Extreme-Value Criticality and Gain Decomposition at the Integer Quantum Hall Transition

Extreme-value fluctuations at quantum critical points remain poorly understood in the presence of strong correlations and openness. At the integer quantum Hall transition in the open Chalker–Coddington network, we show that the maximal wave-function amplitude separates into a global gain and an intrinsic extreme component, $\mathrm{\mid }\psi {\mathrm{\mid }}_{\max }=A\mathrm{\mid }\tilde{\psi }{\mathrm{\mid }}_{\max }$. We introduce extreme-moment scaling for $\mathrm{\mid }\psi {\mathrm{\mid }}_{\max }$ and observe an approximately parabolic exponent function ${\tau }_{\max }(q)$ over moderate $q$, while $\ln \mathrm{\mid }\psi {\mathrm{\mid }}_{\max }$ displays an almost Gaussian bulk over the studied sizes. The gain factor is close to log-normal and largely controls the raw extremes. Gain normalization reorganizes the statistics: ${\tilde{\tau }}_{\max }(q)$ changes qualitatively and $\mathrm{\mid }\tilde{\psi }{\mathrm{\mid }}_{\max }$ does not support a single-parameter generalized extreme-value collapse under standard centering/scaling in the accessible size window. Extreme observables thus provide a robust probe of correlated criticality in open quantum systems.