In this paper, we analytically discuss the scaling properties of the average square end-to-end distance < R-2 > for anisotropic random walk in D-dimensional space (D >= 2), and the returning probability P-n(r...In this paper, we analytically discuss the scaling properties of the average square end-to-end distance < R-2 > for anisotropic random walk in D-dimensional space (D >= 2), and the returning probability P-n(r(0)) for the walker into a certain neighborhood of the origin. We will not only give the calculating formula for < R-2 > and P-n(r(0)), but also point out that if there is a symmetric axis for the distribution of the probability density of a single step displacement, we always obtain < R-perpendicular to n(2) > similar to n, where perpendicular to refers to the projections of the displacement perpendicular to each symmetric axes of the walk; in D-dimensional space with D symmetric axes perpendicular to each other, we always have < R-n(2)> similar to n and the random walk will be like a purely random motion; if the number of inter-perpendicular symmetric axis is smaller than < R-n(2)> similar to n(2) the dimensions of the space, we must have n for very large n and the walk will be like a ballistic motion. It is worth while to point out that unlike the isotropic random walk in one and two dimensions, which is certain to return into the neighborhood of the origin, generally there is only a nonzero probability for the anisotropic random walker in two dimensions to return to the neighborhood.展开更多
The recurrence properties of random walks can be characterized by P61ya number, i.e., the probability that the walker has returned to the origin at least once. In this paper, we consider recurrence properties for a ge...The recurrence properties of random walks can be characterized by P61ya number, i.e., the probability that the walker has returned to the origin at least once. In this paper, we consider recurrence properties for a general 1D random walk on a line, in which at each time step the walker can move to the left or right with probabilities l and r, or remain at the same position with probability o （l ＋ r ＋ o = 1）. We calculate Polya number P of this model and find a simple expression for P as, P = 1 - △, where △ is the absolute difference of l and r （△= ｜l - r｜）. We prove this rigorous expression by the method of creative telescoping, and our result suggests that the walk is recurrent if and only if the left-moving probability l equals to the right-moving probability r.展开更多
We study the open quantum random walk (OQRW) with time-dependence on the one-dimensional lattice space and obtain the associated limit distribution. As an application we study the return probability of the OQRW. We al...We study the open quantum random walk (OQRW) with time-dependence on the one-dimensional lattice space and obtain the associated limit distribution. As an application we study the return probability of the OQRW. We also ask, "What is the average time for the return probability of the OQRW?"展开更多
In this paper the return probability of the one-dimensional discrete-time quantum walk is studied. We derive probabilistic formulas for the return probability related to the quantum walk governed by the Fibonacci coin.
文摘In this paper, we analytically discuss the scaling properties of the average square end-to-end distance < R-2 > for anisotropic random walk in D-dimensional space (D >= 2), and the returning probability P-n(r(0)) for the walker into a certain neighborhood of the origin. We will not only give the calculating formula for < R-2 > and P-n(r(0)), but also point out that if there is a symmetric axis for the distribution of the probability density of a single step displacement, we always obtain < R-perpendicular to n(2) > similar to n, where perpendicular to refers to the projections of the displacement perpendicular to each symmetric axes of the walk; in D-dimensional space with D symmetric axes perpendicular to each other, we always have < R-n(2)> similar to n and the random walk will be like a purely random motion; if the number of inter-perpendicular symmetric axis is smaller than < R-n(2)> similar to n(2) the dimensions of the space, we must have n for very large n and the walk will be like a ballistic motion. It is worth while to point out that unlike the isotropic random walk in one and two dimensions, which is certain to return into the neighborhood of the origin, generally there is only a nonzero probability for the anisotropic random walker in two dimensions to return to the neighborhood.
基金Supported by National Natural Science Foundation of China under Grant No. 10975057Doctor Fund Project of Ministry of Education under Contract 20103201120003+1 种基金the New Teacher Foundation of Soochow University under Contracts Q3108908, Q4108910the Extracurricular Pesearch Foundation of Undergraduates under Grant No. KY2010056A
文摘The recurrence properties of random walks can be characterized by P61ya number, i.e., the probability that the walker has returned to the origin at least once. In this paper, we consider recurrence properties for a general 1D random walk on a line, in which at each time step the walker can move to the left or right with probabilities l and r, or remain at the same position with probability o （l ＋ r ＋ o = 1）. We calculate Polya number P of this model and find a simple expression for P as, P = 1 - △, where △ is the absolute difference of l and r （△= ｜l - r｜）. We prove this rigorous expression by the method of creative telescoping, and our result suggests that the walk is recurrent if and only if the left-moving probability l equals to the right-moving probability r.
文摘We study the open quantum random walk (OQRW) with time-dependence on the one-dimensional lattice space and obtain the associated limit distribution. As an application we study the return probability of the OQRW. We also ask, "What is the average time for the return probability of the OQRW?"
文摘In this paper the return probability of the one-dimensional discrete-time quantum walk is studied. We derive probabilistic formulas for the return probability related to the quantum walk governed by the Fibonacci coin.
