifying the gain mechanism and removing it.
Extreme observables probe criticality, but only after you subtract the amplifier. The signal was always there; the non-universal gain was the noise.