7 problems

Endpoint equality conjecture. The equality

Let $\Phi$ be the Dyson interaction with standard couplings of the form $J(k)=O(k^{-\alpha})$, and let $\mathcal{L}_{\beta\phi}$ be its transfer operator. In the regime…

Johansson–Öberg–Pollicott's conjecture. The critical inverse temperatures are equal:

Let $L$ be the transfer operator for the Dyson potential, let $alpha>1$ be the interaction exponent, and let $\beta_c(alpha)$ be the critical inverse temperature. Dyson eigenfuncti…

Let $\beta_c(alpha)$ be the critical inverse temperature of the Dyson model, and let $L$ be its transfer operator. Subcritical eigenfunction conjecture. In the regime corresponding…

Let $L$ be the transfer operator associated with the Dyson model, with interaction exponent $alpha$. Sharpness conjecture. For $alpha greater than or equal to 0$?

Let $\beta_c^+$ denote the critical inverse temperature of the half-line Dyson model and let $\beta_c$ denote the critical inverse temperature of the whole-line Dyson model. Equali…