The two-thirds passage-time variance exponent
On the nearest-neighbour graph of $\mathbb{Z}^2$ let the edge weights be independent exponential random variables of mean one, and let $T_n$ be the least total weight of a path from $(0, 0)$ to $(n, 0)$, paths moving in all four lattice directions: undirected first-passage percolation, not a directed last-passage model. Physicists predict $\operatorname{Var}(T_n)$ to grow like $n^{2\chi}$ with fluctuation exponent $\chi = 1/3$ in two dimensions, from the scaling theory of Kardar, Parisi and Zhang and from simulations. Kesten proved $\operatorname{Var}(T_n) \le Cn$, and Benaïm and Rossignol $\operatorname{Var}(T_n) \le Cn / \log n$ for a class of weight distributions that includes the exponential. Newman and Piza, and for exponential weights also Pemantle and Peres, proved $\operatorname{Var}(T_n) \ge c \log n$. Basu, Sidoravicius and Sly proved a bound $O(n^{1 - \varepsilon})$ for a class of rotationally invariant Riemannian first-passage models in the plane.
Problem. Is $$\lim_{n \to \infty} \frac{\log \operatorname{Var}(T_n)}{\log n} = \frac{2}{3}?$$
Source: Auffinger, Damron and Hanson (2017), Question 13 (in d = 2, show that χ = 1/3), after the scaling theory of Kardar, Parisi and Zhang (1986); stated by AIM — Open Applied Problems for exponential weights, with the exponent defined through log Var(Tₙ)/log n.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- PR — Probability
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, first-passage percolation, fluctuation exponents, KPZ universality
References
@book{auffinger2017,
author = {Auffinger, Antonio and Damron, Michael and Hanson, Jack},
title = {50 Years of First-Passage Percolation},
series = {University Lecture Series},
volume = {68},
publisher = {American Mathematical Society},
address = {Providence, RI},
year = {2017},
doi = {10.1090/ulect/068},
eprint = {1511.03262},
archivePrefix = {arXiv},
}
@article{kardar1986,
author = {Kardar, Mehran and Parisi, Giorgio and Zhang, Yi-Cheng},
title = {Dynamic scaling of growing interfaces},
journal = {Phys. Rev. Lett.},
volume = {56},
number = {9},
year = {1986},
pages = {889--892},
doi = {10.1103/PhysRevLett.56.889},
}
@article{kesten1993,
author = {Kesten, Harry},
title = {On the speed of convergence in first-passage percolation},
journal = {Ann. Appl. Probab.},
volume = {3},
number = {2},
year = {1993},
pages = {296--338},
doi = {10.1214/aoap/1177005426},
}
@article{benaim2008,
author = {Bena{\"\i}m, Michel and Rossignol, Rapha{\"e}l},
title = {Exponential concentration for first passage percolation through modified {P}oincar{\'e} inequalities},
journal = {Ann. Inst. Henri Poincar{\'e} Probab. Stat.},
volume = {44},
number = {3},
year = {2008},
pages = {544--573},
doi = {10.1214/07-AIHP124},
eprint = {math/0609730},
archivePrefix = {arXiv},
}
@article{newman1995,
author = {Newman, Charles M. and Piza, Marcelo S. T.},
title = {Divergence of shape fluctuations in two dimensions},
journal = {Ann. Probab.},
volume = {23},
number = {3},
year = {1995},
pages = {977--1005},
doi = {10.1214/aop/1176988171},
}
@incollection{pemantle1994,
author = {Pemantle, Robin and Peres, Yuval},
title = {Planar first-passage percolation times are not tight},
booktitle = {Probability and Phase Transition},
editor = {Grimmett, Geoffrey},
series = {NATO ASI Series C: Mathematical and Physical Sciences},
volume = {420},
publisher = {Kluwer},
address = {Dordrecht},
year = {1994},
pages = {261--264},
doi = {10.1007/978-94-015-8326-8_16},
eprint = {math/0404050},
archivePrefix = {arXiv},
}
@misc{basu2026,
author = {Basu, Riddhipratim and Sidoravicius, Vladas and Sly, Allan},
title = {Rotationally invariant first passage percolation: breaking the $n/\log n$ variance barrier},
year = {2026},
eprint = {2604.01214},
archivePrefix = {arXiv},
}
@misc{aim-fpp-kpz-variance,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{The two-thirds passage-time variance exponent}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/170-fpp-kpz-variance.md},
note = {Entry 170 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-00433, "The two-thirds passage-time variance exponent", clopen.solutions.