We define a new class of random walk processes which maximize entropy. This maximal entropy random walk is equivalent to generic random walk if it takes place on a regular lattice, but it is not if the underlying lattice is irregular. In particular, we consider a lattice with weak dilution. We show that the stationary probability of finding a particle performing maximal entropy random walk localizes in the largest nearly spherical region of the lattice which is free of defects. This localization phenomenon, which is purely classical in nature, is explained in terms of the Lifshitz states of a certain random operator.
Maximum entropy random walks (MERWs) are maximally dispersing and play a key role in optimizing information spreading in various contexts. However, building MERWs comes at the cost of knowing beforehand the global structure of the network, a requirement that makes them totally inadequate in real case scenarios. Here, we propose an adaptive random walk (ARW), which instead maximizes dispersion by updating its transition rule on the local information collected while exploring the network. We show how to derive ARW via a large-deviation representation of MERW and study its dynamics on synthetic a…
Comment on "Localization Transition of Biased Random Walks on Random Networks"
苏和Grassberger研究[物理学评论快报。99，098701（2007）]随机图上的随机游动偏向固定目标点。他们提出了一个临界偏置强度b_c，使得当b>b_c时，无限图上的随机游走者最终以概率1到达目标，而当b<b_c时，有限部分游走者漂移到无穷大。他们依赖于Galton-Watson（GW）树上有偏行走的严格结果来计算b_c，并给出了一些论据，表明这一结果对于Erdos-Renyi（ER）图和Molloy-Reed（MR）图等随机图也是成立的.为了验证他们的预测，他们通过数值模拟表明，有限ER图上的平均返回时间（MRT），作为图大小N的函数，在期望的b_c附近表现出一个过渡。在这里，我们表明，在GW树的MRT实际上可以计算分析。这使我们能够（i）分析地表明，MRT确实在b_c处显示出跃迁，（ii）阐明MRT的N依赖性，这与[Phys. Rev. Lett. 99，098701（2007）]。
We study the interplay of two interacting discrete time quantum walks in the presence of disorder. Each walk is described by a Floquet unitary map defined on a chain of two-level systems. Strong disorder induces a novel Anderson localization phase with a gapless Floquet spectrum and one unique localization length $ξ_1$ for all eigenstates for noninteracting walks. We add a local contact interaction which is parametrized by a phase shift $γ$. A wave packet is spreading subdiffusively beyond the bounds set by $ξ_1$ and saturates at a new length scale $ξ_2 \gg ξ_1$. In particular we find $ξ_2 \si…
We study Anderson localization in a generalized discrete time quantum walk - a unitary map related to a Floquet driven quantum lattice. It is controlled by a quantum coin matrix which depends on four angles with the meaning of potential and kinetic energy, and external and internal synthetic flux. Such quantum coins can be engineered with microwave pulses in qubit chains. The ordered case yields a two-band eigenvalue structure on the unit circle which becomes completely flat in the limit of vanishing kinetic energy. Disorder in the external magnetic field does not impact localization. Disorder…
First we report that the adjacency matrices of real-world complex networks systematically have null eigenspaces with much higher dimensions than that of random networks. These null eigenvalues are caused by duplication mechanisms leading to structures with local symmetries which should be more present in complex organizations. The associated eigenvectors of these states are strongly localized. We then evaluate these microstructures in the context of quantum mechanics, demonstrating the previously mentioned localization by studying the spread of continuous-time quantum walks. This null-eigenval…
We study transport within a spatially heterogeneous one-dimensional quantum walk with a combination of hierarchical and random barriers. Recent renormalization group calculations for a spatially disordered quantum walk with a regular hierarchy of barriers alone have shown a gradual decrease in transport but no localization for increasing (but finite) barrier sizes. In turn, it is well-known that extensive random disorder in the spatial barriers is sufficient to localize a quantum walk on the line. Here we show that adding only a sparse (sub-extensive) amount of randomness to a hierarchy of bar…
Distribution of the time at which N vicious walkers reach their maximal height
研究了一维单位时间内N个不相交的布朗运动（恶性步行者）的极值统计。利用路径积分技术，我们精确地计算了最大值M和到达最大值的时间τ_M的联合分布。我们特别关注不相交的布朗桥（“西瓜没有墙”）和不相交的布朗游览（“西瓜有墙”）。我们详细讨论了这种恶性步行者模型在西瓜配置和随机增长模型在弯曲的几何形状，一方面和定向聚合物在无序介质（DPRM）与一个自由端点的另一方面之间的关系。我们还检查我们的结果使用戴森的布朗运动的数值模拟和面对他们的多核生长模型（PNG）和离散晶格上的DPRM模型的数值模拟。这里展示的一些结果是在最近的一封信中宣布的[J. Rambeau和G. Schehr，Europhys. Lett. 91，60006（2010）]。
