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 d-dimensional hyper-cubic lattice. In this short paper, we report the exact results for d = 1 and approximation formulae for d ≥ 2.

• Biased Random Walk
• Monte Carlo Simulations
• Stochastic Process

1. 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, is used. When (i.e.,), 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 (), the standard deviation of the N-step

RW is

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]

where 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.

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.

2. 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 are the polynomials of p to be discussed below.

Furthermore, we obtain the following relationship (via Mathematica):

The 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:

If we let and, we can see that the following is an even function for x:

When, 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 x²:

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 Figure 2):

In the range for which the linear approximation is invalid (the neighborhood of), a 3-term polynomial is a fairly good approximation. In the neighborhood of, it can be expressed as

where

Figure 1

The normalized plot of the averaged absolute distance vs. for in the range of [0, 0.5]. The reference line is also shown for comparison

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 , we have only plotted the range for

The relationship between and is given by, where is defined

by Equation (7). The second term coefficient can be expressed as. The third term

coefficient can be expressed as. We compute these Greek letter coefficients, , and as functions of N, which are shown in Figure 3. It is easy to see that these Greek letter coefficients are of order of one. The asymptotic values of them are obtained as:

We also compute the next three terms of Equation (18):, and. The asymptotic values for the three coefficients are obtained as:

From Figure 3, it is easy to see that, for a reasonable large N, the coefficients are very close to their asymptotic values. Therefore, in the neighborhood of, Equation (19) can be expressed as:

Figure 3

The coefficients, , , and as functions of the total step number N for . The asymptotic lines are also presented for comparison

To verify the validity of the approximation (25), we plot and

with exact results vs. the 3-term approximation Equation (25) for small values of x in Figure 4. It is easy to see that formula (25) is a very good approximation in the neighborhood of.

3. For

For a dimensional hyper-cubic lattice, let () be the probability of walking forward

along the i'th coordinate. We define, where () and

are the unit vectors. The absolute value of can be expressed as:. The angle of each compo-

nent is: (). For a reasonable large N and off the neighborhood of, this is similar to the case of d = 1, and the displacement of the BRW should be the linear relationship, i.e.,

In the neighborhood of, the three term approximation may be expressed as:

Here, are given in Equation (8).

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, 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

Figure 4

The normalized plot of the averaged absolute distance vs. for and 1000 in the neighborhood of . The 3-term approximations lines are also presented for comparison

distance of the g-dimensional -step RW

, where is given by (8). Additionally, in a d-dimensional hyper-cubic

lattice, the (pure random walk) displacement is perpendicular to the (biased) direction of the BRW. In many problems, we are only concerned with the absolute distance in the (biased) direction, i.e., the projection portion of the total absolute distance. For this reason, we can model it as a modified 1d approach as follows: a probability q walking in the g-dimensional hyper-cubic lattice, a probability of going forward and a probability of going backward in the 1-dimension. Because the g-dimensional lattice is perpendicular to the biased direction, the (projected) absolute distance can be expressed as:

We study two cases, one for reasonably large p and the other in the neighborhood of, separately. For the first case, when n is large enough, we have obtained the results in the previous section:

Substituting this into (28) and using the results of the Appendix yields:

It is not surprising that the modification of higher dimensions on the 1d result requires only multiplication of the probability factor.

For the second case, i.e., in the neighborhood of, we can express the following:

where, and are defined by (20-22).

Using the 1d results, (31) can be estimated to be a very simple formula:

where, , and are defined by (23). In the last step, (A1) in the Appendix was used.

4. Discussion and Concluding Remarks

The biased random walk has widely applications in various fields: for examples, a pressured diffusion process, an ionic injection with bombardment, a ballistic transport, financial market data, etc. For most natural phenomena and engineering processes, the particle number is about the order or a fraction of the Avogadro’s constant (~10²³), the traditional treatment is good enough. However, the financial data and some high precision experimental data are far away from a large number, say 10¹⁰. For example in financial industry, the most active index futures, SP500, has only the order of 10⁵ open interest contracts before rolling the date. The daily trading volume is one or two order smaller than the open interest. Therefore, when the particle number is not large enough, one has to consider the new behavior. The present results are just the better quantitative descriptions for those phenomena. In some high precision experiments in physical sciences, one may have to measure parameters with small amount of particles. To quantify the property, the present results can provide better mathematical expressions.

Acknowledgements

The authors would like to Angel Yang for her helpful discussion.

Appendix

In this appendix, we present a very useful approximation formula: for a large enough M and,

It is easy to see that and are continuous and greater than zero, so that is a smooth and restricted monotonic increasing function. Let us compute some special values of:

Numerically, we computed the values of, , , , and concluded that (A1) is a

very good approximation.

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