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Dynamics for Spherical Spin Glasses: Disorder Dependent Initial Conditions

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Abstract

We derive the thermodynamic limit of the empirical correlation and response functions in the Langevin dynamics for spherical mixed p-spin disordered mean-field models, starting uniformly within one of the spherical bands on which the Gibbs measure concentrates at low temperature for the pure p-spin models and mixed perturbations of them. We further relate the large time asymptotics of the resulting coupled non-linear integro-differential equations, to the geometric structure of the Gibbs measures (at low temperature), and derive their FDT solution (at high temperature).

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Notes

  1. The conditioning on (1.16) is interpreted in the usual way: the conditional law of \(\mathbf{J}\) has density given, up to normalization, by the restriction of its original density to the appropriate affine subspace, and the conditional law of the independent \(\mathbf{B}\) is identical to the unconditional one.

  2. In the pure case, i.e. having \(\nu (r)=b_m^2 r^m\), one has that \(\partial _{\perp } H_{\mathbf{J}} ({\varvec{\sigma }}) = \frac{m}{\Vert {\varvec{\sigma }}\Vert } H_{\mathbf{J}}({\varvec{\sigma }})\), hence necessarily \({G_\star }=m {E_\star }/q_\star ^{2}\), whereas in the mixed case the vector \(({E_\star },{G_\star })\) can take any value.

  3. Alternatively \(\nabla H_{\mathbf{J}} ({\varvec{\sigma }}) = - {G_\star }{\varvec{\sigma }}\).

  4. It is easy to verify that in the mixed case the matrix in (1.22) is positive definite for any \(q_\star >0\), while in the pure case taking \(G=m E/q_\star ^2\) yields \(b_m^2 \langle {{\mathsf {v}}}_m, (E,G) \rangle = q_\star ^{-2m} E\).

  5. Except for \(\alpha = - q_\star \) equivalently holding whenever \(\nu (\cdot )\) is an even polynomial

  6. Which in the pure case is restricted to \(G=m E/ q_\star ^2\); see Footnote 2.

  7. In the sense that the law of this array, when interpreting \(\nabla _{\mathrm{sp}}^{2}H_{\mathbf{J}}({\varvec{\sigma }})\) as the corresponding upper triangular matrix, is absolutely continuous w.r.t. the Lebesgue measure on \(\mathbb {R}\times \mathbb {R}\times {\mathbb {R}}^{N-1}\times {\mathbb {R}}^{N(N-1)/2}\).

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Authors and Affiliations

  1. Department of Statistics, Stanford University, Stanford, CA, 94305, USA

    Amir Dembo

  2. Department of Mathematics, Stanford University, Stanford, CA, 94305, USA

    Amir Dembo

  3. Courant Institute, New York University, New York, NY, 10012, USA

    Eliran Subag

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  1. Amir Dembo
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  2. Eliran Subag
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Correspondence to Eliran Subag.

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Communicated by Ivan Corwin.

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Research partially supported by BSF Grant 2014019 (A.D. & E.S.), NSF Grants #DMS-1613091, #DMS-1954337 (A.D), and the Simons Foundation (E.S.).

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Dembo, A., Subag, E. Dynamics for Spherical Spin Glasses: Disorder Dependent Initial Conditions. J Stat Phys 181, 465–514 (2020). https://doi.org/10.1007/s10955-020-02587-z

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