Moderator: Jorge Kurchan (Paris)
Denis Bernard (Paris),
Jean-Philippe Bouchaud (Paris),
Laura Foini (Saclay),
Alice Guionnet (Lyon),
Frédéric Patras (Nice),
Marc Potters (Paris),
Roland Speicher (Saarbrücken),
Denis Bernard (Paris): The Quantum SSEP and the emergence of free probability in many-body mesoscopic quantum systems. pdf
An alternative title could have been "How to characterise fluctuations in diffusive out-of-equilibrium many-body quantum systems?" In general, the difficulty to characterise non-equilibrium systems lies in the fact that there is no analog of the Boltzmann distribution to describe thermodynamic variables and their fluctuations. Over the last 20 years, however, it was observed that fluctuations of diffusive transport show universal properties that do not depend on the microscopic details. The general framework to characterise these systems from a macroscopic point of view is now called the "Macroscopic Fluctuation Theory". A natural question is whether this framework can be extended to quantum mechanics to describe the statistics of purely quantum mechanical effects such as interference or entanglement in diffusive out-of-equilibrium systems. With this aim in mind, I will introduce the Quantum Symmetric Simple Exclusion Process (Q-SSEP), a microscopic model system of fluctuating quantum diffusion. I will in particular present the recent observation that fluctuations of coherences in Q-SSEP have a natural interpretation as free cumulants, a concept from free probability theory, and heuristic arguments why we expect free probability theory to be an appropriate framework to describe coherent fluctuations in generic mesoscopic systems.
Jean-Philippe Bouchaud (Paris): Covariance stability & eigenvector overlaps : a (free) RMT approach. pdf
The time evolution of the eigenvectors of random matrices is an important problem that appears in different contexts, for example covariance matrices in (possibly time dependent) environments. We will review different recent ideas to separate a genuine evolution of the structure of the matrix from a mere disorder realization effect.
Laura Foini (Saclay): Eigenstate Thermalization Hypothesis and Free Probability. pdf
The Eigenstate Thermalization Hypothesis (ETH) implies a form for the matrix elements of local operators between eigenstates of the Hamiltonian, expected to be valid for chaotic systems, which can be understood by analogy with Random Matrix Theory. In fact within this ansatz these matrix elements are modelled as pseudo random variables. In this talk I will first describe how a complete description of high order thermal correlation functions requires correlations between matrix elements. I will then argue how, by analogy with Random Matrix Theory and "typicality arguments", one can assume a certain hierarchy between these correlations. The talk is based on our recent work in which we unveil a relation between ETH and Free Probability.
Refs.: L. Foini, J. Kurchan, The Eigenstate Thermalization Hypothesis and Out of Time Order Correlators, Phys. Rev. E 99, 042139 (2019).
S. Pappalardi, L. Foini, J. Kurchan, Eigenstate Thermalization Hypothesis and Free Probability, Phys. Rev. Lett. 129, 170603 (2022)
Alice Guionnet (Lyon): Matrix models at low temperature.
In the nineties, Voiculescu showed that the non-commutative distribution of independent GUE matrices converges towards free semi-circular variables. In this talk, we will discuss the convergence of more general matrix models as well as the properties of the limit when it exists, in particular in the so-called low temperature region. This talk is based on a recent work with Edouard Morel-Segala.
Frédéric Patras (Nice): Algebraic structures underlying free probability.
The talk will survey various key features of an algebraic approach to free probability that has been systematically developped during the last decade to account for fundamental notions of the theory (free cumulants, various notions of independence, CLTs...). It is based on the combinatorial properties of sentences (sequences of words) and provides another way to organize arguments and computations. It had proved particularly efficient to approach existing combinatorial formulas and derive new ones.
Based on joint works with K. Ebrahimi-Fard.
Marc Potters (Paris): Free probabilities in action: the spectral method for phase retrieval. pdf
Roland Speicher (Saarbrücken): Free probability and free cumulants. pdf
I will talk about some aspects of free probability; in particular, in relation with free cumulants.