I will discuss the classical infinite Coulomb gas and its extension to Riesz interaction potentials $|x|^{-s}$. I will particularly focus on the problem of defining infinite equilibrium states and their potential. This has caused a lot of confusion in the literature, even in the simplest case of periodic lattices. The general case is in fact largely open. Based on the review paper Coulomb and...
Anomalies are the breaking of classical symmetries by quantum effects, and their non-renormalization properties play a crucial role in a wide range of phenomena. I present some rigorous theorems on the (non-perturbative) anomaly non-renormalization in QFT models, based on Renormalization Group, cluster or tree expansion and determinant bounds, proving the exact cancellation of the terms coming...
I propose a conceptual frame to interpret the prolate differential operator, which appears in Communication Theory, as an entropy operator; indeed, I write its expectation values as a sum of terms, each subject to an entropy reading by an embedding suggested by Quantum Field Theory. This adds meaning to the classical work by Slepian et al. on the problem of simultaneously concentrating a...
We consider a random walk moving on a Lévy random medium, namely a one-dimensional renewal point process with i.i.d. inter-distances in the domain of attraction of a stable law. The focus is on the characterization of the law of the first-ladder height and length of the process. The study relies on the construction of a broader class of processes, denoted as Random Walks in Random Scenery on...
Quantum trajectories model quantum systems repeatedly (indirectly) measured. The resulting evolution is a Markov process. In this talk I will discuss their purification property. Proved in 2005 by Kummerer and Maassen, it shows that, in absence of so called dark subspaces, the system state has a tendency to get closer and closer to a pure state along a quantum trajectories. I will revisit this...
During this talk we shall discuss the construction of the massive Sine-Gordon field in the ultraviolet finite regime when the background is a two-dimensional Minkowski spacetime. The correlation functions of the model in the adiabatic limit will be obtained combining recently developed methods of perturbative algebraic quantum field theory with techniques developed in the realm of constructive...
Inspired by the successes of algebraic quantum statistical mechanics in dealing with some fundamental nonequilibrium questions, we investigate the relation between “approach to equilibrium” (sometimes called the Zeroth Law) and the Second Law. Short of being able to provide a new example of non-trivial and physically pertinent system approaching equilibrium, we bring some partial answers to a...
I will talk about slowly varying and non-autonomous quantum dynamics of a translation invariant spin or fermion system on the lattice. This system is assumed to be initially in thermal equilibrium, and we consider realizations of quasi-static processes in the adiabatic limit. By combining the Gibbs variational principle with the notion of quantum weak Gibbs states, I will present some general...
We consider a slowly varying time dependent d-level atom interacting with a photon field. Restricted to the single excitation atom-field sector, the model is a time-dependent generalization of the Wigner-Weisskopf model describing spontaneous emission of an atomic excitation into the radiation field. We analyze the dynamics of the atom and of the radiation field in the adiabatic and small...
The evolution of a quantum system undergoing repeated indirect measurements naturally leads to a Markov chain on the set of states, and which is called a quantum trajectory. When the system under consideration is finite dimensional, and under some natural assumption related to the non-existence of so-called dark subspaces, the state of the systems tends to become pure along the trajectory, a...
Non-equilibrium statistical mechanics has seen some impressive developments in the last three decades, thank to the pioneering works of Evans, Cohen, Morris and Searles on the violation of the second law, soon followed by the ground-breaking formulation of the Fluctuation Theorem by Gallavotti and Cohen for entropy fluctuation in the early nineties.The extension of these results to the quantum...
Poincaré’s theorem and Kac’s lemma on recurrence are basic results that one typically encounters quite early on when studying dynamical systems. Perhaps less well known is a 1993 result of Ornstein and Weiss on the time $R_n$ it takes for the $n$ first symbols in a sequence sampled from an ergodic measure $\mathbb P$ on a one-sided shift to reappear down this same sequence: they build on ideas...
Once the sequence $(n^{-1}\ln R_n)$ of return times introduced in Renaud's talk has been shown to satisfy a law of large numbers, a natural question is to study its large deviations. Quite surprisingly, very limited results were available. In a recent paper with Renaud Raquépas, we proved that the return times satisfy the full large deviation principle, again under some quite mild decoupling...
Introduced in 1993 by Merhav and Ziv, the Merhav-Ziv estimator $Q_n$ is an analogue of the well-known Lempel-Ziv estimator, which estimates the Cross Entropy of two unknown probability measures $\mathbb{P}$ and $\mathbb{Q}$. The algorithm takes as an input two strings $y_1^n$ and $x_1^n$ and does the following: it starts by considering the largest word $y_{1}^m$ which appears inside $x_1^n$,...
I will discuss the dynamics of short-ranged, weakly interacting fermionic lattice models, exposed to extensive perturbations slowly varying in time. We shall focus on the evolution of the expectation of local observables, starting from a positive temperature equilibrium state. At zero temperature, in the last years there has been important progress in the derivation of a many-body adiabatic...
We review the quasi-classical limit of the Nelson model, describing nucleons interacting with a scalar bosonic field, i.e., when the field degrees of freedom becomes classical while the nucleons retain their quantum nature. It is well known that such a model admits a simple energy renormalization of the ultraviolet divergence via the so-called dressing transformation. We then investigate the...
Gibbs measures of nonlinear Schrödinger equations are a fundamental object used to study low-regularity solutions with random initial data. In the dispersive PDE community, this point of view was pioneered by Bourgain in the 1990s. On the other hand, the nonlinear Schrödinger equation can be viewed a classical limit of many-body quantum theory. We are interested in the problem of the...
