# Clustered Attractor Manifolds and Dynamical Condensation in Self-Attention

arXiv: https://arxiv.org/abs/2608.08922 · HTML: https://mlsanity.com/paper/2608.08922/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 27 references that are in the map.
- First posted on arXiv 9 Aug 2026.

## Builds on (most cited first, 21 of 21)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [Learning Transferable Visual Models From Natural Language Supervision](https://mlsanity.com/paper/2103.00020/) (2021)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [Efficient Streaming Language Models with Attention Sinks](https://mlsanity.com/paper/2309.17453/) (2023)
- [Transformers are RNNs: Fast Autoregressive Transformers with Linear Attention](https://mlsanity.com/paper/2006.16236/) (2020)
- [Rethinking Attention with Performers](https://mlsanity.com/paper/2009.14794/) (2020)
- [Deep Equilibrium Models](https://mlsanity.com/paper/1909.01377/) (2019)
- [Attention is Not All You Need: Pure Attention Loses Rank Doubly Exponentially with Depth](https://mlsanity.com/paper/2103.03404/) (2021)
- [A mathematical perspective on Transformers](https://mlsanity.com/paper/2312.10794/) (2023)
- [Stabilizing Transformer Training by Preventing Attention Entropy Collapse](https://mlsanity.com/paper/2303.06296/) (2023)
- [Signal Propagation in Transformers: Theoretical Perspectives and the Role of Rank Collapse](https://mlsanity.com/paper/2206.03126/) (2022)
- [The emergence of clusters in self-attention dynamics](https://mlsanity.com/paper/2305.05465/) (2023)
- [Understanding and Improving Transformer From a Multi-Particle Dynamic System Point of View](https://mlsanity.com/paper/1906.02762/) (2019)
- [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)
- [Clustering in Causal Attention Masking](https://mlsanity.com/paper/2411.04990/) (2024)
- [The Mean-Field Dynamics of Transformers](https://mlsanity.com/paper/2512.01868/) (2025)
- [A multiscale analysis of mean-field transformers in the moderate interaction regime](https://mlsanity.com/paper/2509.25040/) (2025)
- [Two failure modes of deep transformers and how to avoid them: a unified theory of signal propagation at initialisation](https://mlsanity.com/paper/2505.24333/) (2025)
- [Multistability of Self-Attention Dynamics in Transformers](https://mlsanity.com/paper/2511.11553/) (2025)

## Built on by (most cited first, 1 of 1)

- [Nonequilibrium Phases of Repulsive Self-Attention: Chaos, Attention Condensation, and Emergent Locality](https://mlsanity.com/paper/2609.28448/) (2026)

## Abstract (arXiv)

Transformer layers generate state-dependent interaction networks: token representations determine the attention matrix, which in turn updates the representations. We study this feedback in a minimal normalized self-attention dynamics and identify the overlap gap as the central quantity governing its attractor structure in the thermodynamic limit. When tokens form internally aligned clusters and their similarity to members of the same cluster exceeds that to every other cluster by a nonvanishing amount, inter-cluster attention is exponentially suppressed as the dimension increases. This mechanism produces a high-dimensional manifold of clustered fixed points, ranging from a few macroscopic clusters to extensive microscopic fragmentation, and also controls their stability against perturbations. Starting from an unstructured Gaussian state, we find that clustered states nucleate from the diffuse background only above a finite threshold in attention sharpness, giving rise to a dynamical attention-condensation transition.