# Spin Glass Transitions in Random Bond Ising Model (RBIM)

In brief, is there a list of spin glass transition properties for the RBIM on different lattices? Is there any know results about the relationships between these probabilities for a graph and its dual?

In more detail:

I'm interested in RBIMs on a multitude of lattices. Specifically, I'm interested in those models in which the magnitude of the coupling on all links is some value $J$, but the sign is +1 with probability $p$ and $-1$ with $(1-p)$.

On any lattice there is a transition at $T=0$ from the ferromagnetic phase (at and around $p=0$) to a spin glass phase, that occurs at some critical probability $p_c$. The value of $p_c$ depends on the lattice. For the square it's around 11%.

I can find various resources here and there reporting $p_c$ for various lattices. But they are very hard to find and not exhaustive. Does anyone know of a complete list of all such probabilities ever calculated?

Also, for the square and triangular/hexagonal lattices, the $p_c$ for the primal lattice and p_c' for the dual satisfy

$$H(p_c) + H(p_c') = 1$$

Where $$H(p) = -p \log_2(p) - (1-p)\log_2(1-p),$$ the binary Shannon entropy, where logs are taken base 2. Is this known to always hold true, or to hold true under certain conditions? Are there known results for which the rhs is > 1?

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