Washington Experimental Mathematics Lab
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1 Washington Experimental Mathematics Lab Rotation Random Walks Alex Tsun, Michelle Prawiro, Peter Gylys-Colwell Faculty Mentors: Jayadev Athreya, Chris Hoffman Grad Mentor: Anthony Sanchez Department of Mathematics University of Washington Autumn 2017 Alex Tsun, Michelle Prawiro, Peter Gylys-Colwell Washington Faculty Mentors: Experimental Jayadev Mathematics Athreya, Chris Lab Hoffman Grad Mentor: Autumn Anthony 2017Sanchez 1 / 11 (Uni
2 Rotation Random Walk Motivation Random walks are a well studied and interesting mathematical topic. We wish to expand the study of random walks on S 1. Problem On next slide Methods We began with a simpler case (rational case) and now are looking to extend the problem to the irrational case Alex Tsun, Michelle Prawiro, Peter Gylys-Colwell Washington Faculty Mentors: Experimental Jayadev Mathematics Athreya, Chris Lab Hoffman Grad Mentor: Autumn Anthony 2017Sanchez 2 / 11 (Uni
3 Problem Algorithm 1 Rotation Random Walk 1: Fix α = 1 k, k N 2 to be the step size. 2: Set Y 0 = 0, where Y i [0, 1) is the position of the walk at time i. 3: for i = 1, { 2,...n do +1, w.p 1 4: X i = 2 1, w.p 1 2 5: 6: Y i = {Y i 1 + αx i } the new position mod 1. 7: { 8: Z i = +1, 1, if Y i [0, 1 2 ) if Y i [ 1 2, 1) Question: The partial sums S n = n i=1 Z i define a new random walk on Z. Does the mean 1 n n i=1 Z i follow a central limit theorem (the Z i s are not independent!)? Alex Tsun, Michelle Prawiro, Peter Gylys-Colwell Washington Faculty Mentors: Experimental Jayadev Mathematics Athreya, Chris Lab Hoffman Grad Mentor: Autumn Anthony 2017Sanchez 3 / 11 (Uni
4 The (Standard) Central Limit Theorem Let X 1,..., X n be independent and identically distributed (iid) random variables with common mean E[X i ] = µ < and common variance Var[X i ] = σ 2 <. Then, as n, Z n = 1 n n i=1 X i µ σ/ n d N (0, 1) Alex Tsun, Michelle Prawiro, Peter Gylys-Colwell Washington Faculty Mentors: Experimental Jayadev Mathematics Athreya, Chris Lab Hoffman Grad Mentor: Autumn Anthony 2017Sanchez 4 / 11 (Uni
5 Visual Explanation Alex Tsun, Michelle Prawiro, Peter Gylys-Colwell Washington Faculty Mentors: Experimental Jayadev Mathematics Athreya, Chris Lab Hoffman Grad Mentor: Autumn Anthony 2017Sanchez 5 / 11 (Uni
6 Rational Case (α Q) What happens if the steps we take are rational? The answer is much simpler, and can be reduced to a Markov chain with finite state space, which there already exist CLT s for. Alex Tsun, Michelle Prawiro, Peter Gylys-Colwell Washington Faculty Mentors: Experimental Jayadev Mathematics Athreya, Chris Lab Hoffman Grad Mentor: Autumn Anthony 2017Sanchez 6 / 11 (Uni
7 Irrational Case The α trajectory is dense and uniformly distributed. Alex Tsun, Michelle Prawiro, Peter Gylys-Colwell Washington Faculty Mentors: Experimental Jayadev Mathematics Athreya, Chris Lab Hoffman Grad Mentor: Autumn Anthony 2017Sanchez 7 / 11 (Uni
8 Extending Rational Case to Irrationals We can use the CLT result of the rational case for extending the results to the irrationals. For the number of steps = N we can choose a rational p q with p q α < ε N. This ε N is chosen so that for all n < N Y α,n [0, 1/2) Y p,n [0, 1/2) q Thus our Z i s will be the same. Since Z p q,i follows CLT we can bound its distribution close to the normal distribution. Thus we can bound Z α,i close to the normal distribution as N (in Analysis this is sometimes called a triple epsilon argument) Alex Tsun, Michelle Prawiro, Peter Gylys-Colwell Washington Faculty Mentors: Experimental Jayadev Mathematics Athreya, Chris Lab Hoffman Grad Mentor: Autumn Anthony 2017Sanchez 8 / 11 (Uni
9 Continued Fractions Continued Fractions give a very good approximation to any irrational number. Given 0 < α < 1, we can write α = [a 1, a 2, a 3,...] = 1 a a a Of course, we can terminate after any time [a 1,..., a n ]. Each [a 1,..., a n ] can be written as pn q n, where gcd(p n, q n ) = 1. We are guaranteed that α pn q n < 1, a quadratic bound in the denominator q qn 2 n. Alex Tsun, Michelle Prawiro, Peter Gylys-Colwell Washington Faculty Mentors: Experimental Jayadev Mathematics Athreya, Chris Lab Hoffman Grad Mentor: Autumn Anthony 2017Sanchez 9 / 11 (Uni
10 Continued Fractions (example) 2 1 = [2, 2, 2, 2, 2, 2,...] [2] = 1 2 = 0.5 [2, 2] = 1 = = [2, 2, 2] = [2, 2, 2, 2] = = 5 12 = = Alex Tsun, Michelle Prawiro, Peter Gylys-Colwell Washington Faculty Mentors: Experimental Jayadev Mathematics Athreya, Chris Lab Hoffman Grad Mentor: Autumn Anthony 2017 Sanchez 10 / 11 (Uni
11 Future goals Next steps We must understand how fast the rational case converge to the normal distribution. We will conduct more experiments now using continued fractions to approximate the irrational case. Challenges It is hard to find literature on the rate of convergence for the random walk on a cyclic graph following CLT. Alex Tsun, Michelle Prawiro, Peter Gylys-Colwell Washington Faculty Mentors: Experimental Jayadev Mathematics Athreya, Chris Lab Hoffman Grad Mentor: Autumn Anthony 2017 Sanchez 11 / 11 (Uni
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