Dear all,

In addition to the already announced seminar by Tiozzo today (*Harmonic functions on groups, random walks, and the identification of the Poisson boundary*), we will have three seminars in Probability at the Department of Mathematics of Sapienza University in the coming weeks. The speakers will be Andreas Klippel (TU Darmstadt), Milton Jara (IMPA, Rio de Janeiro), and Lucas D’Alimonte (LPSM – Sorbonne Université).

Below you can find the titles and abstracts.

Buona giornata

Alessandra **************************************************

*Monday, November 3 2025*

*16:15, Sala di Consiglio, Dipartimento di Matematica*

*Speaker: *Andreas Klippel (TU Darmstadt)

*Title*: Two Sides of One Coin: Approaches to the Random Loop Model

*Abstract: *In the last decades, graphical representations have emerged as powerful tools to study models from statistical physics. One prominent example is the random loop model, whose two-point function encodes various spin correlations of quantum systems such as the Heisenberg ferromagnet, the antiferromagnet, and the quantum XY model.

Since the model lacks monotonicity properties and correlation inequalities such as FKG, alternative techniques have been developed to analyze it. We will discuss two complementary approaches suitable for different regimes. The first concerns the absence of long loops, corresponding to the regime without a phase transition. This situation can be studied by comparing the model to a percolation process and showing that certain structural features effectively reduce the percolation parameter, leading to exponentially small loops.

The second approach addresses the occurrence of a phase transition, which is typically established through the method of reflection positivity. The presentation will provide an introduction to the random loop model, outline these two analytical techniques, and present several results illustrating their application.

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*Monday, November 10 2025*

*16:15, Sala di Consiglio, Dipartimento di Matematica*

*Speaker: *Milton Jara (Impa, Rio de Janeiro)

*Title*: Quantitative Hydrodynamics for a generalized contact model

*Abstract: *We use the formalism of Quantitative Hydrodynamics to improve the quantitative hydrodynamic limit obtained in Chariker, De Masi, Lebowitz and Presutti (2023) for an interacting particle system inspired by integrate-and-fire neuron models. Our estimates are good enough to show that the typical fluctuations around the aforementioned hydrodynamic limit are Gaussian, and governed by an inhomogeneous stochastic linear equation.

Joint work with Julian Amorim (IMPA) and Yangrui Xiang (Louisiana State University)

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*Monday, November 17 2025*

*16:15, Sala di Consiglio, Dipartimento di Matematica*

*Speaker: *Lucas D’Alimonte (LPSM - Sorbonne Université)

*Title:* Ornstein—Zernike theory for the near-critical planar random cluster model

*Abstract:* In this talk, we will discuss the classical Ornstein-Zernike theory for the random-cluster model (also known as FK percolation). In its modern form, it is a very robust theory, which most celebrated output is the computation of the asymptotically polynomial corrections to the pure exponential decay of the two-points correlation function of the random-cluster model in the subcritical regime. We will present an ongoing project that extends this theory to the near-critical regime of the two-dimensional random-cluster model, thus providing a precise understanding of the Ornstein-Zernike asymptotics when p approaches the critical parameter p_c. The output of this work is a formula encompassing both the critical behaviour of the system when looked at a scale negligible with respect to its correlation length, and its subcritical behaviour when looked at a scale way larger than its correlation length. Based on a joint work with Ioan Manolescu.