# Stochastic Scaling Limits and Synchronization by Noise in Deep Transformer Models

arXiv: https://arxiv.org/abs/2604.26898 · HTML: https://mlsanity.com/paper/2604.26898/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Apr 2026.

## Built on by (most cited first, 7 of 7)

- [Quantifying Concentration Phenomena of Mean-Field Transformers in the Low-Temperature Regime](https://mlsanity.com/paper/2605.10931/) (2026)
- [On the Diverse Dynamical Behaviors Arising in Deep Linear Transformers](https://mlsanity.com/paper/2607.18584/) (2026)
- [Training-Induced Escape from Token Clustering in a Mean-Field Formulation of Transformers](https://mlsanity.com/paper/2605.07772/) (2026)
- [L2 over Wasserstein: Statistical Analysis for Optimal Transport](https://mlsanity.com/paper/2605.21365/) (2026)
- [Quantitative Gaussian-Process limits of Tensor Programs](https://mlsanity.com/paper/2607.06290/) (2026)
- [Uniform Scaling Limits in AdamW-Trained Transformers](https://mlsanity.com/paper/2605.11059/) (2026)
- [Random Quadratic Form with random forcing: Metastable synchronization by noise](https://mlsanity.com/paper/2608.16664/) (2026)

## Abstract (arXiv)

We prove pathwise convergence of the layerwise evolution of tokens in a finite-depth, finite-width transformer model with MultiLayer Perceptron (MLP) blocks to a continuous-time stochastic interacting particle system. We also identify the stochastic partial differential equation describing the evolution of the tokens' distribution in this limit and prove propagation of chaos when the number of such tokens is large. The bounds we establish are quantitative and the limits we consider commute. We further prove that the limiting stochastic model displays synchronization by noise and establish exponential dissipation of the interaction energy on average, provided that the common noise is sufficiently coercive relative to the deterministic self-attention drift. We finally characterize the activation functions satisfying the former condition.