Persi Diaconis (Stanford)

will speak on

Finite Fields Meet Markov Chains

Time: 2:00PM
Date: Thu 1st October 2020
Location: Online [map]

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Abstract: Let $p$ be a prime. Consider a random walk on $F(p)$ that moves from $j$ to $j^2 +1$ or $j^2-1$ with probability 1/2 (the square and add Markov chain). It is an open problem to analyze this simple-sounding scheme. We don't even know the support of the stationary distribution. When $p \equiv 3$ (mod 4) Jimmy He has determined the stationary distribution but this is open for $p\equiv 1$ (mod 4). Squaring is an automorphism for a field of characteristic $2$ and, with He and Marty Isaacs we study the `square and add' chain over $F(2^d)$. For some choice of generators, we can show that $(1/2)d \log d$ steps are necessary and sufficient for mixing BUT the generating set seems to matter and many mysteries remain. This mix of algebra and probability poses simple to state open problems.

(This talk is part of the Algebra and Number Theory series.)

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