Mean-**field** reflected backward stochastic differential equations

**mean**-

**field**type, where the

**mean**-

**field**interaction in terms of the distribution of the $Y$-component of the solution enters in both the driver and the lower obstacle. We consider in details the case where the lower obstacle is a deterministic function of $(Y,\E[Y])$ and discuss the more general dependence on the distribution of $Y$. Under mild Lipschitz and integrability conditions on the coefficients, we obtain the well-posedness of such a class of equations. Under further monotonicity conditions, we show convergence of the standard penalization scheme to the solution of the equation, which hence satisfies a minimality property. This class of equations is motivated by applications in pricing life insurance contracts with surrender options.

10/10 relevant

arXiv

Structured **Mean**-**field** Variational Inference and Learning in
Winner-take-all Spiking Neural Networks

**mean**-

**field**variational inference by learning both the feedback and feedforward weights separately. Expand abstract.

**mean**-

**field**variational inference and learning, on hierarchical directed probabilistic graphical models with discrete random variables. In these models, we do away with symmetric synaptic weights previously assumed for \emph{unstructured}

**mean**-

**field**variational inference by learning both the feedback and feedforward weights separately. The resulting online learning rules take the form of an error-modulated local Spike-Timing-Dependent Plasticity rule. Importantly, we consider two types of WTA circuits in which only one neuron is allowed to fire at a time (hard WTA) or neurons can fire independently (soft WTA), which makes neurons in these circuits operate in regimes of temporal and rate coding respectively. We show how the hard WTA circuits can be used to perform Gibbs sampling whereas the soft WTA circuits can be used to implement a message passing algorithm that computes the marginals approximately. Notably, a simple change in the amount of lateral inhibition realizes switching between the hard and soft WTA spiking regimes. Hence the proposed network provides a unified view of the two previously disparate modes of inference and coding by spiking neurons.

10/10 relevant

arXiv

**Mean** number and correlation function of critical points of isotropic
Gaussian **fields**

**field**with real values. In a first part we study the

**mean**number of critical points of X with index k using random matrices tools. We obtain an exact expression for the probability density of the eigenvalue of rank k of a N-GOE matrix. We deduce some exact expressions for the

**mean**number of critical points with a given index. In a second part we study attraction or repulsion between these critical points. A measure is the correlation function. We prove attraction between critical points when N > 2, neutrality for N = 2 and repulsion for N = 1. The attraction between critical points that occurs when the dimension is greater than two is due to critical points with adjacent indexes. A strong repulsion between maxima and minima is observed. The correlation function between maxima (or minima) depends on the dimension of the ambient space.

4/10 relevant

arXiv

Some aspects of the inertial spin model for flocks and related kinetic equations

**mean**-

**field**limit of the model in the absence of thermal noise, and explore its macroscopic behavior by analyzing the mono-kinetic solutions. Expand abstract.

**mean**-

**field**limit is described by a non-linear Fokker-Planck equation, whose equilibria are fully characterized. Finally, in the case of non-constant interactions, we derive the kinetic equation for the

**mean**-

**field**limit of the model in the absence of thermal noise, and explore its macroscopic behavior by analyzing the mono-kinetic solutions.

4/10 relevant

arXiv

Superradiant optomechanical phases of cold atomic gases in optical resonators

**field**and atomic degrees of freedom we derive a

**mean**-

**field**model that allows us to determine a threshold temperature, above which thermal fluctuations suppress superradiant emission. We then analyze the dynamics of superradiant emission when the motion is described by a

**mean**-

**field**model. In the semiclassical regime and below the threshold temperature we observe that the emitted light can be either coherent or chaotic, depending on the incoherent pump rate. We then analyze superradiant emission from an ideal Bose gas at zero temperature when the superradiant decay rate $\Lambda$ is of the order of the recoil frequency $\omega_R$. We show that the quantized exchange of mechanical energy between the atoms and the

**field**gives rise to a threshold, $\Lambda_c$, below which superradiant emission is damped down to zero. When $\Lambda>\Lambda_c$ superradiant emission is accompanied by the formation of matter-wave gratings diffracting the emitted photons. The stability of these gratings depends on the incoherent pump rate $w$ with respect to a second threshold value $w_c$. For $w>w_c$ the gratings are stable and the system achieves stationary superradiance. Below this second threshold the coupled dynamics becomes chaotic. We characterize the dynamics across these two thresholds and show that the three phases we predict (incoherent, coherent, chaotic) can be revealed via the coherence properties of the light at the cavity output.

4/10 relevant

arXiv

Inter-component asymmetry and formation of quantum droplets in quasi-one-dimensional binary Bose gases

**mean**-

**field**(MF) energy and beyond-MF correction in a weakly interacting binary Bose gas. Expand abstract.

**mean**-

**field**(MF) energy and beyond-MF correction in a weakly interacting binary Bose gas. We analyze generation and stability of the droplets in the quasi-one-dimensional model, in both symmetric and asymmetric settings with respect to the two components. Asymmetry is represented by unequal densities or unequal intracomponent interaction strengths. We find that the latter factor gives rise to an asymmetric droplet, characterized by the population imbalance between the components. Further, we address the formation of asymmetric droplets in both the symmetric and asymmetric systems through the modulational instability induced by the beyond-MF term. The solutions confirm the stability of known symmetric solutions against symmetry-breaking perturbations.

7/10 relevant

arXiv

Metal-Insulator and Magnetic Phase Diagram of Ca$_2$RuO$_4$ from
Auxiliary **Field** Quantum Monte Carlo and Dynamical **Mean** Field Theory

**Field**Quantum Monte Carlo and Dynamical

**Mean**

**Field**Theory, to determine the low-temperature phase diagram of Ca$_2$RuO$_4$. Both methods predict a low temperature, pressure-driven metal-insulator transition accompanied by a ferromagnetic-antiferromagnetic transition. The properties of the ferromagnetic state vary non-monotonically with pressure and are dominated by the ruthenium $d_{xy}$ orbital, while the properties of the antiferromagnetic state are dominated by the $d_{xz}$ and $d_{yz}$ orbitals. Differences of detail in the predictions of the two methods are analyzed. This work is theoretically important as it presents the first application of the Auxiliary

**Field**Quantum Monte Carlo method to an orbitally-degenerate system with both Mott and Hunds physics, and provides an important comparison of the Dynamical

**Mean**

**Field**and Auxiliary

**Field**Quantum Monte Carlo methods.

10/10 relevant

arXiv

Does one-step replica symmetry breaking occur in p-spin Ising models
outside **mean**-**field** theory?

**mean**-

**field**results and have features more like those which would arise with full replica symmetry breaking (FRSB). To help understand how this might come about we have studied in the fully connected $p$-spin model the state of two-step replica symmetry breaking (2RSB). It has a free energy degenerate with that of 1RSB, but the weight of the additional peak in $P(q)$ vanishes. We expect that the state with full replica symmetry breaking (FRSB) is also degenerate with that of 1RSB. We suggest that finite size effects will give a non-vanishing weight to the FRSB features, as also will fluctuations about the

**mean**-

**field**solution. Our conclusion is that outside the fully connected model in the thermodynamic limit, FRSB is to be expected rather than 1RSB.

10/10 relevant

arXiv

Statistical Inference in **Mean**-**Field** Variational Bayes

**mean**-

**field**variational inference. Expand abstract.

**mean**-

**field**variational inference for approximating posterior distributions in complex Bayesian models that may involve latent variables. We show that the

**mean**-

**field**approximation to the posterior can be well-approximated relative to the Kullback-Leibler divergence discrepancy measure by a normal distribution whose center is the maximum likelihood estimator (MLE). In particular, our results imply that the center of the

**mean**-

**field**approximation matches the MLE up to higher-order terms and there is essentially no loss of efficiency in using it as a point estimator for the parameter in any regular parametric model with latent variables. We also propose a new class of variational weighted likelihood bootstrap (VWLB) methods for quantifying the uncertainty in the

**mean**-

**field**variational inference. The proposed VWLB can be viewed as a new sampling scheme that produces independent samples for approximating the posterior. Comparing with traditional sampling algorithms such Markov Chain Monte Carlo, VWLB can be implemented in parallel and is free of tuning.

10/10 relevant

arXiv

Mean-**field** tricritical polymers

**mean**-

**field**random walk model of a polymer density transition, and clarify the nature of the density transition in this context. We consider a continuous-time random walk model on the complete graph, in the limit as the number of vertices $N$ in the graph grows to infinity. The walk has a repulsive self-interaction, as well as a competing attractive self-interaction whose strength is controlled by a parameter $g$. A chemical potential $\nu$ controls the walk length. We determine the phase diagram in the $(g,\nu)$ plane, as a model of a density transition for a single linear polymer chain. A dilute phase (walk of bounded length) is separated from a dense phase (walk of length of order $N$) by a phase boundary curve. The phase boundary is divided into two parts, corresponding to first-order and second-order phase transitions, with the division occurring at a tricritical point. The proof uses a supersymmetric representation for the random walk model, followed by a single block-spin renormalisation group step to reduce the problem to a 1-dimensional integral, followed by application of the Laplace method for an integral with a large parameter.

10/10 relevant

arXiv