Abstract :
In disordered anisotropic square +/− J Ising models SQ(p, q) at groundstates we investigate the pairs (pc, qc) of critical concentrations of antiferromagnetic bonds with concentrations p,q, respectively in orthogonal coordinate directions. We are led to pc(q) ≈ π(q) with π(q) from the so-called adjoined problem. This approach is well supported by simulations for different values of q on the basis of minimal matchings of frustrated plaquettes. In particular, pc(0) ≈ 0.21 from simulations and π(0) = 0.2113248 …, with the conjecture that pc(0) = π(0). The concept of the adjoined problem is extended to d-dimensional (hyper-) cubic lattices. We hereby obtain for pc,d especially in the sotropic case: pc,3 ≈ 0.154, pc,4 ≈ 0.170, pc,5 ≈ 0.178, pc,6 ≈ 0.182. Moreover, in analogy to SQ(p,q) we used the approach also for honeycomb Ising models HC(p,q,r) with no antiferromagnetic bonds in the third plaquette direction (r = 0).
