A transfer matrix for the backbone exponent of two-dimensional percolation

Rephrasing the backbone of two-dimensional percolation as a monochromatic path crossing problem, we investigate the latter by a transfer matrix approach. Conformal invariance links the backbone dimension D-b to the highest eigenvalue of the transfer matrix T, and we obtain the result D-b = 1.6431 +/- 0.0006, For a strip of width L, T is roughly of size 2(3L), but we manage to reduce it to similar toL!. We find that the value of D-b is stable with respect to inclusion of additional 'blobs' tangent to the backbone in a finite number of points.