The Parisi Formula has a Unique Minimizer

Abstract

In 1979, G. Parisi [14] predicted a variational formula for the thermodynamic limit of the free energy in the Sherrington-Kirkpatrick model and described the role played by its minimizer. This formula was verified in the seminal work of Talagrand [19] and later generalized to the mixed p-spin models by Panchenko [12]. In this paper, we prove that the minimizer in Parisi's formula is unique at any temperature and external field by establishing the strict convexity of the Parisi

Introduction to the SK model

“…As we mentioned above, the fact that the Parisi formula in (10) gives an upper bound on the free energy was proved in a breakthrough work of Guerra, [45]. This was the starting point of the proof of the Parisi formula by Talagrand [90].…”

“…First of all, this means that the functional order parameter ζ that achieves the infimum in the Parisi formula (10), which is called the Parisi measure, is not concentrated on one point. The fact that the minimizer is unique follows from the strict convexity of the functional with respect to ζ, which was conjectured in [66] (where a partial result was proved) and recently proved by Auffinger and Chen in [10].…”

“…Talagrand's description of the high temperature region can now also be obtained directly from the recent result of Auffinger and Chen in [10]. Since they showed that the functional in the Parisi formula is strictly convex in ζ, in order to decide whether the minimum over measures concentrated on one point (say, achieved on ζ 0 ) is the global minimum over all ζ, we only need to check that the derivative of the functional at ζ 0 in the direction of any other measure concentrated on one point is non-negative.…”

“…The limit of the free energy is given by the following variational formula discovered by Parisi, (10) lim (1) The parameter ζ is called the functional order parameter, and it has an important physical meaning that will be discussed below. (2) The entire functional being minimized over ζ is known to be Lipschitz with respect to the L 1 -distance [0,1] |ζ 1 (x) − ζ 2 (x)| dx and, therefore, we can minimize over step functions ζ with finitely many steps.…”

“…We already mentioned the papers of Auffinger, Chen [9,10], where a number of interesting properties of the Parisi measure were obtained. However, most of the conjectures of the physicists (discovered numerically) are still open, and we refer to [9] for an overview.…”