{"schema":"pubphys.bundle/1","record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"ca11c24d6feca6096b200db9cd693266289dedee599426b14aba95794addfaf1","created":"2026-10-03T07:18:11Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["5475cf5dc5b1a72283d26dc95dc6e00d7ea4e00096a7c6492e409a1816eeb011"],"salt":"46e0766091026e610647e612306a0ec3386ad67c6b5c8cf43143ee92c1a15bc2","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"value","assisted_by":[],"external_id":"stat.kpz-higher-d.exponents-2plus1","kind":"well-posed","literature_status":"open","n":"1","parents":[],"plain":"For a growing two-dimensional surface, height fluctuations grow with system size as a power law with exponent $\\alpha$. Simulations give $\\alpha$ near 0.39, but no theory derives it.","posed_since":"1986","precise":"For the KPZ equation dh/dt = nu lap h + (lambda/2)(grad h)^2 + eta(x,t) with Gaussian white noise eta in $d = 2$ spatial dimensions, determine the roughness exponent $\\alpha$ (with $z = 2 - \\alpha$ from Galilean invariance) exactly or to $10^{-4}$, and decide whether it is a simple rational number. Large simulations give $\\alpha$ approximately 0.387 to 0.39; the conjecture $\\alpha = 2/5$ is disfavored.","problem_ref":null,"references":"","settled_by":"An exact solution, or a nonperturbative RG computation with controlled error, agreeing with simulations to $10^{-4}$.","status_note":"","title":"Exact roughness exponent of KPZ surfaces in 2+1 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