Self-Averaged Scaling Limits for Random Parabolic Waves

一类抛物型移动边界问题的尺度不变性与迭代变换方法

Scaling Invariance and the Iterative Transformation Method for a Class of Parabolic Moving Boundary Problems

本文通过尺度不变性分析将一类抛物型移动边界问题简化为由常微分方程控制的自由边界问题，并应用迭代变换方法进行数值求解，得到与精确或近似解一致的结果。

Hydrodynamic limit of asymmetric exclusion processes under diffusive scaling in $d\ge 3$

We consider the asymmetric exclusion process. We start from a profile which is constant along the drift direction and prove that the density profile, under a diffusive rescaling of time, converges to the solution of a parabolic equation.

Crossover between parabolic and hyperbolic scaling, oscillatory modes and resonances near flocking

A stability and bifurcation analysis of a kinetic equation indicates that the flocking bifurcation of the two-dimensional Vicsek model exhibits an interplay between parabolic and hyperbolic behavior. For box sizes smaller than a certain large value, flocking appears continuously from a uniform disordered state at a critical value of the noise. Because of mass conservation, the amplitude equations describing the flocking state consist of a scalar equation for the density disturbance from the homogeneous particle density (particle number divided by box area) and a vector equation for a current d…

Renormalisation of parabolic stochastic PDEs

We give a survey of recent result regarding scaling limits of systems from statistical mechanics, as well as the universality of the behaviour of such systems in so-called cross-over regimes. It transpires that some of these universal objects are described by singular stochastic PDEs. We then give a survey of the recently developed theory of regularity structures which allows to build these objects and to describe some of their properties. We place particular emphasis on the renormalisation procedure required to give meaning to these equations. These are expanded notes of the 20th Takagi lectu…

The averaged characteristic polynomial for the Gaussian and chiral Gaussian ensembles with a source

In classical random matrix theory the Gaussian and chiral Gaussian random matrix models with a source are realized as shifted mean Gaussian, and chiral Gaussian, random matrices with real $(β= 1)$, complex ($β= 2)$ and real quaternion $(β= 4$) elements. We use the Dyson Brownian motion model to give a meaning for general $β> 0$. In the Gaussian case a further construction valid for $β> 0$ is given, as the eigenvalue PDF of a recursively defined random matrix ensemble. In the case of real or complex elements, a combinatorial argument is used to compute the averaged characteristic polynomial. Th…

Scaling Limits for Beam Wave Propagation in Atmospheric Turbulence

We prove the convergence of the solutions of the parabolic wave equation to that of the Gaussian white-noise model widely used in the physical literature. The random medium is isotropic and is assumed to have integrable correlation coefficient in the propagation direction. We discuss the limits of vanishing inner scale and divergent outer scale of the turbulent medium.