# Study of the upper-critical dimension of the East model through the breakdown of the Stokes-Einstein relation

## TL;DR

This study explores how dynamical fluctuations and the breakdown of the Stokes-Einstein relation in supercooled liquids depend on dimensionality, suggesting that the East model exhibits non-mean-field behavior up to very high dimensions, possibly infinite.

## Contribution

It provides evidence that the East model's upper critical dimension is at least above 10, indicating hierarchical dynamics persist in all finite dimensions.

## Key findings

- Decoupling indicates non mean-field behavior in the East model.
- The upper critical dimension of the East model may be infinite.
- Hierarchical dynamics exist in the East model across all finite dimensions.

## Abstract

We investigate the dimensional dependence of dynamical fluctuations related to dynamic heterogeneity in supercooled liquid systems using kinetically constrained models. The $d$-dimensional spin-facilitated East model with embedded probe particles is used as a representative super-Arrhenius glass forming system. We investigate the existence of an upper critical dimension in this model by considering decoupling of transport rates through an effective fractional Stokes-Einstein relation, $D\sim{\tau}^{-1+\omega}$, with $D$ and $\tau$ the diffusion constant of the probe particle and the relaxation time of the model liquid, respectively, and where $\omega > 0$ encodes the breakdown of the standard Stokes-Einstein relation. To the extent that decoupling indicates non mean-field behavior, our simulations suggest that the East model has an upper critical dimension which is at least above $d=10$, and argue that it may be actually be infinite. This result is due to the existence of hierarchical dynamics in the East model in any finite dimension. We discuss the relevance of these results for studies of decoupling in high dimensional atomistic models.