5 problems

Let $u$ be the interpolation parameter, and let $\beta_{\rm c}^{(2)}(u)$ denote the critical inverse temperature for the loop model with the quantum weight…

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…

Aldous's conjecture. This random partition has Poisson-Dirichlet$(1)$ distribution.

Exponential-decay conjecture. This correlation function has exponential decay with respect to $\|x-y\|$.

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…