Fragility in [Formula presented]-spin models

We investigate the relation between fragility and phase space properties — such as the distribution of states — in the mean-field [Formula presented]-spin model, a solvable model that has been frequently used in studies of the glass transition. By direct computation of all the relevant quantities, we find that (i) the recently observed correlation between fragility and vibrational properties at low temperature is present in this model and (ii) the total number of states is a decreasing function of fragility, at variance with what is currently believed. We explain these findings by taking into account the contribution to fragility coming from the transition paths between different states. Finally, we propose a geometric picture of the phase space that explains the correlation between properties of the transition paths, distribution of states, and their vibrational properties. However, our analysis may not apply to strong systems where inflection points in the configurational entropy as a function of the temperature are found.