Percolation of Zero-Weight Paths and the Nature of the Phase Boundary in Random-Bond Ising Systems

The shape of the ferromagnetic--paramagnetic phase boundary in random-bond spin systems has been a subject of a long debate; this question remains relevant to a wide range of problems, from magnetic materials to error-correcting codes. In this talk we focus on the two-dimensional random-bond Ising model, where the shape of the boundary at low temperatures remains controversial: theoretical arguments favor a vertical line below the Nishimori point, whereas numerical studies suggest reentrance. We explore this question from a geometrical standpoint, focusing on a recently introduced new percolation problem, the so-called zero-weight (or negative-weight) percolation. On a lattice with randomly assigned +1 and -1 bonds, where antiferromagnetic bonds occur with probability p, the onset of zero-weight percolation corresponds to the emergence of a percolating path containing equal numbers of +1 and -1 bonds. It has been conjectured that the onset of such percolation signifies the loss of FM order at zero temperature. We analyze the implications of this geometric transition for the ferromagnetic transition at all temperatures.