Upper Critical Dimension of the Kardar-Parisi-Zhang Equation

The strong-coupling regime of Kardar-Parisi-Zhang surface growth driven by short-ranged noise is shown to have an upper critical dimension d> less than or equal to four [where the dynamic exponent z takes the value z(d>)0ex0ex=0ex0ex2]. To derive this, we use the mapping onto directed polymers with quenched disorder. Two such polymers coupled by a small contact attraction of strength u are shown to form a bound state at all temperatures β^-1≤βc^-1, the roughening temperature of a single polymer. Comparing singularities of the (de-)localization transition at u0ex0ex=0ex0ex0 below βc^-1 and at βc^-1 then yields d>≤4.