Glassy states: the free Ising model on a tree
Daniel Gandolfo, Christian Maes, Jean Ruiz, Senya Shlosman
TL;DR
This paper studies the free Ising model on a Cayley tree, revealing it decomposes into uncountably many extremal states below the spin glass temperature, with implications for understanding spin glass behavior.
Contribution
It demonstrates that the free state of the ferromagnetic Ising model on a Cayley tree decomposes into uncountably many extremal states below the spin glass temperature, highlighting complex state structure.
Findings
Uncountably many extremal states in the free state decomposition
Relation of tail observable to the Edwards-Anderson parameter
Free state is not extremal below the spin glass temperature
Abstract
We consider the ferromagnetic Ising model on the Cayley tree and we investigate the decomposition of the free state into extremal states below the spin glass temperature. We show that this decomposition has uncountably many components. The tail observable showing that the free state is not extremal is related to the Edwards-Anderson parameter, measuring the variance of the (random) magnetization obtained from drawing boundary conditions from the free state.
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