Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Random walks and the "Euclidean" association scheme in finite vector spaces

Charles Brittenham, Jonathan Pakianathan

Abstract

In this paper, we provide an application to the random distance-$t$ walk in finite planes and derive asymptotic formulas (as $q \to \infty$) for the probability of return to start point after $\ell$ steps based on the "vertical" equidistribution of Kloosterman sums established by N. Katz. The application of these deep results from number theory allow a determination of the second order terms in the answers that simpler spectral gap/mixing rate methods do not. This work relies on a "Euclidean" association scheme studied in prior work of W.M.Kwok, E. Bannai, O. Shimabukuro and H. Tanaka. We also provide a self-contained discussion of the P-matrix and intersection numbers of this scheme for convenience in our application as well as a more explicit form for the intersection numbers in the planar case.

Random walks and the "Euclidean" association scheme in finite vector spaces

Abstract

In this paper, we provide an application to the random distance- walk in finite planes and derive asymptotic formulas (as ) for the probability of return to start point after steps based on the "vertical" equidistribution of Kloosterman sums established by N. Katz. The application of these deep results from number theory allow a determination of the second order terms in the answers that simpler spectral gap/mixing rate methods do not. This work relies on a "Euclidean" association scheme studied in prior work of W.M.Kwok, E. Bannai, O. Shimabukuro and H. Tanaka. We also provide a self-contained discussion of the P-matrix and intersection numbers of this scheme for convenience in our application as well as a more explicit form for the intersection numbers in the planar case.
Paper Structure (8 sections, 15 theorems, 72 equations)

This paper contains 8 sections, 15 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Association scheme definitions
  3. Eigenvalues of the Distance $t$-graphs
  4. Some properties of (twisted) Kloosterman sums
  5. The P and Q matrices of the Euclidean Association scheme
  6. Equidistribution
  7. An application to the random walk in finite planes of odd order
  8. Intersection Numbers and Matrices of the scheme

Key Result

Theorem 1.1

Let $q$ be an odd prime, $q=3 \text{ mod } 4$. Let $R_{q,\ell,t}$ be the probability that you return to the same vertex after $\ell$ steps in the distance-$t$ walk where $t \neq 0$. Then $R_{q,\ell, t}=R_{q,\ell}$ is independent of $t \neq 0$ and initial state. We have Furthermore, as $q \to \infty$ we have: and

Theorems & Definitions (31)

  • Theorem 1.1: Probability of Return in the Distance $t$ Random Walk in $\mathbb{F}_q$-planes
  • Definition 2.1: Association Scheme
  • Definition 2.2: Bose-Messner Algebra
  • Example 2.3
  • Example 2.4: Euclidean association scheme
  • Example 2.5
  • Definition 2.6
  • Definition 2.7: Intersection Matrices
  • Theorem 3.1
  • Proposition 4.1
  • ...and 21 more