Stein's Method and Minimum Parsimony Distance after Shuffles

TL;DR

This paper extends the analysis of minimum parsimony distance after shuffles, demonstrating a transition from Poisson to normal distribution for riffle shuffles using Stein's method and symmetry techniques.

Contribution

It establishes an analogous transition result for riffle shuffles, employing Stein's method and generating functions, and introduces a symmetry-based approach to analyze complex probabilistic problems.

Findings

Transition from Poisson to normal distribution after riffle shuffles

Application of Stein's method to genome rearrangement models

Use of symmetry to simplify complex probabilistic analysis

Abstract

Motivated by Bourque and Pevzner's simulation study of the parsimony method for studying genome rearrangement, Berestycki and Durrett used techniques from random graph theory to prove that the minimum parsimony distance after iterating the random transposition shuffle undergoes a transition from Poisson to normal behavior. This paper establishes an analogous result for minimum parsimony distance after iterates of riffle shuffles or iterates of riffle shuffles and cuts. The analysis is elegant and uses different tools: Stein's method and generating functions. A useful technique which emerges is that of making a problem more tractable by adding extra symmetry, then using Stein's method to exploit the symmetry in the modified problem, and from this deducing information about the original problem.