The jamming transition in high dimension: an analytical study of the TAP equations and the effective thermodynamic potential

Abstract

We present a parallel derivation of the Thouless-Anderson-Palmer (TAP) equations and of an effective potential for the negative perceptron and soft sphere models in high dimension. Both models are continuous constrained satisfaction problems with a critical jamming transition characterized by the same exponents. Our analysis reveals that a power expansion of the potential up to the second order represents a successful framework to approach the jamming line from the SAT phase (the region of the phase diagram where at least one configuration verifies all the constraints), where the ground-state energy is zero. An interesting outcome is that close to jamming the effective thermodynamic potential has a logarithmic contribution, which turns out to be dominant in a proper scaling regime. Our approach is quite general and can be directly applied to other interesting models. Finally, we study the spectrum of small harmonic fluctuations in the SAT phase recovering the typical scaling $D(\omega )\sim {\omega }^2$ below the cutoff frequency but a different behavior characterized by a non-trivial exponent above it.

11 pages; a few typos corrected

References in corpus (5)

• The Jamming Transition in Granular Systems
• Fractal free energy landscapes in structural glasses
• Excess Vibrational Modes and the Boson Peak in Model Glasses
• Exact theory of dense amorphous hard spheres in high dimension. I. The free energy
• The Gardner transition in finite dimensions

Cited by in corpus (5)

• Higher order corrections to the effective potential close to the jamming transition in the perceptron model
• Introduction to the dynamics of disordered systems: equilibrium and gradient descent
• Jamming, relaxation, and memory in a structureless glass former
• Hard-Sphere Jamming through the Lens of Linear Optimization
• Comment on "Explicit Analytical Solution for Random Close Packing in $d=2$ and $d=3$"