From Large Scale Rearrangements to Mode Coupling Phenomenology in Model Glasses

We consider the equilibrium dynamics of Ising spin models with multispin interactions on sparse random graphs (Bethe lattices). Such models undergo a mean-field glass transition upon increasing the graph connectivity or lowering the temperature. Focusing on the low temperature limit, we identify the large scale rearrangements responsible for the dynamical slowing down near the transition. We are able to characterize exactly the critical dynamics by analyzing the statistical properties of such rearrangements. We obtain a precise crossover description of the role of activation at the transition. Our approach can be generalized to a large variety of glassy models on sparse random graphs, ranging from satisfiability to kinetically constrained models.

Figure 1

Schematic view of the phase diagram of the diluted $p$-spin model, with its dynamical and static transition line. See the text for details on the two type of divergences encountered at the dynamical transition. The scaling hypothesis of Eq. (3) is expected to hold in the shaded region.

Figure 2

Top: an optimal ordering for this rearrangement $F$. The numbers indicate the energy along the trajectory, here $b(F)=2$. Bottom: example of the recursive construction for $l=3$, $j=2$. Bubbles stand for generic rooted subtrees. The numbers give the maximal energy in each epoch.

Figure 3

Bottom: integrated law for the distribution of barrier heights. Top: geometrical susceptibility.

Figure 4

The scaling function of Eq. (3) (notice the large corrections to scaling for $\gamma >{\gamma }_{\mathrm{d}}$). Inset: super-Arrhenius behavior at ${\gamma }_d$.