4 problems

Derrida–Griffiths–Higgs conjecture. The following assertions hold: (i) the specific heat, namely the derivative of the finite-volume free energy, converges $\mathbb P$-almost surel…

Fix $t>0$, $\Delta>0$, two segments $[X,X']$ and $[Y,Y']$ in $\mathbb R$, and real numbers $(x_1,\dots,x_n)$, $(y_1,\dots,y_n)$, $(\tilde x_1,\dots,\tilde x_k)$, and…

Fix $k\geq 1$ and $c\in(0,\infty)$. Let $\alpha_1,\ldots,\alpha_k$ be real numbers and let $F(z_1,\ldots,z_k)$ satisfy $\alpha_k=0$, $\alpha_j>\alpha_{j+1}+c$ for $1\leq j\leq k-1$…

Let $\alpha\in(0,\infty)$, let $\chi:=\sum_{n\in\mathbb{N}}[\textbf{P}(S_n=0)]^2$, and let $\beta_c$ denote the critical inverse temperature separating the annealed and quenched cr…