Tracy–Widom fluctuations for a Gaussian lattice polymer at fixed temperature
Let $(\omega_{ij})_{i, j \ge 0}$ be independent standard Gaussian variables, let $\Pi_n$ be the set of up-right lattice paths from $(0, 0)$ to $(n, n)$, and let $$Z_n = \sum_{\pi \in \Pi_n} \exp\Bigl(\sum_{v \in \pi} \omega_v\Bigr),$$ each visited vertex counted once: the point-to-point partition function of the directed polymer with Gaussian disorder at inverse temperature one, which stays fixed as $n$ grows. The limit $f = \lim_{n \to \infty} n^{-1}\mathbb{E}\log Z_n$ exists (Carmona and Hu). Let $F_2$ be the GUE Tracy–Widom distribution, the limit law of $N^{2/3}(\lambda_{\max}(M_N) - 2)$ for $N \times N$ complex Hermitian random matrices $M_N$ with density proportional to $\exp(-N \operatorname{Tr} M_N^2 / 2)$.
At zero temperature, for point-to-point last-passage percolation with geometric weights, Johansson proved fluctuations of order $n^{1/3}$ with limit law $F_2$. At positive temperature, GUE Tracy–Widom fluctuations of the free energy are proved for Seppäläinen's log-gamma polymer, by Borodin, Corwin and Remenik at low enough temperature and by Krishnan and Quastel at every temperature. In the intermediate disorder regime, with inverse temperature $n^{-\alpha}$ tending to zero, they were proved by Krishnan and Quastel for disorder matching enough moments of the log-gamma weights, and for general classes of disorder by Ransford ($1/5 < \alpha < 1/4$) and Agarwal ($2/17 < \alpha < 1/4$).
Problem. Is there $s > 0$ such that $(\log Z_n - nf)/(s\,n^{1/3})$ converges in distribution to $F_2$?
Source: The conjectured Tracy–Widom universality of positive-temperature directed polymer free energies in 1 + 1 dimensions, as stated by Krishnan and Quastel (2018), Introduction, after Johansson's zero-temperature theorem (2000). Stated by AIM — Open Applied Problems for Gaussian disorder at inverse temperature one.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- PR — Probability
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, directed polymers, KPZ universality, Tracy–Widom distribution
References
@article{krishnan2018,
author = {Krishnan, Arjun and Quastel, Jeremy},
title = {Tracy--{W}idom fluctuations for perturbations of the log-gamma polymer in intermediate disorder},
journal = {Ann. Appl. Probab.},
volume = {28},
number = {6},
year = {2018},
pages = {3736--3764},
doi = {10.1214/18-AAP1404},
eprint = {1610.06975},
archivePrefix = {arXiv},
}
@article{johansson2000,
author = {Johansson, Kurt},
title = {Shape fluctuations and random matrices},
journal = {Comm. Math. Phys.},
volume = {209},
number = {2},
year = {2000},
pages = {437--476},
doi = {10.1007/s002200050027},
}
@article{carmona2002,
author = {Carmona, Philippe and Hu, Yueyun},
title = {On the partition function of a directed polymer in a {G}aussian random environment},
journal = {Probab. Theory Related Fields},
volume = {124},
number = {3},
year = {2002},
pages = {431--457},
doi = {10.1007/s004400200213},
}
@article{seppalainen2012,
author = {Sepp{\"a}l{\"a}inen, Timo},
title = {Scaling for a one-dimensional directed polymer with boundary conditions},
journal = {Ann. Probab.},
volume = {40},
number = {1},
year = {2012},
pages = {19--73},
doi = {10.1214/10-AOP617},
}
@article{borodin2013,
author = {Borodin, Alexei and Corwin, Ivan and Remenik, Daniel},
title = {Log-gamma polymer free energy fluctuations via a {F}redholm determinant identity},
journal = {Comm. Math. Phys.},
volume = {324},
number = {1},
year = {2013},
pages = {215--232},
doi = {10.1007/s00220-013-1750-x},
eprint = {1206.4573},
archivePrefix = {arXiv},
}
@article{ransford2025,
author = {Ransford, Julian},
title = {Universality of directed polymers in the intermediate disorder regime},
journal = {Ann. Probab.},
volume = {53},
number = {3},
year = {2025},
pages = {1019--1048},
doi = {10.1214/24-AOP1724},
eprint = {2304.04871},
archivePrefix = {arXiv},
}
@article{agarwal2026,
author = {Agarwal, Pranay},
title = {Improved universality bounds for directed polymers in the intermediate disorder regime},
journal = {Electron. Commun. Probab.},
volume = {31},
year = {2026},
pages = {1--12},
doi = {10.1214/26-ECP752},
eprint = {2509.21453},
archivePrefix = {arXiv},
}
@misc{aim-gaussian-polymer-tracy-widom,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Tracy–Widom fluctuations for a Gaussian lattice polymer at fixed temperature}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/287-gaussian-polymer-tracy-widom.md},
note = {Entry 287 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-00435, "Tracy–Widom fluctuations for a Gaussian lattice polymer at fixed temperature", clopen.solutions.