  In a previous paper, we defined a version of the percolation triangle
condition that is suitable for the analysis of bond percolation on a finite
connected transitive graph, and showed that this triangle condition implies
that the percolation phase transition has many features in common with the
phase transition on the complete graph. In this paper, we use a new and
simplified approach to the lace expansion to prove quite generally that for
finite graphs that are tori the triangle condition for percolation is implied
by a certain triangle condition for simple random walks on the graph.
  The latter is readily verified for several graphs with vertex set $\{0,1,...,
r-1\}^n$, including the Hamming cube on an alphabet of $r$ letters (the
$n$-cube, for $r=2$), the $n$-dimensional torus with nearest-neighbor bonds and
$n$ sufficiently large, and the $n$-dimensional torus with $n>6$ and
sufficiently spread-out (long range) bonds. The conclusions of our previous
paper thus apply to the percolation phase transition for each of the above
examples.
