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

TL;DR

This paper analytically studies the jamming transition in high-dimensional models using TAP equations and an effective potential, revealing a dominant logarithmic contribution near jamming and analyzing fluctuation spectra.

Contribution

It derives a parallel framework for TAP equations and effective potential in high-dimensional models, highlighting the role of a logarithmic term near jamming and extending applicability to other models.

Findings

Power expansion of the potential effectively approaches the jamming line.

Near jamming, the effective potential exhibits a dominant logarithmic contribution.

The fluctuation spectrum shows a quadratic behavior below a cutoff frequency, with a non-trivial exponent above.

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.