The SK model is infinite step replica symmetry breaking at zero temperature

TL;DR

This paper proves that the Parisi measure for the mixed p-spin model at zero temperature has infinitely many points, confirming that the SK model's order parameter is not a step function and that replica symmetry breaking levels diverge as temperature approaches zero.

Contribution

It establishes that the Parisi measure at zero temperature has infinitely many support points, demonstrating a complex structure of replica symmetry breaking in the SK model.

Findings

Parisi measure at zero temperature has infinitely many points

The number of replica symmetry breaking levels diverges as temperature approaches zero

Confirms non-step function nature of the order parameter at zero temperature

Abstract

We prove that the Parisi measure of the mixed p-spin model at zero temperature has infinitely many points in its support. This establishes Parisi's prediction that the functional order parameter of the Sherrington-Kirkpatrick model is not a step function at zero temperature. As a consequence, we show that the number of levels of broken replica symmetry in the Parisi formula of the free energy diverges as the temperature goes to zero.