How Far Can a Biased Random Walker Go ?

The random walk (RW) is a very important model in science and engineering researches. It has been studied over hundreds years. However, there are still some overlooked problems on the RW. Here, we study the mean absolute distance of an N-step biased random walk (BRW) in a ddimensional hyper-cubic lattice. In this short paper, we report the exact results for d = 1 and approximation formulae for 2 d ≥ .


Introduction
As a mathematical model, the random walk (RW) has been widely used in almost all branches of sciences and engineering [1]- [9].Although the unbiased random walk has been studied extensively in literature, the biased random walk (BRW) has not been studied carefully in some cases.
In this short paper, we first give a brief description of the conventional results, and then report our study with some results on the BRW.
Let us consider the one dimensional BRW: a probability p of going forward and a probability (1 − p) of going backward with uniform step length L. Traditionally, the average distance gone in one step is expressed as: ( )( ) ( ) The variance of a one step BRW can be calculated as: After N such steps, the mean distance becomes ( ) In the last expression, 1 2 x p = − is used.When 1 2 p = (i.e., 0 x = ), the mean distance becomes zero.The variance of the N steps is The standard deviation of the N steps is In the case of the pure random walk (RW), i.e. when 1 2 p = ( 0 x = ), the standard deviation of the N-step This value, known widely in literature, is usually considered as the absolute distance of an N-step RW.This expression is independent of the dimensions of the lattice.
However, the mean absolute distance of the N-step RW in a d-dimensional hyper-cubic lattice cannot be expressed by ( 6), but is the following formula [10] ( ) ( ) d N α is a monotonic increasing function of dimension d with saturation value of one: We compute the absolute distance for the N-step biased random walk (BRW).We find that (3) is a fairly good approximation for a reasonably large N and p away from the neighborhood of 1 2 p = .
In Section 2, the exact results for d = 1 are presented.The approximation results for higher dimensions are shown in Section 3. A brief discussion is given afterward.A warning: it is possible that some of our results might have been already published in earlier literatures unknown to us.
For convenience, without loss of generality, we choose a step length of L = 1 in hereafter expressions.

Exact Results for d = 1
For an N step biased random walker (BRW), if the walker moves forward n steps with probability p, and moves backwards (N -n) steps with probability 1 − p, this is a binomial process with probability p.The absolute distance from the origin will be After taking the weighted configuration average, the mean absolute distance of the one-dimensional BRW can be expressed as: Using Mathematica [11], we obtain the following relationship: where ( ) , f N p are the polynomials of p to be discussed below.Furthermore, we obtain the following relationship (via Mathematica): The ( ) 2 , f m p are the m-term 2m'th order polynomials of p with the lowest term being a (m + 1)'th order term.
For convenience, we have listed some exact results for small values of N as follows: Further algebraic calculations yield the following recursion equations (for an even N, let N = 2 m in the following expressions): i.e.Additionally, because ( ) ( ) we can obtain the following expression: , we can see that the following is an even function for x: When 0 x = , the above becomes the unbiased random walk result [10]: In order to obtain these results, we use the following identity: Therefore, Equation (15) can be expressed as a polynomial of x 2 : , where 2 or 2 1.
for 2,3, 4, In order to see the quantitative behavior of the averaged absolute distance as a function of ( ) , we plot Equation (18) in Figure 1 for three typical examples: N = 10, 100, and 1000, respectively.For comparison, line y = 2x is also presented in this figure.It is easy to see that the linear relationship [expressed by Equation ( 3)] can be used for a reasonable large N. Furthermore, the validity range (x value) of the linear approximation becomes larger and larger as N becomes greater and greater.For reasonable accuracy, the ranges are x > 0.25, 0.05, and 0.02, for N = 10, 100, and 1000, respectively.
We have computed some typical values of the approximation error as follows (and partially shown in In the range for which the linear approximation is invalid (the neighborhood of 0 x = ), a 3-term polynomial is a fairly good approximation.In the neighborhood of 0 x ≈ , it can be expressed as where In the neighborhood of 0 = x , the three term approximation may be expressed as: Alternatively, we can consider the multidimensional BRW by transposing the coordinate system so that only one direction is biased.For instance, we can consider a diffusion process via an interface in which there is a pressure applied in the direction perpendicular to the interface.In this situation, all directions except the (biased) direction perpendicular to the interface follow a pure random walk.This model can be applied to many diffusion problems in physics and chemistry.Generally speaking, we can express the dimension, d as 1 d g = + , where we consider the g dimensions to be unbiased random walks and the additional 1 dimension to be biased.For this model, we can study the g dimensions to be unbiased random walks first.According to [10], the mean absolute n x y z M M f q q n q q xyz n n x y z , and concluded that (A1) is a very good approximation.

Figure 1 .
Figure 1.The normalized plot of the averaged absolute distance

Figure 2 .Figure 3 . 4 .
Figure 2. The semi-log plot of the difference between the normalized absolute distance and the linear approximation vs. the biased probability x.For accuracy reasons ( ) the unit vectors.The absolute value of x can be expressed as: For a reasonable large N and off the neighborhood of 0 = x , this is similar to the case of d = 1, and the displacement of the BRW should be the linear relationship, i.e.,