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Simulations give $\\alpha$ near 0.39, but no theory derives it.","posed_since":"1986","precise":"For the KPZ equation $dh/dt = \\nu \\mathrm{lap}\\ h + (\\lambda/2)(\\mathrm{grad}\\ h)^2 + \\eta(x,t)$ 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":"ab32b3f7a386733e616b0caf3f270e449f2e56b7ea17a78177cd8d74dbf321a2","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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