# Some properties of the phase diagram for mixed *p*-spin glasses

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## Abstract

In this paper we study the Parisi variational problem for mixed *p*-spin glasses with Ising spins. Our starting point is a characterization of Parisi measures whose origin lies in the first order optimality conditions for the Parisi functional, which is known to be strictly convex. Using this characterization, we study the phase diagram in the temperature-external field plane. We begin by deriving self-consistency conditions for Parisi measures that generalize those of de Almeida and Thouless to all levels of Replica Symmetry Breaking (RSB) and all models. As a consequence, we conjecture that for all models the Replica Symmetric phase is the region determined by the natural analogue of the de Almeida–Thouless condition. We show that for all models, the complement of this region is in the RSB phase. Furthermore, we show that the conjectured phase boundary is exactly the phase boundary in the plane less a bounded set. In the case of the Sherrington–Kirkpatrick model, we extend this last result to show that this bounded set does not contain the critical point at zero external field.

## Keywords

Parisi formula Sherrington–Kirkpatrick model First order optimality conditions de Almeida–Thouless line## Mathematics Subject Classification

60K35 82B44 82D30 49S05 49K21## Notes

### Acknowledgments

We would like to thank our advisors G. Ben Arous and R.V. Kohn for their support. We would like to thank anonymous referees for their very helpful suggestions regarding the exposition of this paper. We would like to thank the New York University GRI Institute in Paris for its hospitality during the preparation of this paper. This research was conducted while A.J. was supported by a National Science Foundation Graduate Research Fellowship DGE-0813964; and National Science Foundation grants DMS-1209165 and OISE-0730136, and while I.T. was supported by a National Science Foundation Graduate Research Fellowship DGE-0813964; and National Science Foundation grants OISE-0967140 and DMS-1311833.

### Compliance with Ethical Standards

### Conflicts of interest

The authors declare that they have no conflicts of interest.

### Funding

A.J. was supported by a National Science Foundation Graduate Research Fellowship DGE-0813964; and National Science Foundation grants DMS-1209165 and OISE-0730136. I.T. was supported by a National Science Foundation Graduate Research Fellowship DGE-0813964; and National Science Foundation grants OISE-0967140 and DMS-1311833.

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