  Consider a graph in which each site is endowed with a value called
\emph{fitness}. A path in the graph is said to be "open" or "accessible" if the
fitness values along that path is strictly increasing. We say that there is
accessibility percolation between two sites when such a path between them
exists. Motivated by the so called House-of-Cards model from evolutionary
biology, we consider this question on the $L$-hypercube $\{0,1\}^L$ where the
fitness values are independent random variables. We show that, in the large $L$
limit, the probability that an accessible path exists from an arbitrary
starting point to the (random) fittest site is no more than $x^*_{1/2}=
1-\frac12\sinh^{-1}(2) =0.27818\ldots$ and we conjecture that this probability
does converge to $x^*_{1/2}$. More precisely, there is a phase transition on
the value of the fitness $x$ of the starting site: assuming that the fitnesses
are uniform in $[0,1]$, we show that, in the large $L$ limit, there is almost
surely no path to the fittest site if $x>x^*_{1/2}$ and we conjecture that
there are almost surely many paths if $x<x^*_{1/2}$.
  If one conditions on the fittest site to be on the opposite corner of the
starting site rather than being randomly chosen, the picture remains the same
but with the critical point being now $x^*_1= 1-\sinh^{-1}(1)= 0.11863\ldots$.
Along the way, we obtain a large $L$ estimation for the number of self-avoiding
paths joining two opposite corners of the $L$-hypercube.
