6 problems

Large-activity monochromaticity conjecture. For activities $z$ large enough, every Widom–Rowlinson measure is monochromatic. The paper proves the existence of polychromatic phases…

Consider the symmetric Widom–Rowlinson model in dimension $d\geq 2$, with radius distribution $Q$ satisfying the integrability assumption and with $Q(\{0\})<1$. Small-atom extensio…

Let $WR(\boldsymbol{z},\boldsymbol{Q})$ denote the set of Widom–Rowlinson measures in the non-integrable setting, with $q$ ordered phases. Large-activity ordered-phase conjecture.…

Let $B_0,B_1,\ldots,B_n$ be an isoperimetric progression, and let $S_0$ and $S_n$ denote the seeds of its initial and final sets as specified below. Property of connecting progress…

Weighted occupancy-fraction conjecture. For any $\lambda_1,\lambda_2>0$, the weighted occupancy fraction is maximised over all $d$-regular graphs by $K_{d+1}$.

Distinct-activity partition-function conjecture. For any $\lambda_1,\lambda_2>0$,