Francesco Caravenna

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All papers are also available on arXiv

Lecture notes and surveys

F. Caravenna, R. Sun, N. Zygouras
From disordered systems to the Critical 2D Stochastic Heat Flow
Proceedings of the 2026 International Congress of Mathematicians [pdf] [arXiv]
F. Caravenna, R. Sun, N. Zygouras
The Critical 2d Stochastic Heat Flow and Related Models
CIME Lecture Notes (2025+) [pdf] [arXiv]
F. Caravenna, R. Sun, N. Zygouras
Scaling limits of disordered systems and disorder relevance
Proceedings of the XVIII International Congress of Mathematical Physics, 2016 (volume cancelled) [pdf] [arXiv]

Preprints

Q. Berger, F. Caravenna, N. Turchi, Strong disorder for Stochastic Heat Flow and 2D Directed Polymers, preprint 2025 [pdf] [arXiv]
F. Caravenna, A. Donadini, Enhanced noise sensitivity, 2D directed polymers and Stochastic Heat Flow, preprint 2025 [pdf] [arXiv]
F. Caravenna, R. Sun, N. Zygouras, Singularity and regularity of the critical 2D Stochastic Heat Flow, preprint 2025 [pdf] [arXiv]

Papers

Note: Copyright for published versions is held by the respective publishers.

F. Caravenna, F. Cottini, M. Rossi, Quasi-critical fluctuations for 2d directed polymers, Ann. Appl. Probab. 35 (2025), 2604-2643 [pdf] [journal] [arXiv]
L. Broux, F. Caravenna, L. Zambotti, Hairer's multilevel Schauder estimates without Regularity Structures, Trans. Amer. Math. Soc. 377 (2024), 981-1035 [pdf] [journal] [arXiv]
F. Caravenna, R. Sun, N. Zygouras, The critical 2d Stochastic Heat Flow is not a Gaussian Multiplicative Chaos, Ann. Probab. 51 (2023), 2265-2300 [pdf] [journal] [arXiv]
F. Caravenna, R. Sun, N. Zygouras, The critical 2d Stochastic Heat Flow, Inventiones Math. 233 (2023), 325-460 [pdf] [journal] [arXiv]
F. Caravenna, F. Cottini, Gaussian Limits for Subcritical Chaos, Electron. J. Probab. 27 (2022), article no. 81, 1-35 [pdf] [journal] [arXiv]
F. Caravenna, L. Zambotti, Hairer's Reconstruction Theorem without Regularity Structures, EMS Surv. Math. Sci. 7 (2020), 207-251 [pdf] [journal] [arXiv]
F. Caravenna, F. den Hollander, Phase transitions for spatially extended pinning, Probab. Theory Related Fields 181 (2021), 329-375 [pdf] [journal] [arXiv]
F. Caravenna, R. Sun, N. Zygouras, The two-dimensional KPZ equation in the entire subcritical regime, Ann. Probab. 48 (2020), 1086-1127 [pdf] [journal] [arXiv]
F. Caravenna, R. Sun, N. Zygouras, On the moments of the (2+1)-dimensional directed polymer and stochastic heat equation in the critical window, Commun. Math. Phys. 372 (2019), 385-440 [pdf] [journal] [arXiv]
F. Caravenna, R. Sun, N. Zygouras, The Dickman subordinator, renewal theorems, and disordered systems, Electron. J. Probab. 24 (2019), paper no. 101, 40 pp. [pdf] [journal] [arXiv]
F. Caravenna, On the maximum of conditioned random walks and tightness for pinning models, Electron. Commun. Probab. 23 (2018), paper n. 69, 13 pp. [pdf] [journal] [arXiv]
F. Caravenna, R. Doney, Local large deviations and the strong renewal theorem, Electron. J. Probab. 24 (2019), paper no. 72, 1-48 [pdf] [journal] [arXiv]
F. Caravenna, A. Garavaglia, R. van der Hofstad, Diameter in ultra-small scale-free random graphs, Random Structures Algorithms 54 (2019), 444-498 [pdf] [journal] [arXiv]
F. Caravenna, J. Corbetta, The asymptotic smile of a multiscaling stochastic volatility model, Stochastic Process. Appl. 128 (2018), 1034-1071 [pdf] [journal] [arXiv]
F. Caravenna, R. Sun, N. Zygouras, Universality in marginally relevant disordered systems, Ann. Appl. Probab. 27 (2017), 3050-3112 [pdf] [journal] [arXiv]
F. Caravenna, J. Corbetta, General smile asymptotics with bounded maturity, SIAM J. Financial Math. 7 (2016), 720-759 [pdf] [journal] [arXiv]
F. Caravenna, F. L. Toninelli, N. Torri, Universality for the pinning model in the weak coupling regime, Ann. Probab. 45 (2017), 2154-2209 [pdf] [journal] [arXiv]
F. Caravenna, F. den Hollander, N. Pétrélis, J. Poisat, Annealed scaling for a charged polymer, Math. Phys. Anal. Geom. 19 (2016), article no. 2 [pdf] [journal] [arXiv]
F. Caravenna, R. Sun, N. Zygouras, The continuum disordered pinning model, Probab. Theory Related Fields 164 (2016), 17-59 [pdf] [journal] [arXiv]
F. Caravenna, R. Sun, N. Zygouras, Polynomial chaos and scaling limits of disordered systems, J. Eur. Math. Soc. 19 (2017), 1-65 [pdf] [journal] [arXiv]
F. Caravenna, F. den Hollander, A general smoothing inequality for disordered polymers, Electron. Commun. Probab. 18 (2013), article no. 76, 1-15 [pdf] [journal] [arXiv]
F. Caravenna, L. Chaumont, An invariance principle for random walk bridges conditioned to stay positive, Electron. J. Probab. 18 (2013), article no. 60, 1-32 [pdf] [journal] [arXiv]
Q. Berger, F. Caravenna, J. Poisat, R. Sun, N. Zygouras, The critical curve of the random pinning and copolymer models at weak coupling, Commun. Math. Phys. 326 (2014), 507-530 [pdf] [journal] [arXiv]
F. Caravenna, P. Carmona, N. Pétrélis, The discrete-time parabolic Anderson model with heavy-tailed potential, Ann. Inst. H. Poincaré Probab. Statist. 48 (2012), 1049-1080 [pdf] [journal] [arXiv]
A. Andreoli, F. Caravenna, P. Dai Pra, G. Posta, Scaling and multiscaling in financial series: a simple model, Adv. in Appl. Probab. 44 (2012), 1018-1051 [pdf] [journal] [arXiv]
M. Borecki, F. Caravenna, Localization for (1+1)-dimensional pinning models with (∇ + Δ) interaction, Electron. Commun. Probab. 15 (2010), 534-548 [pdf] [journal] [arXiv]
F. Caravenna, G. Giacomin, The weak coupling limit of disordered copolymer models, Ann. Probab. 38 (2010), 2322-2378 [pdf] [journal] [arXiv]
F. Caravenna, N. Pétrélis, Depinning of a polymer in a multi-interface medium, Electron. J. Probab. 14 (2009), 2038-2067 [pdf] [journal] [arXiv]
F. Caravenna, N. Pétrélis, A polymer in a multi-interface medium, Ann. Appl. Probab. 19 (2009), 1803-1839 [pdf] [journal] [arXiv]
F. Caravenna, G. Giacomin, M. Gubinelli, Large scale behavior of semiflexible heteropolymers, Ann. Inst. H. Poincaré Probab. Statist. 46 (2010), 97-118 [pdf] [journal] [arXiv]
F. Caravenna, J.-D. Deuschel, Scaling limits of (1+1)-dimensional pinning models with Laplacian interaction, Ann. Probab. 37 (2009), 903-945 [pdf] [journal] [arXiv]
E. Bolthausen, F. Caravenna, B. de Tilière, The quenched critical point of a diluted disordered polymer model, Stochastic Process. Appl. 119 (2009), 1479-1504 [pdf] [journal] [arXiv]
F. Caravenna, J.-D. Deuschel, Pinning and wetting transition for (1+1)-dimensional fields with Laplacian interaction, Ann. Probab. 36 (2008), 2388-2433 [pdf] [journal] [arXiv]
F. Caravenna, L. Chaumont, Invariance principles for random walks conditioned to stay positive, Ann. Inst. H. Poincaré Probab. Statist. 44 (2008), 170-190 [pdf] [journal] [arXiv]
F. Caravenna, G. Giacomin, L. Zambotti, Sharp asymptotic behavior for wetting models in (1+1)-dimension, Electron. J. Probab. 11 (2006), 345-362 [pdf] [journal] [arXiv]
F. Caravenna, G. Giacomin, L. Zambotti, A renewal theory approach to periodic copolymers with adsorption, Ann. Appl. Probab. 17 (2007), 1362-1398 [pdf] [journal] [arXiv]
F. Caravenna, G. Giacomin, M. Gubinelli, A numerical approach to copolymers at selective interfaces, J. Stat. Phys. 122 (2006), 799-832 [pdf] [journal] [arXiv]
F. Caravenna, G. Giacomin, On constrained annealed bounds for pinning and wetting models, Electron. Commun. Probab. 10 (2005), 179-189 [pdf] [journal] [arXiv]
F. Caravenna, A local limit theorem for random walks conditioned to stay positive, Probab. Theory Related Fields 133 (2005), 508-530 [pdf] [journal] [arXiv]

Selected slides

2025

On (ir)reducible Central Limit Theorems (Talk, 11 Dec 2025)
Workshop 20 Years of the Fourth Moment Theorem, Luxembourg (L) [pdf]
The Critical 2D Stochastic Heat Flow in the strong disorder limit (Talk, 30 Oct 2025)
BIRS Workshop Emerging Synergies between Stochastic Analysis and Statistical Mechanics, Banff (CDN) [pdf]
Noise sensitivity for 2D Stochastic Heat Equation and directed polymers (Talk, 24 Jun 2025)
Workshop on Irregular Stochastic Analysis, Palazzone di Cortona (I) [pdf]
Noise Sensitivity and Critical 2D Directed Polymers (Talk, 4 Apr 2025)
Mark Kac Seminar, Utrecht (NL) [pdf]
The 2D Stochastic Heat Equation and related critical models (Talk, 6 Mar 2025)
Colloquium du LPSM, Paris (F) [pdf]
Noise sensitivity for the 2D Stochastic Heat Equation (Talk, 11 Feb 2025)
Workshop Stochastic Equations and Stochastic Dynamics, SwissMap Research Station, Les Diablerets (CH) [pdf]

2024

The critical 2d Stochastic Heat Flow (Mini-Course with R. Sun and N. Zygouras, 15 May 2024)
Spring School on Critical Singular SPDEs, Peking University (CN) [pdf (part 1)]
The 2d Stochastic Heat Equation and related critical models (Talk, 14 Jun 2024)
4th Italian Meeting on Probability and Mathematical Statistics, Rome (I) [pdf]
Singolarità e Regolarità Aleatorie (Talk, 23 Jan 2024)
Colloquium di Matematica, Fisica e Informatica, University of Parma (I) [pdf]
Enhanced Schauder estimates for families of distributions (Talk, 11 Jan 2024)
Workshop Common trends and challenges in QFT and stochastic PDEs, University of Pavia (I) [pdf]

2023

The critical 2d Stochastic Heat Flow (Talk, 13 Dec 2023)
Workshop Critical phenomena in statistical physics, continuum theories and SPDEs, University of Warwick (UK) [pdf]
The critical 2d Stochastic Heat Flow (Talk, 23 Oct 2023)
21st Symposium Stochastic Analysis on Large Scale Interacting Systems, RIMS, Kyoto University (JP) [pdf]
The critical 2d Stochastic Heat Flow (Mini-Course with R. Sun and N. Zygouras, 12 Sep 2023)
CIME Summer School Statistical Mechanics and Stochastic PDEs, Cetraro (I) [pdf (part 1)]
Renewal Theory, Disordered Systems, and Stochastic PDEs (Talk, 1 Jun 2023)
Workshop Polymers and self-avoiding walks, University of Nantes (F) [pdf]
The critical 2d Stochastic Heat Flow (Talk, 12 June 2023)
Probability Seminar, University of Oxford (UK) [pdf]
Singolarità e Regolarità Aleatorie (Talk, 26 Jan 2023)
I Seminari del Centenario - Cento anni di UMI 1922-2022, University of Milano-Bicocca (I) [pdf] [video]
On the 2d Stochastic Heat Equation and delta Bose gas (Talk, 9 Jan 2023)
Workshop Mathematical Quantum Matter, University of Milano (I) [pdf]

2022

Renewal Theory, Disordered Systems, and Stochastic PDEs (Talk, 23 Sep 2022)
Workshop A journey through complex systems in honor of Paolo Dai Pra's 60th birthday, L'Aquila (I) [pdf]
The critical 2d Stochastic Heat Flow (Talk, 12 Sep 2022)
Workshop Large Scale Stochastic Dynamics, Oberwolfach (D) [pdf]
Renewal Theory, Disordered Systems, and Stochastic PDEs (Talk, 26 Jul 2022)
Workshop Lévy processes and random walks, Manchester (UK) (online) [pdf]
The critical 2d Stochastic Heat Flow (is not a GMC) (Talk, 2 May 2022)
Workshop Random Walk, Reinforcement and Localization, CIRM, Marseille (F) [pdf]
The critical 2d Stochastic Heat Flow (Talk, 19 Mar 2022)
Mathematics Colloquium, GSSI, L'Aquila (I) [pdf] [video]

2021

Gaussian Limits for Subcritical Chaos (Talk, 8 Sep 2021)
Workshop Directed Polymers and Folding, CIRM, Marseille (F) [pdf]
Central Limit Theorems in Disordered Systems and Stochastic PDEs (Talk, 7 Jun 2021)
UMI - PRISMA Seminar (online) [pdf]
Gaussian Limits for 2d Directed Polymers, KPZ and SHE (Mini-course, 19 and 22 Mar 2021)
Berlin-Oxford IRTG (online) [pdf]
Hairer's Reconstruction Theorem without Regularity Structures (Talk, 21 Jan 2021)
One World Probability Seminar (online) [pdf] [video]

2020

On the 2d KPZ and Stochastic Heat Equation via Directed Polymers (Talk, 8 Dec 2020)
Vienna Probability Seminar (online) [pdf]
The two-dimensional directed polymer and the Dickman subordinator (Talk, 11 Sep 2020)
Workshop Random Polymers and Networks, IGESA center, Porquerolles (F) [pdf]
Hairer's Reconstruction Theorem without Regularity Structures (Talk, 11 Jun 2020)
TU Berlin SPDE seminar (online) [pdf]
On the 2d KPZ and Stochastic Heat Equation (Talk, Jan 27 2020)
Probability Seminar, University of Roma Sapienza (I) [pdf]

2019

On the 2d KPZ Equation (Talk, 30 Oct 2019)
Workshop Half day in Stochastic Analysis and applications, University of Padova (I) [pdf]
On the 2d KPZ and Stochastic Heat Equation (Talk, 24 Oct 2019)
Workshop Singular SPDEs and Related Topics, Hausdorff Institute, Bonn (D) [pdf] [video]

2018

On the 2d KPZ and Stochastic Heat Equation via directed polymers (Mini-course, 10-11 Dec 2018)
State of research: Statistical Mechanics, Institut H. Poincaré, Paris (F) [pdf (part 1)]
Moment asymptotics for 2d directed polymer and stochastic heat equation in the critical window (Talk, March 6, 2018)
Stochastic Analysis Seminar, Imperial College, London (UK) [pdf]
Pinning model, universality and rough paths (Talk, 9 Feb 2018)
Rhein-Main Kolloquium Stochastik, TU Darmstadt (D) [pdf]

2017

Local large deviations and the strong renewal theorem (Talk, 27 Jul 2017)
SPA 2017, Moscow (RU) [pdf]
Marginally relevant polymer models in the critical window (Talk, 6 Jun 2017)
Conference Statistical Mechanics, random planar geometry and interacting random walks, University of Lyon 1 (F) [pdf]

2016

Scaling and universality in probability (Talk, 14 Jun 2016)
Mathematics Colloquium of the University of Luxembourg (L) [pdf]
Universality in marginally relevant disordered systems (Talk, 2 Jun 2016)
University of Paris Diderot (F) [pdf]
Polynomial chaos and scaling limits of disordered systems (Course, March 7-11 2016)
Workshop YEP XIII in Eurandom, TU Eindhoven (NL) [pdf: one, two, three]

2015

Polynomial chaos and scaling limits of disordered systems (Course, 28 Sep - 2 Oct 2015)
Berlin-Potsdam Summer School 2015 at Levico Terme (I) [pdf: one, two, three, four, five]
Un'estensione multilineare del teorema limite centrale e il fenomeno dell'universalità per i sistemi disordinati (Talk, 10 Sep 2015)
XX Congress of the Italian Mathematical Union in Siena (I) [pdf]
Polynomial chaos and scaling limits of disordered systems (Talk, 19 Mar 2015)
Probability Seminar, ENS Lyon (F) [pdf]

2014

The continuum disordered pinning model (Talk, 28 Jul 2014)
SPA 2014 Conference, Buenos Aires (AR) [pdf]

2013

Scaling limits and universality for random pinning models (Talk, 12 Sep 2013)
15th Italian Meeting on Hyperbolic Equations, University of Milano-Bicocca (I) [pdf]
Scaling limits and universality for random pinning models (Talk, 19 Aug 2013)
Workshop Universality and Scaling Limits in Probability and Statistical Mechanics, Hokkaido University, Sapporo (JP) [pdf]

2012

Scaling and multiscaling in financial series: a simple model (Talk, 12 Mar 2012)
Department of Mathematics, University of Modena and Reggio Emilia (I) [pdf]

2011

The weak coupling limit of disordered copolymer models (Talk, 15 Feb 2011)
Workshop on Interacting Processes in Random Environments, Fields Institute, Toronto (CDN) [pdf]

2010

A polymer in a multi-interface medium (Talk, 16 Dec 2010)
Institute for Applied Mathematics, University of Bonn (D) [pdf]

2008

Pinning and wetting transition for (1+1)-dimensional fields with Laplacian interaction (Talk, 11 Jun 2008)
University of Warwick (GB) [pdf]

2007

Modelli di polimeri e passeggiate aleatorie (Talk, 26 Sep 2007)
XVIII Congress of the Italian Mathematical Union in Bari (I) [pdf]
On the phase diagram of random copolymers at selective interfaces (Talk, 22 Jun 2007)
Eurandom in Eindhoven (NL) [pdf]

Proceedings

F. Caravenna, F. den Hollander, N. Pétrélis, Lectures on Random Polymers, in: Probability and Statistical Physics in Two and more Dimensions, Clay Mathematics Proceedings 15 (2012), 319-393 [pdf] [arXiv]
F. Caravenna, G. Giacomin, F. L. Toninelli, Copolymers at selective interfaces: settled issues and open problems, in: Probability in complex physical systems, Springer Proceedings in Mathematics 11 (2012), 289-312 [pdf] [journal] [arXiv]
A. Andreoli, F. Caravenna, P. Dai Pra, G. Posta, A model for multiscaling and clustering of volatility in financial indexes, Proceedings of the 19th International Symposium on Mathematical Theory of Networks and Systems - MTNS, vol. 5, no. 9, 2010 [pdf]
F. Caravenna, Modelli di polimeri e passeggiate aleatorie, Boll. Unione Mat. Ital. Serie IX, vol. I (2008), 559-571 [pdf]
F. Caravenna, G. Giacomin, L. Zambotti, Infinite volume limits of polymer chains with periodic charges, Markov Process. Related Fields 13 (2007), 697-730 [pdf] [journal] [arXiv]

Other

Popular science article (in Italian and French )
- Q. Berger, F. Caravenna
  Il paradosso di Simpson e i dati sulle vaccinazioni anticovid
  Internazionale.it, 9 Dec 2021 [website] [pdf + appendix]
- Q. Berger, F. Caravenna
  Le paradoxe de Simpson illustré par des données de vaccination contre le Covid-19
  The Conversation, 4 Nov 2021 [website]
F. Caravenna, The strong renewal theorem, Unpublished manuscript (2015). Superseded by this joint paper with R. Doney [pdf] [arXiv]
F. Caravenna, A note on directly Riemann integrable functions, Unpublished manuscript (2012) [pdf] [arXiv]
F. Caravenna, G. Giacomin, L. Zambotti, Tightness conditions for polymer measures, Unpublished manuscript (2007). Superseded by this paper [pdf] [arXiv]
F. Caravenna, Random walk models and probabilistic techniques for inhomogeneous polymer chains, Ph.D. Thesis (2005), Università di Milano-Bicocca and Université Paris 7 [pdf - extended version] [arXiv]