Quantifying Concentration Phenomena of Mean-Field Transformers in the Low-Temperature Regime
Abstract
Transformers with self-attention modules as their core components have become an integral architecture in modern large language and foundation models. In this paper, we study the evolution of tokens in deep encoder-only transformers at inference time which is described in the large-token limit by a mean-field continuity equation. Leveraging ideas from the convergence analysis of interacting multi-particle systems, with particles corresponding to tokens, we prove that the token distribution rapidly concentrates onto the push-forward of the initial distribution under a projection map induced by the key, query, and value matrices, and remains metastable for moderate times. Specifically, we show that the Wasserstein distance of the two distributions scales like √(log(β+1)/β)exp(Ct)+exp(−ct) in terms of the temperature parameter β−1→ 0 and inference time t≥ 0. For the proof, we establish Lyapunov-type estimates for the zero-temperature equation, identify its limit as t→∞, and employ a stability estimate in Wasserstein space together with a quantitative Laplace principle to couple the two equations. Our result implies that for time scales of order logβ, which correspond to the early-phase behavior most relevant for practical finite-depth models, the token distribution concentrates at the identified limiting distribution. Numerical experiments confirm this and, beyond that, complement our theory by showing that for finite β and large t the dynamics enter a different terminal phase, dominated by the spectrum of the value matrix.
Related
- Concentration phenomena and the Vanishing Mass Conjecture
- A large deviation principle for the Gaussian beta ensemble in the high temperature regime
- The fractional Hénon-Lane-Emden conjecture and some critical nonexistence phenomena
- Bifurcation phenomena in one-phase quasilinear equations with nonstandard growth
- The double descent and Runge phenomena in overparametrized polynomial interpolation
BibTeX
@misc{alcalde2026quantifying,
title = {{Quantifying Concentration Phenomena of Mean-Field Transformers in the Low-Temperature Regime}},
author = {Albert Alcalde and Leon Bungert and Konstantin Riedl and Tim Roith},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2605.10931},
eprint = {2605.10931},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.10931},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Quantifying Concentration Phenomena of Mean-Field Transformers in the Low-Temperature Regime
AU - Albert Alcalde
AU - Leon Bungert
AU - Konstantin Riedl
AU - Tim Roith
PY - 2026
DA - 2026/10/30
PB - arXiv
DO - 10.48550/arxiv.2605.10931
UR - https://arxiv.org/abs/2605.10931
AB - Transformers with self-attention modules as their core components have become an integral architecture in modern large language and foundation models. In this paper, we study the evolution of tokens in deep encoder-only transformers at inference time which is described in the large-token limit by a mean-field continuity equation. Leveraging ideas from the convergence analysis of interacting multi-particle systems, with particles corresponding to tokens, we prove that the token distribution rapidly concentrates onto the push-forward of the initial distribution under a projection map induced by the key, query, and value matrices, and remains metastable for moderate times. Specifically, we show that the Wasserstein distance of the two distributions scales like √(log(β+1)/β)exp(Ct)+exp(−ct) in terms of the temperature parameter β^−1→ 0 and inference time t≥ 0. For the proof, we establish Lyapunov-type estimates for the zero-temperature equation, identify its limit as t→∞, and employ a stability estimate in Wasserstein space together with a quantitative Laplace principle to couple the two equations. Our result implies that for time scales of order logβ, which correspond to the early-phase behavior most relevant for practical finite-depth models, the token distribution concentrates at the identified limiting distribution. Numerical experiments confirm this and, beyond that, complement our theory by showing that for finite β and large t the dynamics enter a different terminal phase, dominated by the spectrum of the value matrix.
ER -