Probability and Random Walks on Groups

Abstract

One of the most important applications of group representation theory is to probability and statistics, via the study of random walks on groups. In a famous paper [1], Bayer and Diaconis gave very precise estimates on how many riffle shuffles it takes to randomize a deck of n cards; the riffle shuffle is the shuffle where you cut the pack of cards into two halves and then interleave them. Bayer and Diaconis concluded based on their results that, for a deck of 52 cards, seven riffle shuffles are enough. Any fewer shuffles are too far from being random, whereas the net gain in randomness for doing more than seven shuffles is not sufficient to warrant the extra shuffles.