Sequential Dynamics in Ising Spin Glasses
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866911001402671104 |
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| author | Dandi, Yatin Gamarnik, David Pernice, Francisco Zdeborová, Lenka |
| author_facet | Dandi, Yatin Gamarnik, David Pernice, Francisco Zdeborová, Lenka |
| contents | We present the first exact asymptotic characterization of sequential dynamics for a broad class of local update algorithms on the Sherrington-Kirkpatrick (SK) model with Ising spins. Focusing on dynamics implemented via systematic scan -- encompassing Glauber updates at any temperature -- we analyze the regime where the number of spin updates scales linearly with system size. Our main result provides a description of the spin-field trajectories as the unique solution to a system of integro-difference equations derived via Dynamical Mean Field Theory (DMFT) applied to a novel block approximation. This framework captures the time evolution of macroscopic observables such as energy and overlap, and is numerically tractable. Our equations serve as a discrete-spin sequential-update analogue of the celebrated Cugliandolo-Kurchan equations for spherical spin glasses, resolving a long-standing gap in the theory of Ising spin glass dynamics. Beyond their intrinsic theoretical interest, our results establish a foundation for analyzing a wide variety of asynchronous dynamics on the hypercube and offer new avenues for studying algorithmic limitations of local heuristics in disordered systems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_09877 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sequential Dynamics in Ising Spin Glasses Dandi, Yatin Gamarnik, David Pernice, Francisco Zdeborová, Lenka Disordered Systems and Neural Networks Mathematical Physics Probability We present the first exact asymptotic characterization of sequential dynamics for a broad class of local update algorithms on the Sherrington-Kirkpatrick (SK) model with Ising spins. Focusing on dynamics implemented via systematic scan -- encompassing Glauber updates at any temperature -- we analyze the regime where the number of spin updates scales linearly with system size. Our main result provides a description of the spin-field trajectories as the unique solution to a system of integro-difference equations derived via Dynamical Mean Field Theory (DMFT) applied to a novel block approximation. This framework captures the time evolution of macroscopic observables such as energy and overlap, and is numerically tractable. Our equations serve as a discrete-spin sequential-update analogue of the celebrated Cugliandolo-Kurchan equations for spherical spin glasses, resolving a long-standing gap in the theory of Ising spin glass dynamics. Beyond their intrinsic theoretical interest, our results establish a foundation for analyzing a wide variety of asynchronous dynamics on the hypercube and offer new avenues for studying algorithmic limitations of local heuristics in disordered systems. |
| title | Sequential Dynamics in Ising Spin Glasses |
| topic | Disordered Systems and Neural Networks Mathematical Physics Probability |
| url | https://arxiv.org/abs/2506.09877 |