Experimental analysis of lattice walks

TL;DR

This paper introduces a computational approach to analyze lattice walks by calculating finite polynomials for fixed lengths and inferring asymptotic behavior, extending analysis to higher dimensions and more complex steps.

Contribution

It presents a new dynamic programming method for enumerating lattice walks of fixed length, enabling analysis of more general and complex walk models beyond algebraic solutions.

Findings

Method effectively computes walk statistics for fixed lengths.

Conjectures about asymptotic behavior of walk statistics.

Applicable to higher-dimensional and more complex step walks.

Abstract

Feller's book An Introduction to Probability Theory and Its Application discusses statistics corresponding to sequences of coin tosses, with a dollar being won or lost depending on the outcome of each toss. This is equivalent to analyzing walks in the plane with each step being one unit up or right. In his paper "Fully AUTOMATED computerized redux of Feller's (v.1) Ch. III (and much more!)" (http://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/feller.html) and the accompanying Maple package, Zeilberger computes the "grand generating function," which, in a single blow, captures information about all of the walk statistics discussed by Feller. In this paper, we continue to investigate walks using computer methods. However, we shall introduce an approach different from that of Zeilberger, who used computer algebra to exactly compute the weight enumerator over all walks--an infinite sum expressed as an algebraic function. Our procedures input a numeric value of $n$ and use dynamic programming methods to find the weight enumerator over all walks of length $n$--a finite polynomial. Then, by generating these polynomials for many values of $n$, we are able to conjecture behavior of the moments of certain statistics as the walk length tends to $\infty$. The advantage of this method is that it is easily applied to more general problems not amenable to an analytic approach. For example, we use it to analyze walks in three dimensions and walks where more general steps (e.g., diagonal steps) are allowed.