4 problems

Product conjecture. The product of the $u_i$-tilted matrices

Let $P^\lambda$ be an $n\times n$ stochastic matrix of the form $P^\lambda$. Let $J(n)$ denote the $n\times n$ matrix whose entries are all equal to $1/n$, and let the matrices in…

Let $S_{\star}$ be the set of sites on which the condensate may reside, and let $r(x,y)$ be the jump rates of the underlying random walk. Let $Y^{\mathrm{rv}}(\cdot)$ be the Markov…

Necessity conjecture. This $n$-way recombination is necessary for exact reversibility: if a child is bred from two parents, then the child will be more similar to each parent than…