A continuous strictly decreasing moment spectrum for Gaussian polymers
Let $(S_n)$ be simple random walk on $\mathbb{Z}^3$ started at $0$, and let $\omega_{n,x}$, $n \ge 1$, $x \in \mathbb{Z}^3$, be independent standard Gaussians independent of the walk. For $\beta > 0$ let $$W_N^\beta = E_S \exp\Bigl(\sum_{n=1}^{N} \bigl(\beta\omega_{n,S_n} - \beta^2/2\bigr)\Bigr), \qquad p^*(\beta) = \sup\Bigl\{p \ge 1 : \sup_N \mathbb{E}_\omega\bigl[(W_N^\beta)^p\bigr] < \infty\Bigr\}.$$ The martingale $(W_N^\beta)_N$ is nonnegative; let $W_\infty^\beta$ be its limit and $\beta_c = \sup\{\beta : P(W_\infty^\beta > 0) = 1\}$.
For the directed polymer based on simple random walk on $\mathbb{Z}^d$, $d \ge 3$, in an i.i.d. environment bounded from above (with $\beta^2/2$ replaced by $\log \mathbb{E} e^{\beta\omega}$), Junk and Lacoin proved that, when $\beta_c < \infty$, $W_\infty^{\beta_c} > 0$ almost surely and that $p^*(\beta)$ tends to $p^*(\beta_c) = 1 + 2/d$ as $\beta \uparrow \beta_c$ (2026). For $d = 3$, $1 + 2/d = 5/3$.
Problem. Is $\beta \mapsto p^*(\beta)$ a continuous strictly decreasing bijection from $(0, \beta_c]$ onto $[5/3, \infty)$, continuity at $\beta_c$ meaning continuity from the left?
Source: Berger (2026), Conjecture 3.32 of the lecture notes Random polymers in random environment, for the directed polymer based on simple random walk in dimension d ≥ 3, onto [1 + 2/d, ∞). Stated by AIM — Open Applied Problems for d = 3 and Gaussian disorder.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- PR — Probability
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, directed polymers, disordered systems, martingales
References
@misc{berger2026polymers,
author = {Berger, Quentin},
title = {Random polymers in random environment: an introduction to disordered systems},
howpublished = {Lecture notes},
year = {2026},
note = {Version of 16 April 2026, Section 3.3.2, Conjecture 3.32},
url = {https://www.math.univ-paris13.fr/~quentin.berger/documents/Disordered-Polymers.pdf},
}
@article{junk2026strong,
author = {Junk, Stefan and Lacoin, Hubert},
title = {Strong disorder and very strong disorder are equivalent for directed polymers},
journal = {Ann. Sci. {\'E}c. Norm. Sup{\'e}r.},
year = {2026},
doi = {10.24033/asens.1849},
eprint = {2402.02562},
archivePrefix = {arXiv},
}
@misc{aim-directed-polymer-moment-spectrum,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{A continuous strictly decreasing moment spectrum for Gaussian polymers}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/193-directed-polymer-moment-spectrum.md},
note = {Entry 193 at commit a153f46, 2026-10-13},
}
Resolution claims
For the directed polymer based on simple random walk in $\Z^3$ with iid standard Gaussian disorder, let $p^*(\beta)$ be the supremum of the orders for which the normalized partition function has moments bounded uniformly in the polymer length. We prove that $\beta\mapsto p^*(\beta)$ is a continuous strictly decreasing bijection from $(0,\beta_c]$ onto $[5/3,\infty)$, where $\beta_c$ is the weak-disorder critical point. …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-18 | Open → Solution claimed | The registry's curators | Moment exponents for Gaussian directed polymers in three dimensions v1 recorded from AIM @ 24edaad, 193-directed-polymer-moment-spectrum/paper.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00424, "A continuous strictly decreasing moment spectrum for Gaussian polymers", clopen.solutions.