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Zero-temperature dynamics of Ising systems on hypercubes.

R Chen¹, J Machta², C M Newman³

¹University of California, San Diego, New York University, New York, New York 10012, USA and Department of Mathematics, La Jolla, California 92093, USA.

Physical Review. E

|December 23, 2025

Summary

This summary is machine-generated.

We investigated Ising ferromagnet dynamics on hypercubes, finding final states depend on dimension. Ground, frozen, and blinker states emerge, with blinkers appearing in even dimensions.

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Area of Science:

Statistical physics
Condensed matter physics
Complex systems

Background:

Ising models are fundamental in statistical mechanics.
Glauber dynamics simulate magnetic systems.
Hypercubes offer a scalable network structure for studying emergent phenomena.

Purpose of the Study:

To analyze zero-temperature Glauber dynamics of Ising ferromagnets on hypercubes of varying dimensions.
To explore the asymptotic behavior of magnetization and ground state probabilities as dimension and time increase.
To characterize the different types of final states observed in the system.

Main Methods:

Numerical simulations of Glauber dynamics on hypercubes.
Analysis of final states: ground, frozen, and blinker states.
Utilizing k-core decomposition to study frozen state geometry.

Main Results:

Identified three distinct final states: ground, frozen, and blinker states.
Provided an exponential lower bound for the number of frozen states using k-core analysis.
Determined that blinker states, characterized by flipping spins, exist only in even dimensions and require at least d=8 for specific configurations.
Investigated the influence of initial conditions versus dynamical evolution on the final state.

Conclusions:

The dimension of the hypercube significantly impacts the emergent behavior of Ising ferromagnets under Glauber dynamics.
Frozen states exhibit complex geometric structures related to k-cores.
Blinker states represent a unique dynamic phenomenon in even-dimensional hypercubes.
Further research is needed to fully understand the "nature versus nurture" aspect and explore open problems.