The planar self-avoiding-walk displacement exponent
Let $\mathcal W_N$ be the set of $N$-step self-avoiding walks on $\mathbb Z^2$ from the origin, that is, nearest-neighbour paths $\omega : \{0, \dots, N\} \to \mathbb Z^2$ with $\omega(0) = 0$ and $\omega(i) \ne \omega(j)$ for $i \ne j$, and let $$R_N^2 = \frac{1}{|\mathcal W_N|} \sum_{\omega \in \mathcal W_N} |\omega(N)|^2$$ be the mean-square end-to-end distance under the uniform measure, with $|\cdot|$ the Euclidean norm. Nienhuis's Coulomb-gas computation predicts $R_N^2 = N^{3/2 + o(1)}$, that is, displacement exponent $\nu = 3/4$. Duminil-Copin and Hammond proved that for every $v > 0$ the probability that a uniform walk in $\mathcal W_N$ leaves the ball of radius $vN$ is exponentially small in $N$, so that $R_N^2 / N^2 \to 0$. Madras proved that $R_N^2 \ge c N^{2/3}$ for some constant $c > 0$.
Problem. Is $$\lim_{N \to \infty} \frac{\log R_N^2}{\log N} = \frac32 \, ?$$ Only the exponent is asked for, not a multiplicative asymptotic constant or convergence of the walk to $\mathrm{SLE}_{8/3}$.
Source: After Flory's approximation (1949); Nienhuis (1982), the Coulomb-gas prediction ν = 3/4; stated as a conjecture by Duminil-Copin and Hammond (2013), §1.2.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- PR — Probability
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, critical exponents, self-avoiding walks
References
@article{nienhuis1982,
author = {Nienhuis, Bernard},
title = {Exact critical point and critical exponents of {$O(n)$} models in two dimensions},
journal = {Phys. Rev. Lett.},
volume = {49},
number = {15},
year = {1982},
pages = {1062--1065},
doi = {10.1103/PhysRevLett.49.1062},
}
@article{duminilcopin2013,
author = {Duminil-Copin, Hugo and Hammond, Alan},
title = {Self-avoiding walk is sub-ballistic},
journal = {Comm. Math. Phys.},
volume = {324},
number = {2},
year = {2013},
pages = {401--423},
doi = {10.1007/s00220-013-1811-1},
eprint = {1205.0401},
archivePrefix = {arXiv},
}
@article{madras2014,
author = {Madras, Neal},
title = {A lower bound for the end-to-end distance of the self-avoiding walk},
journal = {Canad. Math. Bull.},
volume = {57},
number = {1},
year = {2014},
pages = {113--118},
doi = {10.4153/CMB-2012-022-6},
}
@misc{aim-planar-self-avoiding-walk-exponent,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{The planar self-avoiding-walk displacement exponent}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/091-planar-self-avoiding-walk-exponent.md},
note = {Entry 091 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00295, "The planar self-avoiding-walk displacement exponent", clopen.solutions.