Self-Avoiding Walk is Sub-Ballistic

@article{DuminilCopin2013SelfAvoidingWI,
  title={Self-Avoiding Walk is Sub-Ballistic},
  author={Hugo Duminil-Copin and Alan Hammond},
  journal={Communications in Mathematical Physics},
  year={2013},
  volume={324},
  pages={401-423}
}
We prove that self-avoiding walk on $${\mathbb{Z}^d}$$Zd is sub-ballistic in any dimension d ≥ 2. That is, writing $${\| u \|}$$‖u‖ for the Euclidean norm of $${u \in \mathbb{Z}^d}$$u∈Zd, and $${\mathsf{P_{SAW}}_n}$$PSAWn for the uniform measure on self-avoiding walks $${\gamma : \{0, \ldots, n\} \to \mathbb{Z}^d}$$γ:{0,…,n}→Zd for which γ0 = 0, we show that, for each v > 0, there exists $${\varepsilon > 0}$$ε>0 such that, for each $${n \in \mathbb{N}, \mathsf{P_{SAW}}_n \big( {\rm max}\big… 

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