4 problems

Let $d\ge 1$, let $E_d$ be the set of unoriented nearest-neighbor edges of $\mathbb{Z}^d$, and let $\omega=(\omega_e)_{e\in E_d}\in(0,\infty)^{E_d}$ be a translation-invariant ergo…

Let $p\in(p_c,1)$, where $p_c=\frac{1}{2}$ is the critical probability for bond percolation on $\mathbb{Z}^2$. For the biased random walk on the infinite percolation cluster, let…

$L^1$ convergence conjecture. There exists a constant $M\in(-\infty,+\infty)$ such that

Let $p(\beta,\nu)$ be the quenched free energy, let $\lambda(\beta)$ be the corresponding annealed free-energy function, let ${\cal D}(\nu)$ be the domain of admissible inverse tem…