5 problems

Clay's position-matrix eigenvalue conjectures. The following assertions hold:

Multi-stack and restricted shuffle cutoff conjecture. Both chains exhibit cutoff, with the multi-stack chain around time

Let $H_{2m}$ be the $2m$th harmonic number, and let $F^{\mathbf{G}}(n,m)$ denote the expected number of correct guesses made by the strategy $\mathbf{G}$ for a deck of $n$ cards su…

Let $M$ be the position matrix of the one-shelf shuffling machine. Its eigenvalues describe the associated Markov chain on card positions; the previously established eigenvalues in…

The quadratic eigenvector conjecture. $M$ has an eigenvalue-eigenvector pair $(\frac{1}{4},\vec{v})$ with multiplicity at least $1$. Numerical computation suggests this eigenvector…