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We present a parallel derivation of the Thouless-Anderson-Palmer (TAP)equations and of an effective potential for the negative perceptron and softsphere models in high dimension. Both models are continuous constrainedsatisfaction problems with a critical jamming transition characterized by thesame exponents. Our analysis reveals that a power expansion of the potential upto the second order represents a successful framework to approach the jammingline from the SAT phase (the region of the phase diagram where at least oneconfiguration verifies all the constraints), where the ground-state energy iszero. An interesting outcome is that close to jamming the effectivethermodynamic potential has a logarithmic contribution, which turns out to bedominant in a proper scaling regime. Our approach is quite general and can bedirectly applied to other interesting models. Finally, we study the spectrum ofsmall harmonic fluctuations in the SAT phase recovering the typical scaling$D(\omega) \sim \omega^2$ below the cutoff frequency but a different behaviorcharacterized by a non-trivial exponent above it.

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