Higher-order corrections to the effective potential close to the jamming transition in the perceptron model

In view of the results achieved in a previously related work [A. Altieri, S. Franz, and G. Parisi, J. Stat. Mech. (2016) 093301], regarding a Plefka-like expansion of the free energy up to the second order in the perceptron model, we improve the computation here focusing on the role of third-order corrections. The perceptron model is a simple example of constraint satisfaction problem, falling in the same universality class as hard spheres near jamming and hence allowing us to get exact results in high dimensions for more complex settings. Our method enables to define an effective potential (or Thouless-Anderson-Palmer free energy), namely a coarse-grained functional, which depends on the generalized forces and the effective gaps between particles. The analysis of the third-order corrections to the effective potential reveals that, albeit irrelevant in a mean-field framework in the thermodynamic limit, they might instead play a fundamental role in considering finite-size effects. We also study the typical behavior of generalized forces and we show that two kinds of corrections can occur. The first contribution arises since the system is analyzed at a finite distance from jamming, while the second one is due to finite-size corrections. We nevertheless show that third-order corrections in the perturbative expansion vanish in the jamming limit both for the potential and the generalized forces, in agreement with the isostaticity argument proposed by Wyart and coworkers. Finally, we analyze the relevant scaling solutions emerging close to the jamming line, which define a crossover regime connecting the control parameters of the model to an effective temperature.

Figure 1

Schematic illustration of the perceptron model for $\sigma >0$, with only one constraint. The scalar product between $\vec{x}$ and ${\vec{\xi }}_1$ should be bigger than a positive threshold, hence excluding all the vectors in the region below the dashed blue cone. The resulting space of solutions is convex. In the presence of more constraints the space of allowed configurations would reduce but always preserving its convexity. By contrast, for $\sigma <0$ a completely different situation occurs given by the intersection of nonconvex domains: to go from one set of allowed solutions to another one should contemplate the possibility to pass through a positive energy region.

Figure 2

Generalized forces as a function of $\sigma -v_{\mu }$ plotted for different values of the overlap $q$. In the jamming limit, as $q\to 1$, the function approaches the vertical axis (green line), in agreement with the expected divergence of the forces.

Figure 3

Scaling function showing the generalized forces as a function of the gaps, in linear and log-log scales, well fitted via a power law $a+b{\left(\frac{h_{\mu }}{\sqrt{1-q}}\right)}^{-c}$ in the small-gap regime.