Duminilcopin 2013 self avoiding walk is sub ballistic

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corollary_1_2: States that for d at least 2 the mean-square Euclidean distance of the endpoint of the uniform n-step self-avoiding walk on Z^d, divided by n^2, tends to 0.

corollary_1_3: States that for a domain whose boundary is smooth near the two marked boundary points, the critical-weight self-avoiding walk between their lattice approximations at mesh delta has length at most K/delta with probability tending to 0, for every K > 0.

theorem_1_1: States that for d at least 2 and every v > 0 there is an epsilon > 0 such that, for every n, the uniform n-step self-avoiding walk from the origin in Z^d reaches Euclidean distance at least vn with probability at most exp(-epsilon n).

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Hugo Duminil-Copin, Alan Hammond, Self-avoiding walk is sub-ballistic. Comm. Math. Phys. 324 (2013), no. 2, 401-423. DOI 10.1007/s00220-013-1811-1. arXiv:1205.0401. The copy read for this card is the arXiv preprint arXiv:1205.0401v1; labels below follow it.

Theorem 1.1 (p. 1) proves that self-avoiding walk on $\mathbb Z^d$ for $d\ge2$ is sub-ballistic: for every $v>0$ there is $\varepsilon>0$ such that, for each $n\in\mathbb N$, under the uniform measure $\mathsf P_{\mathrm{SAW}_n}$ on $n$-step self-avoiding walks from the origin, $\mathsf P_{\mathrm{SAW}_n}(\max_{0\le k\le n}\|\gamma_k\|\ge vn)\le e^{-\varepsilon n}$. Corollary 1.2 (p. 1), which the paper calls an immediate consequence, gives $n^{-2}\langle\|\gamma_n\|^2\rangle\to0$ for the mean-square endpoint displacement. Corollary 1.3 (p. 4) applies Theorem 1.1, with a sketched argument, to the critical-weight walk between two boundary points of a domain. The introduction recalls the conjectured exponents $\langle\|\gamma_n\|^2\rangle=n^{2\nu+o(1)}$ with $\nu=1$ for $d=1$, $3/4$ for $d=2$, about $0.59$ for $d=3$, and $1/2$ for $d=4$ (with a poly-logarithmic correction) and for $d\ge5$, where Hara and Slade proved it with a Brownian scaling limit (p. 2), and poses Questions 1 to 5 as open (pp. 4--5).

Results

Labels and pages are those of arXiv:1205.0401v1.

• Theorem 1.1 (p. 1): sub-ballisticity with an exponential bound, for $d\ge2$ and every $v>0$.
• Corollary 1.2 (p. 1): $\lim_{n\to\infty}n^{-2}\langle\|\gamma_n\|^2\rangle=0$.
• Corollary 1.3 (p. 4): at $z=\mu_c^{-1}$ the walk between the approximations of two boundary points has length at most $K/\delta$ with probability tending to $0$, for every $K>0$; proved in the paper by a sketch.

Read status. Claims checked for the three results above, read clause by clause on the print; the proofs were read for their structure only.

Source: https://arxiv.org/abs/1205.0401. The arXiv record names arXiv's non-exclusive distribution license (arXiv:1205.0401), every other right reserved.

Bears on

• Problem 529: the problem asks whether the expected endpoint distance $d_2(n)$ of the uniform $n$-step self-avoiding walk satisfies $d_2(n)/n^{1/2}\to\infty$, and whether $d_k(n)\ll n^{1/2}$ for $k\ge3$. Theorem 1.1, or Corollary 1.2 with Jensen's inequality, gives $d_k(n)=o(n)$ for every $k\ge2$ (an observation of the result pages). That is an upper bound far weaker than $n^{1/2}$ and no lower bound, so the paper decides neither question.

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