3 problems

Let $\ell_1,\ell_2,\dots$ be the loop lengths ordered decreasingly, let $m(\beta)$ be the long-loop mass, and let $u$ be the model parameter. Define … Write…

Consider a random loop configuration in a box $\Lambda$ of side length $L$, with loop lengths $\ell_j(\omega)$ ordered decreasingly. Let $m(\beta)$ denote the limiting density of p…

Let $(\Lambda,\mathcal E)$ be a finite periodic box, let $\beta>0$, and let $L_1(\omega),L_2(\omega),\ldots$ be the loop lengths in decreasing order. The normalized loop lengths ar…