2025-26

The conference focuses on recent progress in the study of stochastic dynamics, including contributions from a theoretical perspective as well as the mathematical analysis of numerical methods for the efficient simulation of these dynamics, with particular emphasis on their linear response properties.

Please check the program here

https://sites.google.com/view/sineq-finalconference/home/

We would be grateful if you could circulate it among potentially interested students and researchers.

We are looking forward to seeing you during the week.

The renormalization group is one of the most important tools for the description of critical points in theoretical physics. Most mathematically rigorous treatments are based on implementing the renormalization group as a discrete dynamical system, which introduces a number of theoretical complications, and in particular obscures the emergence of full scale invariance, which is one of the most important features of a critical point. In the last few years, however, there have been several new results about solutions of the Polchinski equation (a nonlinear differential equation which implements the renormalization group as a continuous dynamical system) for Fermionic systems. I will present one in particular, the construction of a nontrivial fixed point of a family of continuous renormalization group flows corresponding to certain weakly interacting Fermionic quantum field theories with a parameter in the propagator allowing the scaling dimension to be tuned in a manner analogous to dimensional regularization. 

We investigate a passive vector field which is transported and stretched by a divergence-free Gaussian velocity field, delta-correlated in time and poorly correlated in space (spatially nonsmooth). Although the advection of a scalar field (Kraichnan's passive scalar model) is known to enjoy regularizing properties, the potentially competing stretching term in vector advection may induce singularity formation. We establish that the regularization effect is actually retained in certain regimes. While this is true in any dimension $d\ge 3$, it notably implies a regularization result for linearized 3D Euler equations with stochastic modeling of turbulent velocities, and for the induction equation in magnetohydrodynamic turbulence.

The presentation is based on a joint work with Francesco Grotto and Mario Maurelli. 

Hour: 14.30

Speaker: Federica Iacovissi (Univaq)

Title: The Matrix Product Ansatz from a probabilistic viewpoint

Abstract: 

We give a systematic exploration of the probabilistic structure of the Matrix Product Ansatz. By suitably enlarging the state space, we show that a probability measure can be described in terms of non-negative matrices via the Matrix Product Ansatz if and only if it can be written as a mixture of inhomogeneous product measures, where the mixing law is a Markov bridge. We illustrate the result with examples and derive large deviation results. Joint work with Davide Gabrielli

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Hour: 15.00

Speaker: Giuseppe Lipardi (GSSI)

Title: The validity of spin-wave theory for quantum spin systems

Abstract: 

The low-temperature behavior of spin systems with a continuous symmetry is expected to be governed by Gaussian fluctuations around the ground state, known as spin waves. For classical models this picture has been rigorously established by Bricmont–Fontaine–Lebowitz–Lieb–Spencer (1982) for the O(2) model and, more recently, by Giuliani–Ott (2025) for general O(N) systems. In the quantum setting, however, the situation is more subtle: the low-energy behavior arises from a nontrivial interplay between thermal and quantum excitations, and a rigorous justification of spin-wave theory, understood as a semiclassical expansion in the spin size S, has remained open.

In this talk I will discuss a constructive approach to this problem for the ground state of the three-dimensional quantum XY model. After introducing the Holstein–Primakoff bosonic representation and the resulting quadratic (spin-wave) approximation, I will apply multiscale renormalization-group methods to construct a formal expansion of ground-state spin correlation functions in powers of 1/S. The coefficients of this expansion are finite and satisfy uniform factorial bounds, and the leading term agrees with the spin-wave prediction. This is based on joint work with S. Cenatiempo(GSSI) and A. Giuliani (Roma Tre).

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Hour: 15.30

Speaker: Giulia Pallotta (Univaq)

Title: Freidlin-Wentzell solutions of discrete Hamilton Jacobi equations

Abstract: 

We study finite irreducible Markov chains on a fixed directed graph with transition rates decaying exponentially in a parameter N. At the exponential scale, as N goes to infinity, the invariant measure satisfies a discrete Hamilton-Jacobi equation, which may admit multiple solutions. We define a notion of discrete viscosity solutions and characterize them geometrically via Lipschitz functions. A distinguished "vanishing viscosity" solution, selected through the matrix tree theorem, provides a combinatorial realization of the minimal action principle from Freidlin and Wentzell’s theory of large deviations, paralleling the weak KAM framework in the discrete setting.

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Hour: 16.00

Speaker: Vishnu Sanjay (GSSI)

Title: On the weak coupling limit for the periodic quantum Lorentz gas

Abstract: 

The quantum Lorentz gas is a fundamental model in kinetic theory, where one studies the effective behaviour of a single quantum particle interacting with its environment. In mathematical terms, the particle evolves according to a linear Schrödinger equation with a potential term representing the environment. In the weak coupling scaling, the particle interacts weakly with the potential, but spacetime is rescaled in such a way that the cumulative effect of the potential becomes significant. For Gaussian random potentials, it is known that under this scaling the rescaled Wigner transform of the wavefunction (which plays the role of a phase space density) converges weakly, on average, to the solution of a linear Boltzmann equation with an energy-preserving collision kernel determined by the covariance of the field.

In this talk, we consider instead a smooth deterministic potential that is periodic on the unit lattice in d space dimensions. Using Wigner series representations and rough path techniques, we show that for a certain class of observables the weak coupling limit yields free transport. For others, we show that the existence of the limit hinges on the continuity of a certain generalized phase-space object at energy band crossings of the free Hamiltonian. This is work done under the supervision of Prof. Massimiliano Gubinelli.