6 problems

Polyakov's conjecture. This $\beta$-dependent quenched disorder should destroy the power-law decay of the induced $XY$ model for every $\beta>0$.

Polyakov's conjecture. The two-dimensional Heisenberg model should never enter a BKT phase at large $\beta$.

Consider the angular variable $theta$ of the XY or Villain model at scales much larger than the microscopic scale, at a temperature below the critical temperature. Spin-wave conjec…

Metastable random-walk conjecture. (i) The metastable time satisfies $\tau^{(N)}=C(J_N,\sigma_N)e^{J_N/\sigma_N^2-\log N}$, where $C(J,\sigma)$ is polynomial in $J$ and $\sigma$. (…

Let $D$ be the domain, let $\theta^{\varepsilon}$ have the Gibbs measure with external field $h>0$ given by, and let $\eta^{\varepsilon}$ be the associated gradient field defined i…

XY-model scaling-limit conjecture. This object should describe the scaling limit of the magnetization of the XY model in the plasma phase. This conjecture connects complex Gaussian…