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Abstract A random deposition model with surface relaxation, so-called the Family model is studied in higher dimensions. In three dimensions, the surface width W (t) characterizing the roughness of a surface grows as 2 b log t at the beginning and becomes saturated at 2 a log L for t ≫ L z , where L is the system size. The dynamic exponent z = 1. 99 (2) is estimated from the relation z = a ∕ b and a nice data collapse of the scaling plot W 2 (L , t) ∼ log L 2 a g t ∕ L z is given with z = 2. In four dimensions, the surface width approaches an intrinsic width W int with a small correction term W 2 (L , t) = W int 2 − L 2 α f t ∕ L z , where z ≈ 1. 97 and negative exponent α ≈ − 0. 52 are obtained. Our results support that the Family model belongs to the Edwards–Wilkinson universality class even in higher dimensions. Highlights • The random deposition model with surface diffusion, the Family model, is studied. • We measure critical exponents by using calculating the surface width. • In three dimensions, the Family model shows the logarithmic growth. • In four dimensions, we measure negative values of critical exponents. • The Family model corresponds to the Edwards–Wilkinson equation below four dimensions. [ABSTRACT FROM AUTHOR]