Noé Cuneo
Université Paris Cité,
UFR de Mathématiques,
Bâtiment Sophie Germain,
8 place Aurélie Nemours,
75205 Paris CEDEX 13
+33 1 57 27 91 11
noe.cuneo [at] u-paris.fr
I am a Maître de conférences (~ tenured assistant professor) in Mathematics at the Laboratoire de Probabilités, Statistique et Modélisation (LPSM), Université Paris Cité (formerly Paris Diderot, Paris 7), France.
Here is my CV (EN|FR) and here are my arXiv, ORCID and Google Scholar pages.
I am a member of the ANR projects PERISTOCH and QTraj, and an associate member of the ENIC project.
I am mainly interested in stochastic processes, dynamical systems and mathematical physics. More specifically, I have been working on the following topics:
Large deviations of return times and related entropy estimators on shift spaces, with R. Raquépas, 2023. Accepted in Communications in Mathematical Physics. Abstract, preprint
We prove the large deviation principle for several entropy and cross entropy estimators based on return times and waiting times on shift spaces over finite alphabets. We consider shift-invariant probability measures satisfying some decoupling conditions which imply no form of mixing nor ergodicity. We establish precise relations between the rate functions of the different estimators, and between these rate functions and the corresponding pressures, one of which is the Rényi entropy function. For the most commonly used definition of return times, the large-deviation rate function is proved to be nonconvex, except in marginal cases. The results apply in particular to irreducible Markov chains, equilibrium measures for Bowen-regular potentials, g-measures, invariant Gibbs states for absolutely summable interactions in statistical mechanics, and also to probability measures which may be far from Gibbsian, including some hidden Markov models and repeated quantum measurement processes.
On entropy production of repeated quantum measurements II. Examples, with T. Benoist, , Journal of Statistical Physics 182(3): 1-71, 2021. Abstract, published version, preprint
We illustrate the mathematical theory of entropy production in repeated quantum measurement processes developed in a previous work by studying examples of quantum instruments displaying various interesting phenomena and singularities. We emphasize the role of the thermodynamic formalism, and give many examples of quantum instruments whose resulting probability measures on the space of infinite sequences of outcomes (shift space) do not have the (weak) Gibbs property. We also discuss physically relevant examples where the entropy production rate satisfies a large deviation principle but fails to obey the central limit theorem and the fluctuation-dissipation theorem. Throughout the analysis, we explore the connections with other, a priori unrelated topics like functions of Markov chains, hidden Markov models, matrix products and number theory.
Additive, almost additive and asymptotically additive potential sequences are equivalent. Communications in Mathematical Physics 377(3): 2579-2595, 2020. Abstract, published version, preprint
Motivated by various applications and examples, the standard notion of potential for dynamical systems has been generalized to almost additive and asymptotically additive potential sequences, and the corresponding thermodynamic formalism, dimension theory and large deviations theory have been extensively studied in the recent years. In this paper, we show that every such potential sequence is actually equivalent to a standard (additive) potential in the sense that there exists a continuous potential with the same topological pressure, equilibrium states, variational principle, weak Gibbs measures, level sets (and irregular set) for the Lyapunov exponent and large deviations properties. In this sense, our result shows that almost and asymptotically additive potential sequences do not extend the scope of the theory compared to standard potentials, and that many results in the literature about such sequences can be recovered as immediate consequences of their counterpart in the additive case. A corollary of our main result is that all quasi-Bernoulli measures are weak Gibbs.
Large Deviations and Fluctuation Theorem for Selectively Decoupled Measures on Shift Spaces, with 1950036-1-54, 2019. Abstract, published version, preprint
. Reviews in Mathematical Physics 31(10):We establish the Level-1 and Level-3 Large Deviation Principles (LDPs) for invariant measures on shift spaces over finite alphabets under very general decoupling conditions for which the thermodynamic formalism does not apply. Such decoupling conditions arise naturally in multifractal analysis, in Gibbs states with hard-core interactions, and in the statistics of repeated quantum measurement processes. We also prove the LDP for the entropy production of pairs of such measures and derive the related Fluctuation Relation. The proofs are based on Ruelle-Lanford functions, and the exposition is essentially self-contained.
Non-Equilibrium Steady States for Networks of Oscillators, with 1-28, 2018. Abstract, published version, preprint
. Electronic Journal of Probability 23(55):Non-equilibrium steady states for chains of oscillators (masses) connected by harmonic and anharmonic springs and interacting with heat baths at different temperatures have been the subject of several studies. In this paper, we show how some of the results extend to more complicated networks. We establish the existence and uniqueness of the non-equilibrium steady state, and show that the system converges to it at an exponential rate. The arguments are based on controllability and conditions on the potentials at infinity.
Fluctuation Theorem and Thermodynamic Formalism, with arXiv:1712.05167, 2017. Abstract, preprint
.We study the Fluctuation Theorem (FT) for entropy production in chaotic discrete-time dynamical systems on compact metric spaces, and extend it to empirical measures, all continuous potentials, and all weak Gibbs states. In particular, we establish the FT in the phase transition regime. These results hold under minimal chaoticity assumptions (expansiveness and specification) and require no ergodicity conditions. They are also valid for systems that are not necessarily invertible and involutions other than time reversal. Further extensions involve asymptotically additive potential sequences and the corresponding weak Gibbs measures. The generality of these results allows to view the FT as a structural facet of the thermodynamic formalism of dynamical systems.
Energy Dissipation in Hamiltonian Chains of Rotators, with R81, 2017. Selected in the Nonlinearity Highlights of 2017 collection. Abstract, published version, preprint
. Nonlinearity 30 (11),We discuss, in the context of energy flow in high-dimensional systems and Kolmogorov-Arnol'd-Moser (KAM) theory, the behavior of a chain of rotators (rotors) which is purely Hamiltonian, apart from dissipation at just one end. We derive bounds on the dissipation rate which become arbitrarily small in certain physical regimes, and we present numerical evidence that these bounds are sharp. We relate this to the decoupling of non-resonant terms as is known in KAM problems.
On the relaxation rate of short chains of rotors interacting with Langevin thermostats, with 1-8, 2017. Abstract, published version, preprint
. Electronic Communications in Probability 22(35):In this short note, we consider a system of two rotors, one of which interacts with a Langevin heat bath. We show that the system relaxes to its invariant measure (steady state) no faster than a stretched exponential $\exp(-c t^{1/2})$. This indicates that the exponent $1/2$ obtained earlier by the present authors and J.-P. Eckmann for short chains of rotors is optimal.
Ph.D thesis: Non-equilibrium Steady States for Hamiltonian Chains and Networks, 2016. Supervisor: Manuscript
.Non-equilibrium steady states for chains of four rotors, with 185–221, 2016. Abstract, published version, preprint
. Communications in Mathematical Physics, 345(1):We study a chain of four interacting rotors (rotators) connected at both ends to stochastic heat baths at different temperatures. We show that for non-degenerate interaction potentials the system relaxes, at a stretched exponential rate, to a non-equilibrium steady state (NESS). Rotors with high energy tend to decouple from their neighbors due to fast oscillation of the forces. Because of this, the energy of the central two rotors, which interact with the heat baths only through the external rotors, can take a very long time to dissipate. By appropriately averaging the oscillatory forces, we estimate the dissipation rate and construct a Lyapunov function. Compared to the chain of length three (considered previously by C. Poquet and the current authors), the new difficulty with four rotors is the appearance of resonances when both central rotors are fast. We deal with these resonances using the rapid thermalization of the two external rotors.
Non-equilibrium steady state and subgeometric ergodicity for a chain of three coupled rotors, with 2397–2421, 2015. Abstract, published version, preprint
. Nonlinearity 28 (2015),We consider a chain of three rotors (rotators) whose ends are coupled to stochastic heat baths. The temperatures of the two baths can be different, and we allow some constant torque to be applied at each end of the chain. Under some non-degeneracy condition on the interaction potentials, we show that the process admits a unique invariant probability measure, and that it is ergodic with a stretched exponential rate. The interesting issue is to estimate the rate at which the energy of the middle rotor decreases. As it is not directly connected to the heat baths, its energy can only be dissipated through the two outer rotors. But when the middle rotor spins very rapidly, it fails to interact effectively with its neighbours due to the rapid oscillations of the forces. By averaging techniques, we obtain an effective dynamics for the middle rotor, which then enables us to find a Lyapunov function. This and an irreducibility argument give the desired result. We finally illustrate numerically some properties of the non-equilibrium steady state.
Controlling General Polynomial Networks, with 1255–1274, 2014. Abstract, published version, preprint
. Communications in Mathematical Physics, 328(3):We consider networks of massive particles connected by non-linear springs. Some particles interact with heat baths at different temperatures, which are modeled as stochastic driving forces. The structure of the network is arbitrary, but the motion of each particle is 1D. For polynomial interactions, we give sufficient conditions for Hörmander's "bracket condition" to hold, which implies the uniqueness of the steady state (if it exists), as well as the controllability of the associated system in control theory. These conditions are constructive; they are formulated in terms of inequivalence of the forces (modulo translations) and/or conditions on the topology of the connections. We illustrate our results with examples, including "conducting chains" of variable cross-section. This then extends the results for a simple chain obtained in [Eckmann, Pillet, Rey-Bellet, 1999].
Comments, questions, and mistake reports are more than welcome.