Fluctuation limits for **mean**-**field** interacting nonlinear Hawkes processes

**mean**spike-activity thereby characterizing the asymptotic fluctuations in terms of a stochastic Volterra integral equation. Expand abstract.

**mean**spike-activity thereby characterizing the asymptotic fluctuations in terms of a stochastic Volterra integral equation. Our approach differs from previous approaches in making use of the associated resolvent in order to represent the fluctutations as Skorokhod continuous mappings of weakly converging martingales. Since the Lipschitz properties of the resolvent are explicit, our analysis in principle also allows to derive approximation errors in terms of driving martingales.

8/10 relevant

arXiv

Beyond bogoliubov dynamics

**mean**

**field**, as well as the reduced densities of the N-body system to arbitrary precision, given only the knowledge of the two-point functions of a quasi-free state and the solution of the Hartree equation. Expand abstract.

**mean**

**field**scaling regime. We construct corrections to the Bogoliubov dynamics that approximate the true N-body dynamics in norm to arbitrary precision. The corrections are such that they can be explicitly computed in an N-independent way from the solutions of the Bogoliubov and Hartree equations and satisfy a generalized form of Wick's theorem. We determine the n-point correlation functions of the excitations around the

**mean**field, as well as the reduced densities of the N-body system to arbitrary precision, given only the knowledge of the two-point functions of a quasi-free state and the solution of the Hartree equation. In this way, the complex problem of computing all n-point correlation functions for an interacting N-body system is essentially reduced to the problem of solving the Hartree equation and the PDEs describing the two-point functions of the Bogoliubov time evolution, which are equations for a two-particle system.

5/10 relevant

arXiv

Orientation Dependence of the Magnetic Phase Diagram of Yb$_2$Ti$_2$O$_7$

**field**-temperature phase diagram. We find it to be richly anisotropic with a critical endpoint for $\vec{B}\,\parallel\,\langle 100\rangle$, while

**field**parallel to $\langle 110 \rangle$ and $\langle 111 \rangle$ enhances the critical temperature by up to a factor of two and shifts the onset of the

**field**-polarized state to finite

**fields**. Landau theory shows that Yb$_2$Ti$_2$O$_7$ in some ways is remarkably similar to pure iron. However, it also pinpoints anomalies that cannot be accounted for at the classical

**mean**-

**field**level including a dramatic enhancement of $T_{\mathrm{C}}$ and reentrant phase boundary by

**fields**with a component transverse to the easy axes, as well as the anisotropy of the upper critical

**field**in the quantum limit.

4/10 relevant

arXiv

**Mean** **field** theory for deep dropout networks: digging up gradient
backpropagation deeply

**mean**

**field**theory shows that the existence of depth scales that limit the maximum depth of signal propagation and gradient backpropagation. Expand abstract.

**mean**

**field**theory has been applied to the study of neural networks and has achieved a great deal of success. The theory has been applied to various neural network structures, including CNNs, RNNs, Residual networks, and Batch normalization. Inevitably, recent work has also covered the use of dropout. The

**mean**

**field**theory shows that the existence of depth scales that limit the maximum depth of signal propagation and gradient backpropagation. However, the gradient backpropagation is derived under the gradient independence assumption that weights used during feed forward are drawn independently from the ones used in backpropagation. This is not how neural networks are trained in a real setting. Instead, the same weights used in a feed-forward step needs to be carried over to its corresponding backpropagation. Using this realistic condition, we perform theoretical computation on linear dropout networks and a series of experiments on dropout networks. Our empirical results show an interesting phenomenon that the length gradients can backpropagate for a single input and a pair of inputs are governed by the same depth scale. Besides, we study the relationship between variance and

**mean**of statistical metrics of the gradient and shown an emergence of universality. Finally, we investigate the maximum trainable length for deep dropout networks through a series of experiments using MNIST and CIFAR10 and provide a more precise empirical formula that describes the trainable length than original work.

10/10 relevant

arXiv

Mean-**field** solution of the weak-strong cluster problem for quantum
annealing with stoquastic and non-stoquastic catalysts

**mean**-

**field**version as proposed by Albash [Phys. Rev. A 99 (2019) 042334] who showed by numerical diagonalization that non-stoquastic $XX$ interactions (non-stoquastic catalysts) remove the problematic first-order phase transition. We solve the problem exactly in the thermodynamic limit by analytical methods and show that the removal of the first-order transition is successfully achieved either by stoquastic or non-stoquastic $XX$ interactions depending on whether the $XX$ interactions are introduced within the weak cluster, within the strong cluster, or between them. We also investigate the case where the interactions between the two clusters are sparse, i.e. not of the

**mean**-

**field**all-to-all type. The results again depend on where to introduce the $XX$ interactions. We further analyze how inhomogeneous driving of the transverse

**field**affects the performance of the system without $XX$ interactions and find that inhomogeneity in the transverse

**field**removes the first-order transition if appropriately implemented.

10/10 relevant

arXiv

Magnetic phase transitions in quantum spin-orbital liquids

**field**that provides an approximate description of our model within

**mean**

**field**theory.

4/10 relevant

arXiv

An optimal semiclassical bound on certain commutators

**mean**-

**field**version of bounds introduced as an assumption by N. Benedikter, M. Porta and B. Schlein in a study of the

**mean**-

**field**evolution of a fermionic system.

4/10 relevant

arXiv

Small-scale spatial structure affects predator-prey dynamics and coexistence

**mean**-

**field**model predicts the coexistence of both species. Expand abstract.

**mean**-

**field**approximation, where interactions between individuals are assumed to occur in proportion to their average density. Such

**mean**-

**field**approximations amount to ignoring spatial structure. In this work, we consider an individual based model of a two-species community that is composed of consumers and resources. The model describes migration, predation, competition and dispersal of offspring, and explicitly gives rise to varying degrees of spatial structure. We compare simulation results from the individual based model with the solution of a classical

**mean**-

**field**approximation, and this comparison provides insight into how spatial structure can drive the system away from

**mean**-

**field**dynamics. Our analysis reveals that mechanisms leading to intraspecific clustering and interspecific segregation, such as short-range predation and short-range dispersal, tend to increase the size of the resource species relative to the

**mean**-

**field**prediction. We show that under certain parameter regimes these mechanisms lead to the extinction of consumers whereas the classical

**mean**-

**field**model predicts the coexistence of both species.

7/10 relevant

bioRxiv

Social Optima in **Mean** **Field** Linear-Quadratic-Gaussian Control with
Volatility Uncertainty

**mean**

**field**approximation, we design a set of decentralized strategies, which are further shown to be asymptotically social optimal by perturbation analysis. Expand abstract.

**mean**

**field**linear-quadratic-Gaussian (LQG) social optimum control with volatility-uncertain common noise. The diffusion terms in the dynamics of agents contain an unknown volatility process driven by a common noise. We apply a robust optimization approach in which all agents view volatility uncertainty as an adversarial player. Based on the principle of person-by-person optimality and a two-step-duality technique for stochastic variational analysis, we construct an auxiliary optimal control problem for a representative agent. Through solving this problem combined with a consistent

**mean**

**field**approximation, we design a set of decentralized strategies, which are further shown to be asymptotically social optimal by perturbation analysis.

10/10 relevant

arXiv

Strong Solutions of **Mean**-**Field** SDEs with irregular expectation
functional in the drift

**mean**-

**field**equations... Expand abstract.

**mean**-

**field**stochastic differential equations where the drift depends on the law in form of a Lebesgue integral with respect to the pushforward measure of the solution. We show existence and uniqueness of Malliavin differentiable strong solutions for irregular drift coefficients, which in particular include the case where the drift depends on the cumulative distribution function of the solution. Moreover, we examine the solution as a function in its initial condition and introduce sufficient conditions on the drift to guarantee differentiability. Under these assumptions we then show that the Bismut-Elworthy-Li formula proposed in Bauer et al. (2018) holds in a strong sense, i.e. we give a probabilistic representation of the strong derivative with respect to the initial condition of expectation functionals of strong solutions to our type of

**mean**-

**field**equations in one-dimension.

10/10 relevant

arXiv