Demonstration of a conjecture for random walks in d-dimensional Sierpinski fractals

Random walks on some fractals can be analysed by renormalization procedures. These techniques make it possible to obtain the Laplace transform of the first-passage time probability density function of a random walker that moves in the fractal. The calculation depends on a function rho(x) that is particular to each kind of fractal. For the Sierpinski family of fractals, it has been conjectured that rho(x) = 2dx(2) - 3(d - 1)x + d - 2, where d is the dimension of the Euclidean space in which the Sierpinski fractal is embedded. We provide a proof of the conjecture that is based on the symmetries of the Sierpinski fractal.