6 problems

Let the quantum generalized random energy model (QGREM) have multiple phase transitions, with critical field strengths denoted by $\Gamma_k$, and let $\pmb{\sigma_k}$ denote the co…

CREM Gibbs-measure threshold conjecture. There should exist a threshold $\beta_G\geq 0$ such that the Gibbs measure can be efficiently approximated for $\beta<\beta_G$ and cannot b…

In the 2-WSBM, let $N$ be the number of vertices, $k$ the community size, and let $\gamma$ denote the signal-to-noise ratio; assume $k=o(\log N)$. For $k=2$, the model is equivalen…

Let $p_{\beta}(x)$ be the Gibbs-measure density in the random landscape, and let $\overline{p_{\beta}(x)}$ denote its disorder average. Write $\mathcal{P}(x)$ for the probability d…

Let $\beta$ denote the inverse temperature and let $\beta_c$ be the critical inverse temperature. In the high-temperature regime, suppose the model has a stationary equation whose…

Consider a directed polymer on a disordered Cayley tree, with independent random potentials on the bonds, partition function $\mathcal{Z}(t)$, and generating function … which satis…