Are genetic algorithms special instances of random search done in an unexpectedly short run-time?
I guess the easiest  way to reason it (this is after thinking about what user:vzn said 


  a genetic algorithm exploits the gradient in the fitness function due
  to crossbreeding. if there is no gradient to exploit, it is indeed
  roughly the same as a random search over the search space.


, um that the chromsomes which embedd some sort of a possible solution has a higher chance of being a solution than a randomly generated chromosome.On mutation and crossbreeding it up its chance of being closer to the answer.I do not know random variable theory or stochastic calculus(if it is ver used in such a thing) to prove it but it's feels pretty intuitive now.

ANd if you think carefully about what the fitness function does probabilistically, because it takes the best fits, the best fits have a higher chance of being crossbred and mutated into a better solution than an unfit solution.

As for a random search based searching , it really doesn't this narrowing down part to give an answer in most cases as compared to a GA.