  In this paper, we study the abundance of self-avoiding paths of a given
length on a supercritical percolation cluster on $\bbZ^d$. More precisely, we
count $Z_N$ the number of self-avoiding paths of length $N$ on the infinite
cluster, starting from the origin (that we condition to be in the cluster). We
are interested in estimating the upper growth rate of $Z_N$, $\limsup_{N\to
\infty} Z_N^{1/N}$, that we call the connective constant of the dilute lattice.
After proving that this connective constant is a.s.\ non-random, we focus on
the two-dimensional case and show that for every percolation parameter $p\in
(1/2,1)$, almost surely, $Z_N$ grows exponentially slower than its expected
value. In other word we prove that $\limsup_{N\to \infty} (Z_N)^{1/N}
<\lim_{N\to \infty} \bbE[Z_N]^{1/N}$ where expectation is taken with respect to
the percolation process. This result can be considered as a first mathematical
attempt to understand the influence of disorder for self-avoiding walk on a
(quenched) dilute lattice. Our method, which combines change of measure and
coarse graining arguments, does not rely on specifics of percolation on
$\bbZ^2$, so that our result can be extended to a large family of two
dimensional models including general self-avoiding walk in random environment.
