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

Abstract

We analyze the perceptron model performing a Plefka-like expansion of the free energy. This model falls in the same universality class as hard spheres near jamming, allowing to get exact predictions in high dimensions for more complex systems. Our method enables to define an effective potential (or TAP free energy), namely a coarse-grained functional depending on the contact forces and the effective gaps between the particles. The derivation is performed up to the third order, with a particular emphasis on the role of third order corrections to the TAP free energy. These corrections, irrelevant in a mean-field framework in the thermodynamic limit, might instead play a fundamental role when considering finite-size effects. We also study the typical behavior of the 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. In our analysis, third order contributions vanish in the jamming limit, both for the potential and the generalized forces, in agreement with the argument proposed by Wyart and coworkers invoking isostaticity. Finally, we analyze the scalings emerging close to the jamming line, which define a crossover regime connecting the control parameters of the model to an effective temperature.