Quentin Berger, Francesca Cottini,
Free energy and phase transition for 2D directed polymers with critical spatial correlations, arXiv:2608.30829 [math.PR]
The directed polymer model is a celebrated model that describes a polymer placed in some heterogeneous medium (or random environment). In this article, we study the directed polymer model in its critical dimension d=2 in a random environment with critical long-range spatial correlations. More precisely, we study the free energy of the model, which is a key physical quantity that encodes some localization properties, and we obtain its sharp high-temperature asymptotics. This answers (and extends) a question raised in a paper by Lacoin in 2010, but more importantly it develops a robust method to estimate the free energy of disordered systems in their critical dimension, which relies only on second moment estimates.

Quentin Berger, Francesco Caravenna, Nicola Turchi,
Strong disorder for Stochastic Heat Flow and 2D Directed Polymers, arXiv:2508.02478 [math.PR]
We study here the directed polymer model in its critical dimension d=2, whose partition functions may be interpreted as a discretized version of the stochastic heat equation with multiplicative noise. Caravenna, Sun and Zygouras showed in 2023 that, in a critical window of parameters, the field of partition functions converges (as a random measure) to a non-trivial, non-Gaussian, limiting measure-valued process called Critical 2D Stochastic Heat Flow (SHF). The main result of the article is to show some local extinction of the SHF in a large time or large disorder intensity limit. In fact, we give a quantitative version of this local extinction, based on novel estimates for directed polymers in the so-called strong disorder regime.

Irene Ayuso Ventura, Quentin Berger,
Non-linear conductances of Galton-Watson trees and application to the (near) critical random cluster model, to appear in Ann. Appl. Probab.,
[arXiv]
We study in this article the non-linear effective resistance (or L^p-resistance) between the root and the leaves of random Bienaymé-Galton-Watson trees. We give the exact asymptotic behavior of this resistance for a tree of height n in some "critical", "quasi-critical" and "sub-critical" regimes of resistances. These results are actually valid in a more general setting of concave recursions on trees, and have applications for instance in the study of connective probabilities of the random cluster model on random trees.

Quentin Berger, Carsten Chong, Hubert Lacoin,
The Stochastic Heat Equation with multiplicative Lévy noise: Existence, moments, and intermittency, Commun. Math. Phys., Vol. 402 (2023), pp. 2215–2299.
[Journal] [arXiv]
This article studies the stochastic heat equation with multiplicative noise of Lévy type, i.e. non-Gaussian and possibly heavy-tailed. It provides some (dimension dependent) integrability conditions on the Lévy measure of the noise for the existence and unicity of a non-trivial solution. These conditions are in fact very close to being necessary and sufficient and largely extend pre-existing results. Additionally, we study the solution in a large time limit and we investigate its so-called intermittency properties: in particular, we show that the solution is intermittent in any dimension, for any disorder intensity and for any Lévy measure! This is in sharp constrast with the case of Gaussian noises, where the solution is known not to be intermittent in dimension d≥3 at small enough disorder intensity.

Quentin Berger,
Notes on Random Walks in the Cauchy domain of attraction, Probab. Theory Relat. Fields, Vol. 175, Issue 1-2 (2019), pp. 1-44.
[Journal] [arXiv]
This article studies random walks on ℤ, in the domain of attraction of some α-stable law, with α∈(0,2), with a special emphasis on the so-called Cauchy case α=1. This case is usually more technical and often left aside: somewhat surprisingly, some basic results where not available in the literature, and this article fills this gap. It shows in particular sharp estimates of large deviation probabilities (including local versions) and several applications (hitting times, renewal theorems) are developed.

Quentin Berger, Niccoló Torri,
Directed polymers in heavy-tail random environment, Ann. Probab., Vol. 47, Num. 6 (2019), pp. 4024-4076.
[Journal] [arXiv]
The directed polymer model is a celebrated model that describes a polymer placed in some heterogeneous medium (or random environment). In certain regimes of temperature, the model exhibits a localization phenomenon: the polymer somehow "stretches" to reach more favorable regions in the environment. However, the precise description of the localized trajectories (super-diffusivity exponents, scaling limits, etc.) remains largely open. In this article, we study the case where the random environment is heavy-tailed and in particular does not have finite second moments. We obtain a remarkably precise description of the scaling limits of the system in all possible temperature regimes, through some energy-entropy variational problem whose well-posedness is non-trivial and treated in a companion article.

Quentin Berger, Hubert Lacoin,
Pinning on a defect line: characterization of marginal disorder relevance and sharp asymptotics for the critical point shift, J. Inst. Math. Jussieu, Vol. 17 (2018), Issue 2, 305-346.
[Journal] [arXiv]
The question of the influence of disorder on physical systems is a central one in physics: the goal is to determine whether some arbitrarily weak disorder modifies the properties of the system and in particular of its phase transition. The disordered pinning model has been studied though this length for over three decades: mathematical results have confirmed for this model the validity of the so-called Harris criterion for the relevance/irrelevance of disorder. In the marginal case, the Harris criterion was subject to contradictory predictions in the physics literature. This article gives a definitive answer for the pinning model and provides a necessary and sufficient condition for disorder relevance.

Articles by anti-chronological order

---

Preprints

---

1. Quentin Berger, Francesco Caravenna, Jeremy Clark, Alexander Dunlap, Yu Gu, Ziyang Liu, Shuta Nakajima, Rongfeng Sun, Li-Cheng Tsai, Zuodi Xie, Nikos Zygouras,
   The critical 2D KPZ process,
   arXiv:2610.31614 [math.PR]
2. Quentin Berger, Francesca Cottini,
   Free energy and phase transition for 2D directed polymers with critical spatial correlations,
   arXiv:2608.30829 [math.PR]

---

3. Quentin Berger, Nicola Turchi, Nikos Zygouras,
   Fractional moments of the Stochastic Heat Flow and 2D Directed Polymers,
   arXiv:2608.13359 [math.PR]

---

4. Quentin Berger, Shuta Nakajima,
   Sharp behavior of the free energy for the two-dimensional directed polymer model,
   arXiv:2605.30707 [math.PR]

---

5. Quentin Berger, Hubert Lacoin,
   The Random Walk Pinning Model I: Lower bounds on the free energy and disorder irrelevance,
   arXiv:2509.08789 [math.PR]

---

6. Quentin Berger, Hubert Lacoin,
   The Random Walk Pinning Model II: Upper bounds on the free energy and disorder relevance,
   arXiv:2509.08769 [math.PR]

---

7. Quentin Berger, Francesco Caravenna, Nicola Turchi,
   Strong disorder for Stochastic Heat Flow and 2D Directed Polymers,
   arXiv:2508.02478 [math.PR]

---

---

To appear

---

8. Quentin Berger, Loïc Béthencourt, Camille Tardif,
   On joint returns to zero of Bessel processes,
   to appear in Stoch. Process. Appl.,
   arXiv:2406.19344 [math.PR]

---

9. Quentin Berger, Alexandre Legrand, Rémy Poudevigne-Auboiron, Christophe Sabot,
   A random polymer approach to the weak disorder phase of the vertex reinforced jump process,
   to appear in Ann. Inst. Henri Poincaré: Probab. Stat.,
   arXiv:2503.10209 [math.PR]

---

10. Irene Ayuso Ventura, Quentin Berger,
    Non-linear conductances of Galton-Watson trees and application to the (near) critical random cluster model,
    to appear in Ann. Appl. Probab.,
    arXiv:2404.11564 [math.PR]

---

11. Quentin Berger, Loïc Béthencourt, Camille Tardif,
    Persistence problems for additive functionals of one-dimensional Markov processes,
    to appear in Ann. Inst. Henri Poincaré: Probab. Stat.,
    arXiv:2304.09034 [math.PR]

---

---

2026

---

12. Quentin Berger, Nicolas Bouchot,
    Some properties of the principal Dirichlet eigenfunction in Lipschitz domains, via probabilistic couplings,
    Electron. J. Probab., Vol. 31, p. 1-37 (2026).
    [Journal] [arXiv]

---

2025

---

13. Quentin Berger, Loïc Béthencourt,
    An application of Sparre Andersen's fluctuation theorem for exchangeable and sign-invariant random variables,
    Séminaire de Probabilités LII (2025), Chapter 9, pp. 367--388
    [Journal] [arXiv]

---

14. Quentin Berger, Céline Bonnet, Lucile Laulin, Kilian Raschel,
    Topics in random walks,
    ESAIM: Proceedings and Surveys, Vol. 80, p. 51 (2025), Modélisation aléatoire et stochastique - Journées MAS 2022.
    [Journal] [PDF]

---

---

2024

---

15. Quentin Berger, Matthias Birkner, Linglong Yuan,
    Collective vs. individual behaviour for sums of i.i.d. random variables: appearance of the one-big-jump phenomenon,
    Ann. Fac. Sci. Toulouse: Math., Série 6, Tome 33 (2024) no. 5, pp. 1413-1485.
    [Journal] [arXiv]

---

16. Irene Ayuso Ventura, Quentin Berger,
    Ising model on a Galton-Watson tree with a sparse random external field,
    Electron. J. Probab., Vol. 29, no 174 (2024) 43pp.
    [Journal] [arXiv]

---

17. Quentin Berger, Alexandre Legrand,
    Scaling limit of the disordered generalized Poland-Scheraga model for DNA denaturation
    Probab. Theory Relat. Fields, Vol. 190 (2024) pp. 179–258.
    [Journal] [arXiv]

---

18. Quentin Berger, Brune Massoulié,
    Wetting on a wall and wetting in a well: Overview of equilibrium properties,
    Stochastic Process. Appl., Vol. 170 (2024), Special issue in tribute to Francis Comets.
    [Journal] [arXiv]

---

---

2023

---

19. Quentin Berger, Carsten Chong, Hubert Lacoin,
    The Stochastic Heat Equation with multiplicative Lévy noise: Existence, moments, and intermittency,
    Commun. Math. Phys., Vol. 402 (2023), pp. 2215–2299.
    [Journal] [arXiv]

---

---

2022

---

20. Quentin Berger, Chien-Hao Huang, Niccoló Torri, Ran Wei,
    One-dimensional polymers in random environments: stretching vs. folding,
    Electron. J. Probab., Vol. 27 (2022), pp. 1-45.
    [Journal] [arXiv]

---

21. Quentin Berger, Hubert Lacoin,
    The continuum directed polymer in Lévy Noise,
    J. Éc. Polytech., Tome 9 (2022), pp. 213-280.
    [Journal] [arXiv]

---

22. Quentin Berger, Niccoló Torri, Ran Wei,
    Non-directed polymers in heavy-tail random environment in dimension d≥2,
    Electron. J. Probab., Vol. 27 (2022), pp. 1-67.
    [Journal] [arXiv]

---

---

2021

---

23. Quentin Berger, Hubert Lacoin,
    The scaling limit of the directed polymer with power-law tail disorder,
    Commun. Math. Phys., Vol. 386 (2021), pp. 1051-1105.
    [Journal] [arXiv]

---

24. Quentin Berger, Niccoló Torri,
    Beyond Hammersley’s Last-Passage Percolation: a discussion on possible local and global constraints,
    Ann. Inst. Henri Poincaré: Comb., Phys. Inter., Vol. 8, Num. 2 (2021), pp. 213-241.
    [Journal] [arXiv]

---

---

2020

---

25. Quentin Berger, Giambattista Giacomin, Maha Khatib,
    Disorder and denaturation transition in the generalized Poland-Scheraga model,
    Ann. Henri Lebesgue, Vol. 3 (2020), pp. 299-339.
    [Journal] [arXiv]

---

26. Quentin Berger, Michele Salvi,
    Scaling limit of sub-ballistic 1D Random Walk among biased conductances: a story of wells and walls,
    Electron. J. Probab., Vol. 25, no 30 (2020), 43 pp.
    [Journal] [arXiv]

---

---

2019

---

27. Quentin Berger,
    Notes on Random Walks in the Cauchy domain of attraction,
    Probab. Theory Relat. Fields, Vol. 175, Issue 1-2 (2019), pp. 1-44.
    [Journal] [arXiv]

---

28. Quentin Berger,
    Strong renewal theorems and local large deviations for multivariate random walks and renewals,
    Electron. J. Probab., Vol. 24, no. 46 (2019), 47 pp.
    [Journal] [arXiv]

---

29. Quentin Berger, Giambattista Giacomin, Hubert Lacoin,
    Disorder and critical phenomena: the α=0 copolymer model,
    Probab. Theory Relat. Fields, Vol. 174, Issue 3-4 (2019), pp. 787–819.
    [Journal] [arXiv]

---

30. Quentin Berger, Michele Salvi,
    Scaling of sub-ballistic 1D Random Walks among biased Random Conductances,
    Markov Processes Relat. Fields, Vol. 25, Issue 1 (2019), pp 171-187.
    [Journal] [arXiv]

---

31. Quentin Berger, Niccoló Torri,
    Directed polymers in heavy-tail random environment,
    Ann. Probab., Vol. 47, Num. 6 (2019), pp. 4024-4076.
    [Journal] [arXiv]

---

32. Quentin Berger, Niccoló Torri,
    Entropy-controlled Last-Passage Percolation,
    Ann. Appl. Probab., Vol. 29, Num. 3 (2019), pp. 1878-1903.
    [Journal] [arXiv]

---

---

2018

---

33. Kenneth S. Alexander, Quentin Berger,
    Geodesics Toward Corners in First Passage Percolation,
    J. Stat. Phys., Vol. 172, Issue 4 (2018), pp 1029-1056.
    [Journal] [arXiv]

---

34. Kenneth S. Alexander, Quentin Berger,
    Pinning of a renewal on a quenched renewal,
    Electron. J. Probab., Vol. 23 (2018), no 6, 48 pp.
    [Journal] [arXiv]

---

35. Quentin Berger, Giambattista Giacomin, Maha Khatib,
    DNA melting structures in the generalized Poland-Scheraga model,
    ALEA, Lat. Am. J. Probab. Math. Stat., Vol. 15 (2018), pp 993-1025.
    [Journal] [arXiv]

---

36. Quentin Berger, Frank den Hollander, Julien Poisat,
    Annealed scaling for a charged polymer in dimensions two and higher,
    J. Phys. A: Math. Theor, Vol. 51, no 5 (2018), special issue in honour of Stuart Whittington’s 75th birthday.
    [Journal] [arXiv]

---

37. Quentin Berger, Hubert Lacoin,
    Pinning on a defect line: characterization of marginal disorder relevance and sharp asymptotics for the critical point shift,
    J. Inst. Math. Jussieu, Vol. 17 (2018), Issue 2, 305-346.
    [Journal] [arXiv]

---

---

2017

---

38. Quentin Berger, Hubert Lacoin,
    The high-temperature behavior for the directed polymer in dimension 1+2,
    Ann. Inst. Henri Poincaré: Probab. Stat., Vol. 53 (2017), no 1, 430–450.
    [Journal] [arXiv]

---

---

2016

---

39. Kenneth S. Alexander, Quentin Berger,
    Local asymptotics for the first intersection of two independent renewals,
    Electron. J. Probab., Vol. 21 (2016), no 68, 1-20.
    [Journal] [arXiv]

---

40. Kenneth S. Alexander, Quentin Berger,
    Local limit theorem and renewal theory with no moments,
    Electron. J. Probab., Vol. 21 (2016), no 66, 1-18.
    [Journal] [arXiv]

---

---

2010–2015

---

41. Quentin Berger, Julien Poisat,
    On the critical curve of the pinning and copolymer models in correlated Gaussian environment,
    Electron. J. Probab., Vol. 20 (2015), no 71, 1-35.
    [Journal] [arXiv]

---

42. Quentin Berger,
    Influence of disorder for polymer pinning models,
    ESAIM: Proceedings and Surveys Vol. 51, p.74 (October 2015), Modélisation aléatoire et stochastique - Journées MAS 2014.
    [Journal] [PDF]

---

43. Quentin Berger,
    Pinning model in random correlated environment: appearance of an infinite disorder regime,
    J. Stat. Phys., Vol. 155, Issue 3 (2014), 544-570.
    [Journal] [arXiv]

---

44. Quentin Berger, Francesco Caravenna, Julien Poisat, Rongfeng Sun, Nikos Zygouras,
    The critical curve of the random pinning and copolymer models at weak coupling,
    Comm. Math. Phys., Vol 326, Issue 2 (2014) 507-530.
    [Journal] [arXiv]

---

45. Quentin Berger,
    Comments on the influence of disorder for pinning model in Gaussian correlated environment,
    ALEA, Lat. Am. J. Probab. Math. Stat. X (2), (2013) 953–977.
    [Journal] [arXiv]

---

46. Quentin Berger, Fabio Toninelli,
    Hierachical pinning model in correlated random environment,
    Ann. Inst. Henri Poincaré Probab. Stat., Vol. 48 Num. 3 (2013) 781-816.
    [Journal] [arXiv]

---

47. Quentin Berger, Hubert Lacoin,
    Sharp critical behavior for random pinning model with correlated environment,
    Stochastic Process. Appl., Vol. 122, Issue 4 (2012) 1397-1436.
    [Journal] [arXiv]

---

48. Quentin Berger, Hubert Lacoin,
    The effect of disorder on the free-energy for the Random Walk Pinning Model: smoothing of the phase transition and low temperature asymptotics,
    J. Stat. Phys., Vol. 142, Issue 2 (2011) 322-341.
    [Journal] [arXiv]

---

49. Quentin Berger, Fabio Toninelli,
    On the critical point of the Random Walk Pinning Model in dimension d=3,
    Electron. J. Probab. Vol. 15 (2010) no 21, 654-683.
    [Journal] [arXiv]