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
In this paper, we consider the stationary probability and first-passage time of biased random walk on 1D chain, where at each step the walker moves to the left and right with probabilities p and q respectively(0 p, q ...In this paper, we consider the stationary probability and first-passage time of biased random walk on 1D chain, where at each step the walker moves to the left and right with probabilities p and q respectively(0 p, q 1,p + q = 1). We derive exact analytical results for the stationary probability and first-passage time as a function of p and q for the first time. Our results suggest that the first-passage time shows a double power-law F ^(N-1)~γ, where the exponent γ = 2 for N < |p-q|^(-1) and γ = 1 for N > |p-q|^(-1). Our study sheds useful insights into the biased random-walk process.展开更多
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.
基金Supported by the National Natural Science Foundation of China under Grant No.11205110Shanghai Key Laboratory of Intelligent Information Processing(IIPL-2011-009)Innovative Training Program for College Students under Grant No.2015xj070
文摘In this paper, we consider the stationary probability and first-passage time of biased random walk on 1D chain, where at each step the walker moves to the left and right with probabilities p and q respectively(0 p, q 1,p + q = 1). We derive exact analytical results for the stationary probability and first-passage time as a function of p and q for the first time. Our results suggest that the first-passage time shows a double power-law F ^(N-1)~γ, where the exponent γ = 2 for N < |p-q|^(-1) and γ = 1 for N > |p-q|^(-1). Our study sheds useful insights into the biased random-walk process.
