A Transfer Matrix for the Backbone Exponent of Two-Dimensional Percolation

Jesper-Lykke Jacobsen ¹, Paul Zinn-Justin ¹

Journal of Physics A 35 (2002) 2131-2144

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 \pm 0.0006. For a strip of width L, T is roughly of size 2^{3^L}, but we manage to reduce it to \sim L!. 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.

• 1. Laboratoire de Physique Théorique et Hautes Energies (LPTHE),
  CNRS : UMR7589 – Université Paris VI – Pierre et Marie Curie – Université Paris VII – Paris Diderot