Abstract Accessibility percolation is a new type of percolation problem inspired by evolutionary biology: a random number, called its fitness, is assigned to each vertex of a graph, then a path in the graph is accessible if fitnesses are strictly increasing through it. In the rough Mount Fuji (RMF) model, the fitness function is defined on the graph as <?CDATA $\omega(v) = \eta(v)+\theta\cdot d(v)$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>ω</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>v</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>η</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>v</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>+</mml:mo> <mml:mi>θ</mml:mi> <mml:mo>⋅</mml:mo> <mml:mi>d</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>v</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> , where θ is a positive number called the drift, d is the distance to the source of the graph and <?CDATA $\eta(v)$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>η</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>v</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> are i.i.d. random variables. In this paper, we determine values of θ for having RMF accessibility percolation on the hypercube and the two-dimensional lattices <?CDATA $\mathbb{L}^2$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">L</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:math> and <?CDATA $\mathbb{L}^2_{alt}$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msubsup> <mml:mrow> <mml:mi mathvariant="double-struck">L</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>a</mml:mi> <mml:mi>l</mml:mi> <mml:mi>t</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msubsup> </mml:math> .