One-dimensional disordered Ising models by replica and cavity methods

Using a formalism based on the spectral decomposition of the replicated transfer matrix for disordered Ising models, we obtain several results that apply both to isolated one-dimensional systems and to locally treelike graph and factor graph ($p$-spin) ensembles. We present exact analytical expressions, which can be efficiently approximated numerically for many types of correlation functions and for the average free energies of open and closed finite chains. All the results achieved, with the exception of those involving closed chains, are then rigorously derived without replicas, using a probabilistic approach with the same flavor of cavity method.

Figure 1

Pictorial representation of the matrix $T_n$ (left), its powers $T_n^{\ell }$ (center), and the matrix ${\tilde{T}}_n^{\left(\ell \right)}$ (right).

Figure 2

Top: The leading eigenvalue ${\lambda }_1$ of the sector $D^{\left(1\right)}$ in the RFIM, as a function of the temperature and of the Gaussian external field with variance ${\sigma }_H^2$. Bottom: The corresponding right eigenfunction $g_1\left(u\right)$ at ${\sigma }_H=0.8$. The random fields $h$ and $\tilde{h}$ are distributed as the cavity fields arriving on a chain embedded in a random regular graph with connectivity $z=3$, therefore the transition point is localized at ${\lambda }_1=\frac{1}{2}$.

Figure 3

Top: The leading eigenvalue ${\lambda }_2$ of the sector $D^{\left(2\right)}$ in a $J=\pm 1$ spin glass as a function of the temperature and of the uniform external field $H$. The phase diagram is also shown in the $H$-$T$ plane. Bottom: The corresponding right eigenfunctions $g_2\left(u\right)$ along the orange line of the top picture. The random fields $h$ and $\tilde{h}$ are distributed as the cavity fields arriving on a chain embedded in a random regular graph with connectivity $z=3$, therefore the transition point is localized at ${\lambda }_2=\frac{1}{2}$.