11 problems

Fix $(s,t,p,q)\in{\mathbb R}^4$. Let ${\cal Q}_n^{(s,t,p,q)}$ be the quadri-tilings of ${\cal Q}_n$ with first height change $(\lfloor ns\rfloor,\lfloor nt\rfloor)$ and second heig…

For $\alpha\in[0,1]$, let $f_{\mathrm{null}}(\alpha)$ be the quenched free energy of the null Random Subsequence Model, let $f^{\mathrm{BMM}}(\alpha)$ be the quenched free energy o…

Let $f_{\mathrm{pl}}(\alpha)$ and $f_{\mathrm{pl}}^{\mathrm{ann}}(\alpha)$ denote the planted quenched and annealed free energies, respectively, for aspect ratio $\alpha$. Planted…

Let $\mathcal Q$ be the space of paths used to parameterize overlaps, let $q=(q_1,q_2)\in\mathcal Q^2$, and let $\psi_2:\mathcal Q^2\to\mathbb R$ be the initial free-energy functio…

Infinite-order criticality conjecture. The free energy satisfies

Let $\mathcal{D}$ be the distribution of the disorder random variable, and suppose that it has finite second moment. Throughout the paper the disorder is assumed to be symmetric, w…

Let $\hat\beta_{\mathrm{eff}}$ be the effective inverse temperature defined in the source, and let $f^{\operatorname{Coul}}(\beta)$, $f^{\operatorname{IV}}(\beta^{-1})$,…

Let $F_q(\beta)$ and $F_a(\beta)$ denote the quenched and annealed free energies of the directed polymer, respectively, let $d_s$ be the spectral dimension of the underlying fracta…

Genus two G-function vanishing conjecture. For every such $M$,

Let $V_{d+1}(x)$ be any potential of degree $d+1$, and let ${\widetilde{\epsilon}}_i$ be the occupation numbers on the cycles ${\widetilde{{\cal A}}}_i$, for $i=1,\ldots,d$. Here…

Uniform Hessian bound. There is a constant $C$, independent of $k$, such that