The statistics of first-passage times of random walks to target sites has proved to play a key role in determining the kinetics of space exploration in various contexts. In parallel, the number of distinct sites visited by a random walker and related observables have been introduced to characterize the geometry of space exploration. Here, we address the question of the joint distribution of the first-passage time to a target and the number of distinct sites visited when the target is reached, which fully quantifies the coupling between kinetics and geometry of search trajectories. Focusing on 1-dimensional systems, we present a general method and derive explicit expressions of this joint distribution for several representative examples of Markovian search processes. In addition, we obtain a general scaling form, which holds also for non Markovian processes and captures the general dependence of the joint distribution on its space and time variables. We argue that the joint distribution has important applications to various problems, such as a conditional form of the Rosenstock trapping model, and the persistence properties of self-interacting random walks.
We propose local-biased random walks on general networks where a Markovian walker can choose between different types of biases in each node to define transitions to its neighbors depending on their degrees. For this ergodic dynamics, we explore the capacity of the random walker to visit all the nodes characterized by a global mean first passage time. This quantity is calculated using eigenvalues and eigenvectors of the transition matrix that defines the dynamics. In the first part, we illustrate how our framework leads to optimal transport for small-size graphs through the analysis of all the …
The territory explored by a random walk is a key property that may be quantified by the number of distinct sites that the random walk visits up to a given time. The extent of this spatial exploration characterizes many important physical, chemical, and ecological phenomena. In spite of its fundamental interest and wide utility, the number of visited sites gives only an incomplete picture of this exploration. In this work, we introduce a more fundamental quantity, the elapsed time $τ_n$ between visits to the $n^{\rm th}$ and the $(n+1)^{\rm st}$ distinct sites, from which the full dynamics abou…
考虑了自排斥两足蜘蛛行走，其中局部随机运动由两个独立的控制参数$ β_d$和$ β_h $控制，前者控制两足位置之间的距离（$ d $），后者控制遍历路径的自交叉统计.局部运动的概率度量应该是"真正的自我回避行走"乘以一个因子，该因子随$ d $呈指数衰减。在短时间的瞬态行为之后，根据$β_d$和$β_h$的值，在长时间内观察到各种行为。我们的统计分析表明，该系统经历了两个（小和大$β_d$）制度之间的交叉确定在大时代（$t$）。在小的$β_d$制度，随机步行者（确定的位置的腿的蜘蛛）保持在一个固定的非零的距离在大的时间限制，而在第二个制度（大$β_d$ s），步行者之间的吸收力占主导地位的其他随机力平均。在后一种情况下，$ d $以幂律方式随时间的对数衰减。当系统被映射到生长过程（由通过每个点的访问次数识别的高度场表示）时，粗糙度和平均高度在两种状态下表现出不同的行为，即，在第一种状态下，它们与$t$呈幂律关系，而在第二种状态下，它们与$\log t$呈幂律关系。随机步行者的痕迹和缠绕角度的分形维数一致地经历了类似的交叉。
