A **mean** **field** approach to model flows of agents with path preferences
over a network

**mean**

**field**equilibrium. Expand abstract.

**mean**

**field**approach where the standard forward backward system of equations is also intertwined with a path preferences dynamics. The path preferences are influenced by the congestion status on the whole network as well as the possible hassle of being forced to run during the tour. We prove the existence of a

**mean**

**field**equilibrium as a fixed point, over a suitable set of time-varying distributions, of a map obtained as a limit of a sequence of approximating functions. Then, a bi-level optimization problem is formulated for an external controller who aims to induce a specific

**mean**

**field**equilibrium.

10/10 relevant

arXiv

On the diffusive-**mean** **field** limit for weakly interacting diffusions
exhibiting phase transitions

**mean**

**field**limit and derive optimal rates of convergence in relative entropy of the Gibbs measure to the (unique) limit of the mean field energy below the critical temperature. Expand abstract.

**mean**

**field**and diffusive(homogenisation)limits. In particular, we show that these two limits do not commute if the

**mean**

**field**system constrained to the torus undergoes a phase transition, that is to say if it admits more than one steady state. A typical example of such a system on the torus is given by the noisy Kuramoto model of

**mean**

**field**plane rotators. As a by-product of our main results, we also analyse the energetic consequences of the central limit theorem for fluctuations around the

**mean**

**field**limit and derive optimal rates of convergence in relative entropy of the Gibbs measure to the (unique) limit of the

**mean**

**field**energy below the critical temperature.

10/10 relevant

arXiv

Linear-Quadratic **Mean**-**Field** Reinforcement Learning: Convergence of
Policy Gradient Methods

**mean**-

**field**linear-quadratic setting. Expand abstract.

**mean**

**field**control problems in discrete time, which can be viewed as Markov decision processes for a large number of exchangeable agents interacting in a

**mean**

**field**manner. Such problems arise, for instance when a large number of robots communicate through a central unit dispatching the optimal policy computed by minimizing the overall social cost. An approximate solution is obtained by learning the optimal policy of a generic agent interacting with the statistical distribution of the states of the other agents. We prove rigorously the convergence of exact and model-free policy gradient methods in a

**mean**-

**field**linear-quadratic setting. We also provide graphical evidence of the convergence based on implementations of our algorithms.

10/10 relevant

arXiv

Role of the **Mean**-**field** in Bloch Oscillations of a Bose-Einstein
Condensate in an Optical Lattice and Harmonic Trap

**mean**

**field**becomes large, soliton and vortex structures appear in the condensate wavefunction. Expand abstract.

**mean**

**field**plays an important role in the Bloch-like oscillations that occur after sufficiently large initial displacement. We find that a moderate

**mean**

**field**significantly suppresses the dispersion of the condensate in momentum space. When the

**mean**

**field**becomes large, soliton and vortex structures appear in the condensate wavefunction.

10/10 relevant

arXiv

Strong ergodicity breaking in aging of **mean** **field** spin glasses

**mean**-

**field**spin glass models has achieved a fundamental role, thanks to the asymptotic analytic solution found by Cugliandolo and Kurchan. However this solution is based on assumptions (e.g. the weak ergodicity breaking hypothesis) which have never been put under a strong test until now. In the present work we present the results of an extraordinary large set of numerical simulations of the prototypical

**mean**-

**field**spin glass models, namely the Sherrington-Kirkpatrick and the Viana-Bray models. Thanks to a very intensive use of GPUs, we have been able to run the latter model for more than $2^{64}$ spin updates and thus safely extrapolate the numerical data both in the thermodynamical limit and in the large times limit. The measurements of the two-times correlation functions in isothermal aging after a quench from a random initial configuration to a temperature $T<T_c$ provides clear evidence that, at large times, such correlations do not decay to zero as expected by assuming weak ergodicity breaking. We conclude that strong ergodicity breaking takes place in

**mean**-

**field**spin glasses aging dynamics which, asymptotically, takes place in a confined configurational space. Theoretical models for the aging dynamics need to be revised accordingly.

10/10 relevant

arXiv

**Mean** **Field** approach to stochastic control with partial information

**Mean**

**Field**Control Theory leads to very complex equations, like the Master equation, which is a nonlinear infinite dimensional P. Expand abstract.

**Mean**

**Field**Control theory. Since

**Mean**

**Field**Control theory is much posterior to the development of Stochastic Control with partial information, the tools, techniques, and concepts obtained in the last decade, for

**Mean**

**Field**Games and

**Mean**

**field**type Control theory, have not been used for the control of Zakai equation. Our objective is to connect the two theories. We get the power of new tools, and we get new insights for the problem of stochastic control with partial information. For

**mean**

**field**theory, we get new interesting applications, but also new problems. Indeed,

**Mean**

**Field**Control Theory leads to very complex equations, like the Master equation, which is a nonlinear infinite dimensional P.D.E., for which general theorems are hardly available, although active research in this direction is performed. Direct methods are useful to obtain regularity results. We will develop in detail the LQ regulator problem, but since we cannot just consider the Gaussian case, well-known results, such as the separation principle is not available. An important result is available in the literature, due to A. Makowsky. It describes the solution of Zakai equation for linear systems with general initial condition (non-gaussian). We show that the separation principle can be extended for quadratic pay-off functionals, but the Kalman filter is much more complex than in the gaussian case. Finally we compare our work to the work of E. Bandini et al. and we show that the example E. Bandini et al. provided does not cover ours. Our system remains nonlinear in their setting.

10/10 relevant

arXiv

Superheavy nuclei in microscopic collective Hamiltonian approach: the impact of beyond **mean** **field** correlations on the ground state and fission properties

**mean**

**field**effects on the ground state and fission properties of superheavy nuclei has been investigated in a five-dimensional collective Hamiltonian based on covariant density functional theory. The inclusion of dynamical correlations reduces the impact of the $Z=120$ shell closure and induces substantial collectivity for the majority of the $Z=120$ nuclei which otherwise are spherical at the

**mean**

**field**level (as seen in the calculations with the PC-PK1 functional). Thus, they lead to a substantial convergence of the predictions of the functionals DD-PC1 and PC-PK1 which are different at the

**mean**

**field**level. On the contrary, the predictions of these two functionals remain distinctly different for the $N=184$ nuclei even when dynamical correlations are included. These nuclei are mostly spherical (oblate) in the calculations with PC-PK1 (DD-PC1). Our calculations for the first time reveal significant impact of dynamical correlations on the heights of inner fission barriers of superheavy nuclei with soft potential energy surfaces, the minimum of which at the

**mean**

**field**level is located at spherical shape. These correlations affect the fission barriers of the nuclei, which are deformed in the ground state at the

**mean**

**field**level, to a lesser degree.

10/10 relevant

arXiv

Low-Energy Effective Theory at a Quantum Critical Point of the
Two-Dimensional Hubbard Model: **Mean**-**Field** Analysis

**mean**-

**field**calculations. The focus falls on Van Hove filling and the the hopping amplitude t'/t=0.341. The fRG data suggest a quantum critical point (QCP) in this region and in its vicinity a singular fermionic self-energy, Im $\Sigma(\omega)/\omega \sim |\omega|^{-\gamma}$ with $\gamma\approx 0.26$. Here we start a more detailed investigation of this QCP using a bosonic formulation for the effective action, where the bosons couple to the order parameter

**fields**. To this end, we use the channel decomposition of the fermionic effective action developed in [Phys. Rev. B 79, 195125 (2009)], which allows to perform Hubbard-Stratonovich transformations for all relevant order parameter

**fields**at any given energy scale. We stop the flow at a scale where the correlations of the order parameter

**field**are already pronounced, but the flow is still regular, and derive the effective boson theory. It contains d-wave superconducting, magnetic, and density-density interactions. We analyze the resulting phase diagram in the

**mean**-

**field**approximation. We show that the singular fermionic self-energy suppresses gap formation both in the superconducting and magnetic channel already at the

**mean**-

**field**level, thus rounding a first-order transition (without self-energy) to a quantum phase transition (with self-energy). We give a simple effective model that shows the generality of this effect. In the two-dimensional Hubbard model, the effective density-density interaction is peaked at a nonzero frequency, so that solving the

**mean**-

**field**equations already involves a functional equation instead of simply a matrix equation (on a technical level, similar to incommensurate phases). Within a certain approximation, we show that such an interaction leads to a short quasiparticle lifetime.

10/10 relevant

arXiv

Convergence Analysis of Machine Learning Algorithms for the Numerical
Solution of **Mean** **Field** Control and Games: II -- The Finite Horizon Case

**mean**

**field**control problems, the second is more general and can also be applied to the FBSDE arising in the theory of mean field games. Expand abstract.

**mean**

**field**control problem. To provide a guarantee on how our numerical schemes approximate the solution of the original

**mean**

**field**control problem, we introduce a new optimization problem, directly amenable to numerical computation, and for which we rigorously provide an error rate. Several numerical examples are provided. Both methods can easily be applied to problems with common noise, which is not the case with the existing technology. Furthermore, although the first approach is designed for

**mean**

**field**control problems, the second is more general and can also be applied to the FBSDE arising in the theory of

**mean**

**field**games.

10/10 relevant

arXiv

**Mean** **field** approximation of a heterogeneous population of plants in competition

**mean**-

**field**distribution, consisting in a semi-Lagrangian scheme with an interpolation step using Gaussian process regression, is illustrated for a heterogeneous population model representing plants in competition for light. Expand abstract.

**field**are still poorly understood. However, they explain a large part of the heterogeneity in a

**field**and may have longer-term consequences, especially in mixed stands. Modeling can help to better understand these phenomena but requires simulating the interactions between different individuals. In the case of large populations, assessing the parameters of a heterogeneous population model from experimental data is intractable computationally. This paper investigates the

**mean**-

**field**approximation of large dynamical systems with random initial conditions and individual parameters, and with interaction being represented by pairwise potentials between individuals. Under this approximation, each individual is in interaction with an infinitely-crowded population, summarized by a probability measure, the

**mean**-

**field**limit distribution, being itself the weak solution of a non-linear hyperbolic partial differential equation. In particular, the phenomenon of chaos propagation implies that the individuals are independent asymptotically when the size of the population tends towards infinity. This result provides perspectives for a possible simplification of the inference problem. The simulation of the

**mean**-

**field**distribution, consisting in a semi-Lagrangian scheme with an interpolation step using Gaussian process regression, is illustrated for a heterogeneous population model representing plants in competition for light.

10/10 relevant

arXiv