  Viewing Transformers as interacting particle systems, we describe the
geometry of learned representations when the weights are not time dependent. We
show that particles, representing tokens, tend to cluster toward particular
limiting objects as time tends to infinity. Cluster locations are determined by
the initial tokens, confirming context-awareness of representations learned by
Transformers. Using techniques from dynamical systems and partial differential
equations, we show that the type of limiting object that emerges depends on the
spectrum of the value matrix. Additionally, in the one-dimensional case we
prove that the self-attention matrix converges to a low-rank Boolean matrix.
The combination of these results mathematically confirms the empirical
observation made by Vaswani et al. [VSP'17] that leaders appear in a sequence
of tokens when processed by Transformers.
