KPZ in a multidimensional random geometry of multiplicative cascades

Fluctuations of two-dimensional stochastic heat equation and KPZ equation in subcritical regime for general initial conditions

The solution of Kardar-Parisi-Zhang equation (KPZ equation) is solved formally via Cole-Hopf transformation $h=\log u$, where $u$ is the solution of multiplicative stochastic heat equation(SHE). In earlier works by Chatterjee and Dunlap, Caravenna, Sun, and Zygouras, and Gu, they consider the solution of two dimensional KPZ equation via the solution $u_\varepsilon$ of SHE with flat initial condition and with noise which is mollified in space on scale in $\varepsilon$ and its strength is weakened as $β_\varepsilon=\hatβ \sqrt{\frac{2π\varepsilon}{-\log \varepsilon}}$, and they prove that when $…

Multiplicative stochastic heat equations on the whole space

We carry out the construction of some ill-posed multiplicative stochastic heat equations on unbounded domains. The two main equations our result covers are, on the one hand the parabolic Anderson model on $\mathbf{R}^3$, and on the other hand the KPZ equation on $\mathbf{R}$ via the Cole-Hopf transform. To perform these constructions, we adapt the theory of regularity structures to the setting of weighted Besov spaces. One particular feature of our construction is that it allows one to start both equations from a Dirac mass at the initial time.

Positive random walks and an identity for half-space SPDEs

The purpose of this article is threefold. First, we introduce a new type of boundary condition for the multiplicative-noise stochastic heat equation on the half space. This is essentially a Dirichlet boundary condition but with a nontrivial normalization near the boundary which leads to inhomogeneous transition densities (roughly, those of a Brownian \textit{meander}) within the associated chaos series. Secondly, we prove a new convergence result of the directed-polymer partition function in an octant to the multiplicative stochastic heat equation with this type of boundary condition, which in…

Universality for multiplicative statistics of Hermitian random matrices and the integro-differential Painlevé II equation

We study multiplicative statistics for the eigenvalues of unitarily-invariant Hermitian random matrix models. We consider one-cut regular polynomial potentials and a large class of multiplicative statistics. We show that in the large matrix limit several associated quantities converge to limits which are universal in both the potential and the family of multiplicative statistics considered. In turn, such universal limits are described by the integro-differential Painlevé II equation, and in particular they connect the random matrix models considered with the narrow wedge solution to the KPZ eq…

Universal aspects of curved, flat & stationary-state Kardar-Parisi-Zhang statistics

Motivated by the recent exact solution of the {\it stationary-state} Kardar-Parisi-Zhang (KPZ) statistics by Imamura & Sasamoto (Phys. Rev. Lett. {\bf 108}, 190603 (2012)), as well as a precursor experimental signature unearthed by Takeuchi (Phys. Rev. Lett. {\bf 110}, 210604 (2013)), we establish here the universality of these phenomena, examining scaling behaviors of directed polymers in a random medium, the stochastic heat equation with multiplicative noise, and kinetically roughened KPZ growth models. We emphasize the value of cross KPZ-Class universalities, revealing crossover effects of …

Multidimensional Lambert-Euler inversion and vector-multiplicative coalescent processes

In this paper we show the existence of the minimal solution to the multidimensional Lambert-Euler inversion, a multidimensional generalization of $[-e^{-1} ,0)$ branch of Lambert W function $W_0(x)$. Specifically, for a given nonnegative irreducible symmetric matrix $V \in \mathbb{R}^{k \times k}$, we show that for ${\bf u}\in(0,\infty)^k$, if equation $$y_j \exp\{-{\bf e}_j^T V {\bf y} \} = u_j ~~~~~~\forall j=1,...,k,$$ has at least one solution, it must have a minimal solution ${\bf y}^*$, where the minimum is achieved in all coordinates $y_j$ simultaneously. Moreover, such ${\bf y}^*$ is t…

Constructing a solution of the $(2+1)$-dimensional KPZ equation

The $(d+1)$-dimensional KPZ equation is the canonical model for the growth of rough $d$-dimensional random surfaces. A deep mathematical understanding of the KPZ equation for $d=1$ has been achieved in recent years, and the case $d\ge 3$ has also seen some progress. The most physically relevant case of $d=2$, however, is not very well-understood mathematically, largely due to the renormalization that is required: in the language of renormalization group analysis, the $d=2$ case is neither ultraviolet superrenormalizable like the $d=1$ case nor infrared superrenormalizable like the $d\ge 3$ cas…

随机环境中随机游走的中等偏差的普适 KPZ 涨落

Universal KPZ Fluctuations for Moderate Deviations of Random Walks in Random Environments

本文研究了随机环境中随机游走（RWRE）模型转移概率的中等偏差，证明了其矩向乘性噪声随机热方程（SHE）及 KPZ 方程的普适收敛性，并基于此引入了极端扩散系数。

KPZ极限的层次结构源于随机介质中的定向随机游走模型

Hierarchy of KPZ limits arising from directed random walk models in random media

本文研究了随机介质中定向随机游走模型的KPZ极限层次结构，通过削弱假设条件，证明了SHE在多种模型中的出现，并揭示了中等偏差指数的层次结构。

Green function of a random walk in a cone

This paper studies the asymptotic behavior of the Green function of a multidimensional random walk killed when leaving a convex cone with smooth boundary. Our results imply uniqueness, up to a multiplicative factor, of the positive harmonic function for the killed random walk.

一般初始数据下SHE的李雅普诺夫指数

Lyapunov exponents of the SHE for general initial data

本文提出了一种精确计算(1+1)维随机热方程(SHE)李雅普诺夫指数的方法，适用于有界确定性正初始数据和平稳初始数据，并由此推导出一般初始数据下KPZ方程上尾大偏差速率函数的精确公式。

On fixed points of a generalized multidimensional affine recursion

Let $G$ be a multiplicative subsemigroup of the general linear group $\Gl(\mathbb{R}^d)$ which consists of matrices with positive entries such that every column and every row contains a strictly positive element. Given a $G$--valued random matrix $A$, we consider the following generalized multidimensional affine equation R\stackrel{\mathcal{D}}{=}\sum_{i=1}^N A_iR_i+B, where $N\ge2$ is a fixed natural number, $A_1,...,A_N$ are independent copies of $A$, $B\in\mathbb{R}^d$ is a random vector with positive entries, and $R_1,...,R_N$ are independent copies of $R\in\mathbb{R}^d$, which have also p…

Critical Mandelbrot Cascades

We study Mandelbrot's multiplicative cascade measures at the critical temperature. As has been recently shown by Barral, Rhodes and Vargas (arXiv:1203.5445), an appropriately normalized sequence of cascade measures converges weakly in probability to a nontrivial limit measure. We prove that these limit measures have no atoms and give bounds for the modulus of continuity of the cumulative distribution of the measure. Using the earlier work of Barral and Seuret (2007), we compute the multifractal spectrum of the measures. We also extend the result of Benjamini and Schramm (2009), in which the KP…

乘积的右尾渐进行为

Right-tail asymptotics for products of independent normal random variables

研究独立正态变量乘积的右尾概率渐进行为，提出显式渐近公式及修正项，采用边界 saddle-point/Laplace 方法证明。

度修正随机块模型：游走表示

Degree corrected stochastic block model: excursion representation

本文分析了度修正随机块模型（DCSBM）的连通分量，提出一种基于游走表示和弱收敛的随机场编码方法，将多维首中时间的极小化问题等价转化为单一实值随机过程问题，并引入了新型类复合算子。