Depinning transition at the upper critical dimension

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We study the effect of quenched random field disorder on a driven elastic interface close to the depinning transition at the upper critical dimension dc=4 using the functional renormalization group. We have found that the displacement correlation function behaves with distance x as ln(x)²/³ for large x. Slightly above the depinning transition the force-velocity characteristics is described by the equation v f |ln(f)|²/⁹, while the correlation length behaves as Lᵥ f⁻¹/²|ln(f)|¹/⁶, where f=F/Fc-1 is the reduced driving force.