TY - JOUR

T1 - Bounding Flows for Spherical Spin Glass Dynamics

AU - Ben Arous, Gérard

AU - Gheissari, Reza

AU - Jagannath, Aukosh

N1 - Publisher Copyright:
© 2019, Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2020/2/1

Y1 - 2020/2/1

N2 - We introduce a new approach to studying spherical spin glass dynamics based on differential inequalities for one-time observables. Using this approach, we obtain an approximate phase diagram for the evolution of the energy H and its gradient under Langevin dynamics for spherical p-spin models. We then derive several consequences of this phase diagram. For example, at any temperature, uniformly over all starting points, the process must reach and remain in an absorbing region of large negative values of H and large (in norm) gradients in order 1 time. Furthermore, if the process starts in a neighborhood of a critical point of H with negative energy, then both the gradient and energy must increase macroscopically under this evolution, even if this critical point is a saddle with index of order N. As a key technical tool, we estimate Sobolev norms of spin glass Hamiltonians, which are of independent interest.

AB - We introduce a new approach to studying spherical spin glass dynamics based on differential inequalities for one-time observables. Using this approach, we obtain an approximate phase diagram for the evolution of the energy H and its gradient under Langevin dynamics for spherical p-spin models. We then derive several consequences of this phase diagram. For example, at any temperature, uniformly over all starting points, the process must reach and remain in an absorbing region of large negative values of H and large (in norm) gradients in order 1 time. Furthermore, if the process starts in a neighborhood of a critical point of H with negative energy, then both the gradient and energy must increase macroscopically under this evolution, even if this critical point is a saddle with index of order N. As a key technical tool, we estimate Sobolev norms of spin glass Hamiltonians, which are of independent interest.

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U2 - 10.1007/s00220-019-03649-4

DO - 10.1007/s00220-019-03649-4

M3 - Article

AN - SCOPUS:85076305977

SN - 0010-3616

VL - 373

SP - 1011

EP - 1048

JO - Communications In Mathematical Physics

JF - Communications In Mathematical Physics

IS - 3

ER -