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[[en:seminaires:SeminaireMod:index|Seminar of Probability and Modeling]]\\
Wednesday May 27, 2026, 2:15PM, Sophie Germain 1013\\
**Saverio Palazzi** (Université Paris Cité) //From sparse graphs to lattices: investigating phase transitions via the M-layer construction//
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This talk introduces the M-layer construction, a recent graph-theoretic and diagrammatic method designed to study critical phenomena on finite-dimensional lattices [1,2]. By performing a perturbative expansion in topological loops around a random regular graph, this framework provides a unified tool to analytically compute correlation functions and critical exponents. This presentation motivates the need for this approach, discussing the limitations of standard field theories in disordered systems, and thoroughly introduces the foundational concepts of the method. A brief recap of the main results is then given [3,4], including its application to the zero-temperature spin glass in a magnetic field, where it provides new analytical insights below the upper critical dimension [5]. The conclusion includes the connections of this approach to concepts in graph theory, such as graph M-coverings, offering a powerful alternative to standard Renormalization Group techniques.

1. Altieri, A., Angelini, M. C., Lucibello, C., Parisi, G., Ricci-Tersenghi, F., & Rizzo, T. (2017). Loop expansion around the Bethe approximation through the M-layer construction. Journal of Statistical Mechanics: Theory and Experiment, 2017(11), 113303.
2. Palazzi, S. "The M-Layer construction: a diagrammatic framework for critical behavior in lattice models.", PhD thesis, Sapienza University of Rome (2025).
3. Angelini, M. C., Palazzi, S., Parisi, G., & Rizzo, T., "Bethe M-layer construction on the Ising model." Journal of Statistical Mechanics: Theory and Experiment 2024.6 (2024): 063301.
4. Angelini, M. C., Palazzi, S., Rizzo, T., & Tarzia, M., "Bethe M-layer construction for the percolation problem." SciPost Physics 18.1 (2025): 030.
5. Angelini, M. C., Palazzi, S., Parisi, G., & Rizzo, T., "Critical exponents of the spin-glass transition in a field at zero temperature." Proceedings of the National Academy of Sciences 122.37 (2025): e2511882122.
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[[en:seminaires:SeminaireMod:index|Seminar of Probability and Modeling]]\\
Wednesday June 17, 2026, 2:15PM, Sophie Germain 1013\\
**Raphael Lefevere** (LPSM) //Hierarchical Lorentz Mirror Model: Normal Transport and a Universal 2/3 Mean--Variance Law//
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Le modèle de miroirs de Lorentz fournit un cadre épuré pour étudier un transport macroscopique engendré uniquement par le désordre figé de l’environnement. Nous introduisons une version hiérarchique pour laquelle la distribution des traversées gauche–droite satisfait une récurrence exacte. En dimension d\geq 3, nous démontrons rigoureusement un transport normal : la conductance moyenne se comporte comme (\text{section transverse})/(\text{longueur}) à toutes les échelles de longueur. Une hypothèse de fermeture gaussienne, étayée par les simulations numériques, prédit que le rapport variance/moyenne de la conductance converge vers la valeur universelle 2/3 pour tout d\geq 2 —ce que nous appelons la « loi des 2/3 ». Nous fournissons des preuves numériques de cette loi des 2/3 dans le modèle original, non hiérarchique, des miroirs de Lorentz en dimension d=3, et conjecturons qu’il s’agit d’une signature universelle du transport normal induit par un appariement aléatoire des courants. Il s’agit d’une collaboration avec Hal Tasaki (Gakushuin University).
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