My research interests mainly focus on Probability Theory, Statistical Mechanics and Stochastic Analysis, in particular I work on projects related to central limit theorems for chaos expansions, random polymers and disordered systems, scaling limits and singular stochastic PDEs.
Some keywords: disordered systems, phase transitions and critical phenomena, directed polymers in random environment, Stochastic Heat Flow, singular stochastic PDEs, central limit theorems for polynomial/Wiener chaos, random walks, Bernoulli percolation, random graphs, etc.
If you are curious to learn more... My research activity lies at the interface between statistical mechanics and stochastic analysis. The unifying theme of my work is the model of the directed polymer in random environment, a probabilistic model with diverse applications in statistical physics, disordered systems, singular stochastic equations, and limit theorems.
This model describes a random walk on ℤ^d perturbed by a random environment. Originally introduced in statistical mechanics, it has recently attracted more interest due to its connection with the stochastic heat equation (SHE) with multiplicative noise, for which it provides a discrete counterpart, and with the KPZ equation via the Hopf–Cole transform. The case d=2 is particularly interesting, as the SHE is singular in this dimension. Several recent works (including some of mine) therefore use the directed polymer in dimension 2 as a discrete framework to study and better understand different notions of solutions to the SHE and their properties.
Starting from the standard model, several generalizations are possible. In particular, I have studied random environments that are spatially correlated, as well as models where the underlying random walk evolves not on ℤ^d but on a random graph, for instance one arising from supercritical Bernoulli percolation.
The directed polymer also motivates a broader interest in central limit theorems for polynomial chaos and in the study of Gaussian fields. Indeed, the partition function of the directed polymer admits a representation in terms of polynomial chaos, that is, multilinear polynomials in independent random variables. Understanding under which conditions such objects converge to a Gaussian limit goes beyond the framework of disordered systems: polynomial chaos provide indeed fundamental examples of degenerate U-statistics, arise in the construction of Wiener chaos (Gaussian or Poisson), and appear naturally in the study of functions defined on discrete structures.
My research focuses on developing probabilistic tools for the study of disordered systems and singular stochastic equations, connecting discrete models, continuous limits, and normal approximation techniques.
If you’re particularly curious... my papers are listed below.
Preprints
F. Cottini and M. Nitzschner. Non-coincidence of critical points for directed polymers on supercritical percolation clusters. arXiv:2510.24532, 2025+. arXiv
F. Caravenna, F. Cottini and G. Peccati. On irreducible central limit theorems. arXiv:2510.00748, 2025+. arXiv
C. Cosco, F. Cottini and A. Donadini. A central limit theorem for two-dimensional directed polymers with critical spatial correlation. arXiv:2509.16694, 2025+. arXiv
Publications
F. Caravenna, F. Cottini and M. Rossi. Quasi-critical fluctuations for 2d directed polymers. Ann. Appl. Probab. 35(4), 2604-2643, 2025. arXiv
F. Caravenna and F. Cottini. Gaussian Limits for Subcritical Chaos. Electron. J. Probab. 27, 1-35, 2022. arXiv
S. Bonaccorsi, F. Cottini and D. Mugnolo. Random evolution equations: well-posedness, asymptotics, and applications to graphs. Appl. Math. Optim. 84, 2849-2887, 2021. arXiv
Others
F. Cottini. Central limit theorems for polynomial chaos and fluctuations for 2d directed polymers. Ph.D. thesis, Università degli Studi di Milano-Bicocca, 2023. PDF
My collaborators
Quentin Berger (Université Sorbonne Paris Nord)
Stefano Bonaccorsi (Università degli Studi di Trento)
Nicolas Bouchot (Universität Innsbruck)
Francesco Caravenna (Università degli Studi di Milano-Bicocca)
Clément Cosco (Université Paris Dauphine)
Anna Donadini (Università degli Studi di Milano-Bicocca)
Delio Mugnolo (University of Hagen)
Maximilian Nitzschner (The Hong Kong University of Science and Technology)
Giovanni Peccati (University of Luxembourg)
Maurizia Rossi (Università degli Studi di Milano-Bicocca)
Selected slides*
Directed Polymer in Critical Spatial Correlated Environment PDF
On (ir)reducible central limit theorems for homogenenous sums PDF
Gaussian limits for polynomial chaos and 2d directed polymers PDF
Gaussian fluctuations for 2d directed polymers in the quasi-critical regime PDF
Gaussian limits for subcritical chaos PDF
The Directed Polymer model: a "bridge" between Statistical Mechanics and Stochastic Analysis PDF
*all handwritten by me :)