3 problems

Let $X$ be the normalized exponential functional introduced in the source, and let $q$ be in the range where the displayed moments exist. Freezing hypothesis. For $\beta>1$, … This…

Let $f_{\beta}$ be the free-energy random variable, let $G$ be an independent Gumbel random variable, and define … Let $V_m$ denote the minimum of the underlying logarithmically co…

Consider a system with a temperature parameter $T$ and a duality transformation $T\mapsto 1/T$. A function is self-dual if it is invariant under this transformation. Freezing-trans…