A section of the phase diagram for the mixture (2, 3, 4) on a Bethe lattice with connectivity z = 5. Thick lines represent discontinuous liquid-glass and glass-glass transitions while thin dashed lines correspond to continuous liquid-glass transition. For r = 0.24 there is only one glass phase, while for r = 0.3 and r = 0.34 there are two glass phases. For r = 0.34 the discontinuous glass-glass transition is disconnected from the liquid phase.
Higher-order singularities in the mixture (2, 3, 4) on a Bethe lattice with z = 5. The transition lines corresponding to generic and degenerate cusp bifurcations are denoted with and , respectively. They coalesce smoothly in a degenerate swallowtail singularity. The dashed and dotted curves belong to the unstable branches of the transition lines.
The order parameter, Φ, representing the fraction of frozen spins vs temperature, T, at r = 0.32 and for several values of q in a mixture (2, 3, 4) with z = 5.
A section of the phase diagram for the mixture f = (2, 5, 7) on a Bethe lattice with connectivity z = 12 for r = 0.4, 0.52, 0.6, and 0.68. The full lines represent discontinuous transitions while the dots correspond to higher-order singularities with cusp and swallowtail structures, respectively denoted with and . Notice that for r = 0.68 the glass-glass transition line is disconnected from the liquid phase.
The order parameter, Φ, representing the fraction of frozen spins vs temperature, T, at r = 0.68 and for several values of q, in the (2, 5, 7) mixture with z = 12. The double jump of Φ for q = 0.74, 0.76, and 0.78 corresponds to a discontinuous liquid-glass transition followed by a glass-glass transition upon lowering the temperature.
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