Running a Random Number generator in parallel? Will this speed up a computation?

I’m calculating basically a multidimensional random walk problem. To get more accuracy, I need larger systems (more dimensions), which requires longer time. To speed up the calculation, I’m delving into parallel computing for the first time.

I call the fortran random number generator frequently. It occurred to me that I might split the random number generator off from the rest of the program and run it in parallel. Of course, the generated random numbers would be stored in a shared memory location. Does it seem like this could potentially speed up the computation? Does a random number generator take up a large fraction of the CPU in a random walk problem?