Numerical results for the Directed Polymer model in 1+4 dimensions in various types of disorder are presented. The results are obtained for system size considerably larger than that considered previously. For the extreme strong disorder case (Min-Max system), associated with the Directed Percolation model, the expected value of the meandering exponent, zeta = 0.5 is clearly revealed, with very week finite size effects. For the week disorder case, associated with the KPZ equation, finite size effects are stronger, but the value of seta is clearly seen in the vicinity of 0.57. In systems with "strong disorder" it is expected that the system will cross over sharply from Min-Max behavior at short chains to weak disorder behavior at long chains. This is indeed what we find. These results indicate that 1+4 is not the Upper Critical Dimension (UCD) in the week disorder case, and thus 4+1 does not seem to be the upper critical dimension for the KPZ equation.
Aiming to investigate the upper critical dimension, $d_u$, of the KPZ class, in [EPL 103 (2013) 10005] some growth models were numerically analyzed using Cayley trees (CTs) as substrates, as a way to access their behavior in the infinite-dimensional limit, and some unexpected results were reported: logarithmic roughness scaling, differing for EW and KPZ models (indicating that even at $d=\infty$ the KPZ nonlinearity is still relevant); beyond asymptotically rough EW surfaces above the upper critical dimension of the EW class. Motivated by these strange findings, I revisit these growth models h…
Transition from KPZ to Tilted Interface Critical Behavior in a Solvable Asymmetric Avalanche Model
我们使用离散时间公式来研究非对称雪崩过程[Phys. Rev. Lett. Vol.87，084301（2001）]，并获得了作为取决于参数$μ$和$α$的倾倒概率的函数的颗粒的平均雪崩尺寸的精确表达式。通过将模型映射到驱动接口问题的临界线以下和以上，我们展示了如何不同的制度的雪崩可能会导致不同类型的临界界面行为的特点是退火或淬火无序并获得确切的相关临界指数违反了一个众所周知的标度关系时，$α\ne 2$。
On the upper critical dimension of the KPZ universality class: KPZ and related equations on a fully connected graph
Universidad de Extremadura（埃斯特雷马杜拉大学）；Instituto de Computación Científica Avanzada de Extremadura (ICCAEx), Universidad de Extremadura（埃斯特雷马杜拉先进计算研究所 (ICCAEx)）；Universidad Carlos III de Madrid（马德里卡洛斯三世大学）
We investigate the thermodynamic uncertainty relation for the $(1+1)$ dimensional Kardar-Parisi-Zhang equation on a finite spatial interval. In particular, we extend the results for small coupling strengths obtained previously to large values of the coupling parameter. It will be shown that, due to the scaling behavior of the KPZ equation, the TUR product displays two distinct regimes which are separated by a critical value of an effective coupling parameter. The asymptotic behavior below and above the critical threshold is explored analytically. For small coupling, we determine this product p…
We investigate the universal behavior of the Kardar-Parisi-Zhang (KPZ) equation with temporally correlated noise. The presence of time correlations in the microscopic noise breaks the statistical tilt symmetry, or Galilean invariance, of the original KPZ equation with delta-correlated noise (denoted SR-KPZ). Thus it is not clear whether the KPZ universality class is preserved in this case. Conflicting results exist in the literature, some advocating that it is destroyed even in the limit of infinitesimal temporal correlations, while others find that it persists up to a critical range of such c…
We establish the first tight bound on the lower tail probability of the half-space KPZ equation with Neumann boundary parameter $A = -1/2$ and narrow-wedge initial data. When the tail depth is of order $T^{2/3}$, the lower bound demonstrates a crossover between a regime of super-exponential decay with exponent $\frac{5}{2}$ (and leading pre-factor $\frac{2}{15 π}T^{1/3}$) and a regime with exponent $3$ (and leading pre-factor $\frac{1}{24}$); the upper bound demonstrates a crossover between a regime with exponent $\frac{3}{2}$ (and arbitrarily small pre-factor) and a regime with exponent $3$ (…
For each $t\geq 1$ we construct an $\mathbf{N}$-indexed ensemble of random continuous curves with three properties: 1. The lowest indexed curve is distributed as the time $t$ Hopf-Cole solution to the Kardar-Parisi-Zhang (KPZ) stochastic PDE with narrow wedge initial data; 2. The entire ensemble satisfies a resampling invariance which we call the $\mathbf{H}$-Brownian Gibbs property (with $\mathbf{H}(x)=e^{x}$); 3. Increments of the lowest indexed curve, when centered by $-t/24$ and scaled down vertically by $t^{1/3}$ and horizontally by $t^{2/3}$, remain uniformly absolutely continuous (i.e. …
The statistics of the fluctuations of quantum many-body systems are highly revealing of their nature. In driven-dissipative systems displaying macroscopic quantum coherence, as exciton polariton condensates under incoherent pumping, the phase dynamics can be mapped to the stochastic Kardar-Parisi-Zhang (KPZ) equation. However, in two dimensions (2D), it was theoretically argued that the KPZ regime may be hindered by the presence of vortices, and a non-equilibrium BKT behavior was reported close to condensation threshold. We demonstrate here that, when a discretized 2D polariton system is consi…
We present an exact solution for the height distribution of the KPZ equation at any time $t$ in a half space with flat initial condition. This is equivalent to obtaining the free energy distribution of a polymer of length $t$ pinned at a wall at a single point. In the large $t$ limit a binding transition takes place upon increasing the attractiveness of the wall. Around the critical point we find the same statistics as in the Baik-Ben--Arous-Péché transition for outlier eigenvalues in random matrix theory. In the bound phase, we obtain the exact measure for the endpoint and the midpoint of the…
Surface growth governed by the Kardar-Parisi-Zhang (KPZ) equation in dimensions higher than two undergoes a roughening transition from smooth to rough phases with increasing the nonlinearity. It is also known that the KPZ equation can be mapped onto quantum mechanics of attractive bosons with a contact interaction, where the roughening transition corresponds to a binding transition of two bosons with increasing the attraction. Such critical bosons in three dimensions actually exhibit the Efimov effect, where a three-boson coupling turns out to be relevant under the renormalization group so as …
We revisit the Lieb-Liniger model for $n$ bosons in one dimension with attractive delta interaction in a half-space $\mathbb{R}^+$ with diagonal boundary conditions. This model is integrable for arbitrary value of $b \in \mathbb{R}$, the interaction parameter with the boundary. We show that its spectrum exhibits a sequence of transitions, as $b$ is decreased from the hard-wall case $b=+\infty$, with successive appearance of boundary bound states (or boundary modes) which we fully characterize. We apply these results to study the Kardar-Parisi-Zhang equation for the growth of a one-dimensional …
Inspired by the recent results on totally asymmetric simple exclusion processes on a periodic lattice with short-ranged quenched hopping rates [A. Haldar, A. Basu, Phys Rev Research 2, 043073 (2020)], we study the universal scaling properties of the Kardar-Parisi-Zhang (KPZ) equation with short-ranged quenched columnar disorder in general d-dimensions. We show that there are generic propagating modes in the system that have their origin in the quenched disorder and make the system anisotropic. We argue that the presence of the propagating modes actually make the effects of the quenched disorde…
Local average height distribution of fluctuating interfaces
生长表面的高度波动可以用有限时刻空间点上高度的概率分布来表征。近年来，关于$1+1$维Kardar-Parisi-Zhang（KPZ）方程的这个量的研究取得了令人瞩目的进展.在这里，我们注意到，在临界尺寸或以上，有限时间的一点高度分布在广泛的线性表面生长模型中是不好定义的，除非模型在小尺度下正则化。通过依赖于系统的小尺度截断的正则化导致普适性的部分损失。作为一种可能的替代方案，我们引入了\emph{local average height}。对于线性模型，这个量的概率密度在任何维度上都是明确定义的。这些模型的弱噪声理论（WNT）产生的“最佳路径”的界面上的局部平均高度的非平衡波动的条件。作为一个例子，我们考虑保守的爱德华-威尔金森（EW）方程，其中，没有正则化，有限时间的一点高度分布在所有的物理维度是不明确的。我们还确定了一个密切相关的问题的有限时间\emph{height-difference}分布的非保守EW方程在$1+1$维的界面的最佳路径。最后，我们讨论了$2+1$维（非正则化）KPZ方程中有限时间单点高度分布的UV突变。
Dynamical Renormalization Group Study for a Class of Non-local Interface Equations
我们提供了一个详细的动态重整化群研究一类随机方程，描述非保守的非局部相互作用介导的界面生长。我们明确地考虑了形态稳定的情况和研究较少的情况，其中发生了斑图形成，其中平坦表面对周期性扰动是线性不稳定的。我们发现，后者导致非平凡的标度行为在适当的参数范围内，当结合Kardar-Parisi-Zhang（KPZ）的非线性，但不对应于KPZ普适类。这种新的渐近行为的特点是两个标度律，固定的临界指数的尺寸无关的值，同意从数值模拟和实验系统以前的报告。我们表明，精确形式的线性稳定化条款不修改这些方程的流体动力学行为。的比例尺法，通常与伽利略不变性，示出来自顶点取消发生（至少一个循环顺序）的任何选择的线性项中的运动方程，是独立的表面上的形态稳定性，因此推广这一众所周知的属性的KPZ方程。此外，该论证也适用于其他系统，如Lai-Das Sarma-Villain方程，其中顶点消去已知{\em not to}意味着方程的相关对称性。
