4 problems

For loop percolation on the integer lattice $\mathbb{Z}^d$, let $\alpha_c(0)$ denote the critical parameter for percolation at $\kappa=0$, and let … denote the associated susceptib…

The algebraic rationality phenomenon. There exists a rational function

The rationality phenomenon for finite submatching events. For $n\ge k$, there is an even polynomial $Q_{\pi_0}(n)$ of degree $k(k+1)$ with dyadic rational coefficients such that

The rationality phenomenon for submatching events. For any $\pi_0\in\textnormal{NC}_{n}$, the probability of the associated submatching event is a dyadic rational number that is co…