Oscillatory correlation of delayed random walks

We investigate analytically and numerically the statistical properties of a random walk model with delayed transition probability dependence (delayed random walk). The characteristic feature of such a model is the oscillatory behavior of its correlation function. We investigate a model whose transient and stationary oscillatory behavior is analytically tractable. The correspondence of the model with a Langevin equation with delay is also considered. Noise and correlative effects (memory) are two elements which are associated with many natural systems. In physics, two main approaches have been developed to study such systems with noise and memory. One approach is formulating the model in physical space with a differential equation of motion such as the “generalized Langevin equation”[1, 2]. The other is to formulate a model in probability space as a nonMarkovian problem as in the “generalized master equation” approach [3]. These two avenues have been developed and applied to various problems in physics. Examples include studies on the Alder-Wainwright effect[4], spin relaxation[5], and driven two-level atoms[6]. The delayed stochastic system we discuss here can be viewed as a special case, where only a single (memory) point at a fixed time interval in the past has influence on the current state of the system. Research of such systems, particularly those with no noise, has been carried out in fields of mathematics[7], biology[8], artificial neural network[9], electrical circuits[10], as well as in physics[11]. Models with both noise and delay have also been considered numerically[12] and analytically as an extention of the Langevin equation[13]. These works represent approaches and formulations in physical space. For the probability space approach, “Delayed random walk” is recently proposed [14] and has been applied to model human posture controls[15]. However, an analytical understanding of this random walk is yet far from being complete. The main theme of this paper is to increase the analytical understanding of the behavior of a delayed random walk model. The oscillatory correlation function is found to be associated with delayed random walks [14, 16]. We show here that such oscillatory behavior of the of the correlation function is analytically tractable. From the study of random walks point of view, this delayed random walk model provides an example whose correlation function behaves differently compared to commonly known random walks with memory, such as self-avoiding, or persistent walks [17]. In addition, we note that oscillatory or chaotic behavior associated with delays are generally difficult to analyze[12]. Hence, this model also serves as one of the rare analytically tractable examples among models with delay. We consider a random walk which takes a unit step in a unit time. The delayed random walk we start with is an extension of a position dependent random walk whose step toward the origin is more likely when no delay exists. Formally, it has the following definition: P (Xt+1 = n;Xt+1−τ = s) = g(s− 1)P (Xt = n− 1;Xt+1−τ = s;Xt−τ = s− 1) + g(s+ 1)P (Xt = n− 1;Xt+1−τ = s;Xt−τ = s+ 1) + f(s− 1)P (Xt = n+ 1;Xt+1−τ = s;Xt−τ = s− 1) + f(s+ 1)P (Xt = n+ 1;Xt+1−τ = s;Xt−τ = s+ 1), (1) f(x) + g(x) = 1, (2) where the position of the walker at time t is Xt, and P (Xt1 = u1;Xt2 = u2) is the joint probability for the walker to be at u1 and u2 at time t1 and t2, respectively. f(x) and g(x) are transition probabilities to take a step to the negative and positive directions respectively at the position x. In this paper, we further place the conditions: f(x) > g(x) (x > 0), f(−x) = g(x) (∀x). (3) These conditions make the delayed random walks symmetric with respect to the origin, which is attractive without delay (τ = 0). We now proceed to obtain a few properties from this general definition. By the symmetry with respect to the origin, the average position of the walker is 0. This symmetry is further used to inductively show [18] in the stationary state (t → ∞) that P (Xt+1 = n;Xt = n+ 1) = P (Xt+1 = n+ 1;Xt = n). (4) We derived the stationary probability distribution for the previously discussed delayed random walk model using this property [14]. Also, the multiplication of Eq. (1) for the stationary state by cos(αn) and summation over n and s yields for the generating function: 〈cos(αXt)〉 = cos(α)〈cos(αXt)〉+ sin(α)〈sin(αXt){f(Xt−τ )− g(Xt−τ )}〉 (5) In particular, we have a following invariant relationship with respect to the delay. 1 2 = 〈Xt{f(Xt−τ )− g(Xt−τ )}〉 (6) This invariant property is used below. We will consider a specialized model for the rest of this paper [19]. We define f(x) and g(x) as f(x) = 1 2 (1 + 2d) (x > a), 1 2 (1 + βx) (−a ≤ x ≤ a), 1 2 (1− 2d) (x < −a), g(x) = 1 2 (1 − 2d) (x > a), 1 2 (1− βx) (−a ≤ x ≤ a), 1 2 (1 + 2d) (x < −a). (7)