Strong disorder versus a shifted pinning threshold
Let $\tau = \{0 = \tau_0 < \tau_1 < \cdots\}$ be the renewal process with independent gaps distributed as $$P(\tau_1 = n) = \frac{n^{-5/4}}{\sum_{m \ge 1} m^{-5/4}} \qquad (n \ge 1),$$ and let $(\omega_n)_{n \ge 1}$ be independent standard Gaussians, independent of $\tau$. For $\beta > 0$ and $h \in \mathbb{R}$ let $$Z_{N,h}^{\beta,\omega} = E_\tau \exp\Bigl(\sum_{n=1}^{N} (\beta\omega_n + h)\,\mathbf{1}_{\{n \in \tau\}}\Bigr), \qquad F(\beta,h) = \lim_{N \to \infty} \frac{1}{N}\, \mathbb{E}_\omega \log Z_{N,h}^{\beta,\omega},$$ and $h_c(\beta) = \inf\{h : F(\beta,h) > 0\}$. The normalized partition function $W_N = Z_{N,-\beta^2/2}^{\beta,\omega}$ is a nonnegative martingale; let $W_\infty$ be its almost-sure limit. By Kolmogorov's zero-one law, $P(W_\infty = 0)$ is $0$ or $1$.
Here $-\beta^2/2$ is the annealed critical point, and $h_c(\beta) \ge -\beta^2/2$. Berger's notes show, adapting an argument of Lacoin, that $P(W_\infty > 0) = 1$ implies $h_c(\beta) = -\beta^2/2$, and that there is $\beta_c \in (0, \infty)$ with $W_\infty > 0$ almost surely for $\beta < \beta_c$ and $W_\infty = 0$ almost surely for $\beta > \beta_c$; the renewal exponent $1/4$ lies in $(0, 1/2)$, where Alexander and Toninelli proved $h_c(\beta) = -\beta^2/2$ for all small enough $\beta > 0$ (2008); Lacoin gave a short martingale proof (2010). Junk and Lacoin proved the corresponding implication for directed polymers in a random environment.
Problem. For every $\beta > 0$, does $P(W_\infty = 0) = 1$ imply $h_c(\beta) > -\beta^2/2$?
Source: Berger (2026), Conjecture 4.9 of the lecture notes Random Interfaces and Pinning Models, for every renewal exponent α ≠ 0 (strong disorder implies very strong disorder). Stated by AIM — Open Applied Problems for Gaussian disorder and the renewal law P(τ₁ = n) ∝ n^{-5/4}, that is α = 1/4.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- PR — Probability
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, disordered systems, pinning models, renewal processes
References
@misc{berger2026pinning,
author = {Berger, Quentin},
title = {Random Interfaces and Pinning Models},
howpublished = {Lecture notes, Sorbonne Universit{\'e}},
year = {2026},
note = {Version of 4 March 2026, Section 4.2, Conjecture 4.9},
url = {https://www.math.univ-paris13.fr/~quentin.berger/documents/Random-Interfaces.pdf},
}
@article{alexander2008,
author = {Alexander, Kenneth S.},
title = {The effect of disorder on polymer depinning transitions},
journal = {Comm. Math. Phys.},
volume = {279},
number = {1},
year = {2008},
pages = {117--146},
doi = {10.1007/s00220-008-0425-5},
}
@article{toninelli2008,
author = {Toninelli, Fabio Lucio},
title = {A replica-coupling approach to disordered pinning models},
journal = {Comm. Math. Phys.},
volume = {280},
number = {2},
year = {2008},
pages = {389--401},
doi = {10.1007/s00220-008-0469-6},
}
@article{lacoin2010,
author = {Lacoin, Hubert},
title = {The martingale approach to disorder irrelevance for pinning models},
journal = {Electron. Commun. Probab.},
volume = {15},
year = {2010},
pages = {418--427},
doi = {10.1214/ECP.v15-1572},
eprint = {1002.4753},
archivePrefix = {arXiv},
}
@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},
}
@article{junk2026coincidence,
author = {Junk, Stefan and Lacoin, Hubert},
title = {Coincidence of critical points for directed polymers for general environments and random walks},
journal = {Orbita Math.},
volume = {3},
number = {1},
year = {2026},
pages = {89--125},
doi = {10.2140/om.2026.3.89},
eprint = {2502.04113},
archivePrefix = {arXiv},
}
@misc{aim-pinning-strong-disorder-critical-shift,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Strong disorder versus a shifted pinning threshold}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/191-pinning-strong-disorder-critical-shift.md},
note = {Entry 191 at commit a153f46, 2026-10-13},
}
Resolution claims
We study renewal pinning with independent standard Gaussian disorder and gap law $K(n)=n^{-5/4}/\zeta(5/4)$. For every inverse temperature $\beta>0$, we prove that almost-sure extinction of the normalized free partition function at the annealed field $-\beta^2/2$ implies a strictly larger quenched critical field. The decreasing renewal hazard allows us to control future intersections through the polymer's age distribution. Under strong disorder, this …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-18 | Open → Solution claimed | The registry's curators | Strong disorder and critical-point shift for a Gaussian pinning model v1 recorded from AIM @ 24edaad, paper/manuscript.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00457, "Strong disorder versus a shifted pinning threshold", clopen.solutions.