# Random Quadratic Form on a Sphere: Synchronization by Common Noise

arXiv: https://arxiv.org/abs/2603.06187 · HTML: https://mlsanity.com/paper/2603.06187/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 47 references that are in the map.
- First posted on arXiv 6 Mar 2026.

## Builds on (most cited first, 18 of 18)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [A mathematical perspective on Transformers](https://mlsanity.com/paper/2312.10794/) (2023)
- [The emergence of clusters in self-attention dynamics](https://mlsanity.com/paper/2305.05465/) (2023)
- [Sinkformers: Transformers with Doubly Stochastic Attention](https://mlsanity.com/paper/2110.11773/) (2021)
- [Dynamic metastability in the self-attention model](https://mlsanity.com/paper/2410.06833/) (2024)
- [Emergence of meta-stable clustering in mean-field transformer models](https://mlsanity.com/paper/2410.23228/) (2024)
- [The Mean-Field Dynamics of Transformers](https://mlsanity.com/paper/2512.01868/) (2025)
- [Quantitative Clustering in Mean-Field Transformer Models](https://mlsanity.com/paper/2504.14697/) (2025)
- [A multiscale analysis of mean-field transformers in the moderate interaction regime](https://mlsanity.com/paper/2509.25040/) (2025)
- [Synchronization on circles and spheres with nonlinear interactions](https://mlsanity.com/paper/2405.18273/) (2024)
- [Clustering in pure-attention hardmax transformers and its role in sentiment analysis](https://mlsanity.com/paper/2407.01602/) (2024)
- [Synchronization of mean-field models on the circle](https://mlsanity.com/paper/2507.22857/) (2025)
- [Perceptrons and localization of attention's mean-field landscape](https://mlsanity.com/paper/2601.21366/) (2026)
- [Clustering in Deep Stochastic Transformers](https://mlsanity.com/paper/2601.21942/) (2026)
- [Attention's forward pass and Frank-Wolfe](https://mlsanity.com/paper/2508.09628/) (2025)
- [On the number of modes of Gaussian kernel density estimators](https://mlsanity.com/paper/2412.09080/) (2024)
- [YuriiFormer: A Suite of Nesterov-Accelerated Transformers](https://mlsanity.com/paper/2601.23236/) (2026)

## Built on by (most cited first, 4 of 4)

- [Homogenized Transformers](https://mlsanity.com/paper/2604.01978/) (2026)
- [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)
- [Random Quadratic Form with random forcing: Metastable synchronization by noise](https://mlsanity.com/paper/2608.16664/) (2026)

## Abstract (arXiv)

We introduce the Random Quadratic Form (RQF): a stochastic differential equation which formally corresponds to the gradient flow of a random quadratic functional on a sphere. While the one-point dynamics of the system is a Brownian motion and thus has no preferred direction, the two-point motion exhibits nontrivial synchronizing behaviour. In this work we study synchronization of the RQF, namely we give both distributional and path-wise characterizations of the solutions by studying invariant measures and random attractors of the system. The RQF model is motivated by the study of the role of linear layers in transformers and illustrates the synchronization by common noise phenomena arising in the simplified models of transformers. In particular, we provide an alternative (independent of self-attention) explanation of the clustering behaviour in deep transformers and show that tokens cluster even in the absence of the self-attention mechanism.