Seminars

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.