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The jamming transition in high dimension: an analytical study of the TAP equations and the effective thermodynamic potential
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
below the cutoff frequency but a different behavior
characterized by a non-trivial exponent above it.Comment: 11 pages; a few typos correcte
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