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Register a new userExact mean first-passage time on generalized Vicsek fractal
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Fractal phenomena may be widely observed in a great number of complex systems. In this paper, we revisit the well-known Vicsek fractal, and study some of its structural properties for purpose of understanding how the underlying topology influences its dynamic behaviors. For instance, we analytically determine the exact solution to mean first-passage time for random walks on Vicsek fractal in a more light mapping-based manner than previous other methods, including typical spectral technique. More importantly, our method can be quite efficient to precisely calculate the solutions to mean first-passage time on all generaliz
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Since diffusion processes arise in so many different fields, efficient tech-nics for the simulation of sample paths, like discretization schemes, represent crucial tools in applied probability. Such methods permit to obtain approximations of the first-passage times as a by-product. For efficiency reasons, it is particularly challenging to simulate directly this hitting time by avoiding to construct the whole paths. In the Brownian case, the distribution of the first-passage time is explicitly known and can be easily used for simulation purposes. The authors introduce a new rejection sampling algorithm which permits to perform an exact simulation of the first-passage time for general one-dimensional diffusion processes. The efficiency of the method, which is essentially based on Girsanovs transformation , is described through theoretical results and numerical examples.
In this paper we study first-passge percolation models on Delaunay triangulations. We show a sufficient condition to ensure that the asymptotic value of the rescaled first-passage time, called the time constant, is strictly positive and derive some upper bounds for fluctuations. Our proofs are based on renormalization ideas and on the method of bounded increments.
The explicit determinations of the mean first-passage time (MFPT) for trapping problem are limited to some simple structure, e.g., regular lattices and regular geometrical fractals, and determining MFPT for random walks on other media, especially complex real networks, is a theoretical challenge. In this paper, we investigate a simple random walk on the the pseudofractal scale-free web (PSFW) with a perfect trap located at a node with the highest degree, which simultaneously exhibits the remarkable scale-free and small-world properties observed in real networks. We obtain the exact solution for the MFPT that is calculated through the recurrence relations derived from the structure of PSFW. The rigorous solution exhibits that the MFPT approximately increases as a power-law function of the number of nodes, with the exponent less than 1. We confirm the closed-form solution by direct numerical calculations. We show that the structure of PSFW can improve the efficiency of transport by diffusion, compared with some other structure, such as regular lattices, Sierpinski fractals, and T-graph. The analytical method can be applied to other deterministic networks, making the accurate computation of MFPT possible.
Continuous-time stochastic processes play an important role in the description of random phenomena, it is therefore of prime interest to study particular variables depending on their paths, like stopping time for example. One approach consists in pointing out explicit expressions of the probability distributions, an other approach is rather based on the numerical generation of the random variables. We propose an algorithm in order to generate the first passage time through a given level of a one-dimensional jump diffusion. This process satisfies a stochastic differential equation driven by a Brownian motion and subject to random shocks characterized by an independent Poisson process. Our algorithm belongs to the family of rejection sampling procedures, also called exact simulation in this context: the outcome of the algorithm and the stopping time under consideration are identically distributed. It is based on both the exact simulation of the diffusion at a given time and on the exact simulation of first passage time for continuous diffusions. It is therefore based on an extension of the algorithm introduced by Herrmann and Zucca [16] in the continuous framework. The challenge here is to generate the exact position of a continuous diffusion conditionally to the fact that the given level has not been reached before. We present the construction of the algorithm and give numerical illustrations, conditions on the recurrence of jump diffusions are also discussed.
We study the time constant $mu(e_{1})$ in first passage percolation on $mathbb Z^{d}$ as a function of the dimension. We prove that if the passage times have finite mean, $$lim_{d to infty} frac{mu(e_{1}) d}{log d} = frac{1}{2a},$$ where $a in [0,infty]$ is a constant that depends only on the behavior of the distribution of the passage times at $0$. For the same class of distributions, we also prove that the limit shape is not an Euclidean ball, nor a $d$-dimensional cube or diamond, provided that $d$ is large enough.
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