Free energy fluctuations scale as t^(1/3) — KPZ universality class, Tracy-Widom distribution
RANDOM POTENTIAL LANDSCAPE + OPTIMAL POLYMER
HEIGHT FUNCTION h(x,t) — KPZ GROWTH
FREE ENERGY FLUCTUATION SCALING
Time step: 0
Endpoint spread ξ: –
F variance t^(1/3): –
KPZ universality (Kardar-Parisi-Zhang 1986): The directed polymer in a random medium has free energy F(t) = -T log Z(t), where Z is the partition function summing over all polymer paths. The key result: the free energy fluctuations scale as δF ~ t^(1/3) (not t^(1/2) as expected from CLT), with the fluctuation distribution converging to the Tracy-Widom GUE distribution (Johansson 2000, confirmed experimentally in liquid crystal turbulence by Takeuchi & Sano 2010). The spatial correlation of the polymer endpoint is ξ ~ t^(2/3) — the KPZ exponents (1, 1/3, 2/3) satisfy the scaling relation χ = 2β − 1. The KPZ class is one of the most studied universality classes in non-equilibrium statistical mechanics.