On the Expected Winding Number of a Random Walk on the Unit Lattice

Overview

Objectives/Goals Some recent studies have focused on the winding number of a random walk. Given a random walk s starting at (1,1) on the unit lattice, the winding number w of s is the number of signed complete rotations the walk has made about (1/2,1/2). Despite the known results on the continuous winding number, the discrete version appears to be unstudied. This project investigates the root mean square expectation of the winding number. Methods/Materials We rephrase the problem in terms of a diagonal lattice and determine the winding number as a function of two variables counting steps beginning and ending on the positive x-axis. We then condition on the values of these variables and examine the change in expectation created by each additional step in the walk to express the desired expectation as a summation of only two smaller expectations. A symmetry that yields a bijection between types of these random walks allows us to determine these unknowns and thus reach our final result. Conclusions/Discussion We have found an explicit expression for the RMS expected winding number after n steps of a random walk beginning at (1,0) on the unit lattice. This expression is in terms of a binomial sum; we first find the expectation recursively and then exploit a symmetry of random walks to solve the recursion. This result gives us a better understanding of the rotational properties of random walks and thus may be useful in further investigations into this field.

Competition history

• CSEF 2006 Mathematics & Software · Entry S1222

Resources

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