3y PhD Position:
Statistical Physics of Generative AI

Scientific advisor Matteo Negri

Scientific theme Statistical Physics of Generative AI

Location LPTM, CY Cergy Paris Université, Cergy-Pontoise

Duration 3 years, starting Autumn 2026 (flexible)

Gross salary Approximatively €2,300 / month (standard national doctoral salary)

Contact matteo.negri1@cyu.fr

The project

Hopfield networks, the classical models of associative memory in physics, have recently been shown to exhibit a surprising generative behaviour [1]: when trained on structured data, they do not just memorise examples but learn to recombine their hidden features, producing attractors that correspond to configurations never seen during training. This generalisation transition reframes spurious states, long considered a flaw, as a source of creativity. Even more recently, the presence of these emergent attractors near test data points has related to the generative capabilities of diffusion models, both in the vision [2] and in the language domain [3].

The project's goal is to leverage this fundamental mechanism to build a theoretical framework for modern generative AI, with a particular focus on language models. Concretely, this means extending the “creative associative memory” picture to transformers, describing tokens as high-dimensional spin variables interacting in a highly non-linear way [4].

The work combines analytical and numerical efforts in equal measure: on the theoretical side, statistical mechanics, mean-field theory, and the replica method; on the numerical side, de-novo design and simulation of simple architectures to test theoretical predictions, and experiments on large-scale existing language models.

Environment and funding

The PhD student joins LPTM, a theoretical physics lab at CY Cergy Paris Université. The lab benefits from close ties with ETIS, the local computer science laboratory, offering natural opportunities for interdisciplinary exchange. CY Cergy Paris Université is also a member of the EUTOPIA alliance, a European network of universities that provides additional mobility and collaboration opportunities across partner institutions. Matteo Negri is also well connected with the community of theory of AI both in Paris and internationally.

Funding for equipment and travel will be available, including active support to attend conferences and schools, and to visit partner labs.

Who should apply

The ideal candidate has a strong background in statistical physics of disordered systems and enjoys the interplay between analytical work and large numerical experiments. That said, related backgrounds are equally welcome:

• Statistical physics, condensed matter, or physics of complex systems.

• Probability, statistics, or applied mathematics.

• Machine learning theory or theoretical computer science.

• Software engineering with a strong interest in fundamental aspects of AI.

No prior knowledge of Hopfield networks or associative memories is required. The working language is English.

How to apply

Send the following to matteo.negri1@cyu.fr:

• CV (2 pages max).

• Cover letter or research statement: what draws you to this project, and what you'd like to explore (1-2 pages).

• Names of one or two people I can contact as academic referees.

Informal contact before applying is very welcome.

Bibliography

[1]  Kalaj, S., Lauditi, C., Perugini, G., Lucibello, C., Malatesta, E. M., & Negri, M. (2025). Random features Hopfield networks generalize retrieval to previously unseen examples. Physica A: Statistical Mechanics and its Applications, 130946.

[2] Pham, B., Raya, G., Negri, M., Zaki, M. J., Ambrogioni, L., & Krotov, D. (2025). Memorization to generalization: Emergence of diffusion models from associative memory. arXiv preprint arXiv:2505.21777.

[3] Pham, B., Zaki, M. J., Ambrogioni, L., Krotov, D., & Negri, M. (2026). Language Diffusion Models are Associative Memories Capable of Retrieving Unseen Data. arXiv preprint arXiv:2604.26841.

[4] D’Amico, F., & Negri, M. (2024, July). Self-attention as an attractor network: transient memories without backpropagation. In 2024 IEEE Workshop on Complexity in Engineering (COMPENG) (pp. 1-6). IEEE.