49 problems

Sharp upper-bound conjecture. Without any further assumption on the model, the sharp upper bound for $p(\beta)$ should be the bound in equation (e117). The source compares this pro…

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

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…

Critical free-energy power-law conjecture. The third derivative of the free energy satisfies

Free-energy singularity conjecture. There exists a constant $\alpha$ such that

Let $\beta$ denote the inverse temperature, and let the free energy limit be the limit computed in the paper. All-temperature free energy conjecture. Although the computations are…

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 $d\geqslant1$, and let $\theta,u,v$ be parameters in either the maximal current regime or the equilibrium regime. Consider the inverse-Wishart polymer on a strip of arbitrary w…

Fix $d\geqslant 1$ and a parameter $\theta$ such that $2\theta>\frac{d-1}{2}$. Consider the inverse-Wishart polymer with delta boundary condition, defined by the step boundary cond…

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…

Let $\xi$ be the model's interaction function, let $\overline F_N(t,0)$ denote the usual free energy without enrichment, and let $f$ be the limit free energy characterized, in the…

In the four-dimensional $O(N)$ model, consider a complex-valued action evaluated at a complex saddle. ABS conjecture. The physical free energy is the real part of the complex-value…

Let $\eta$ be an integrable random variable taking values in $\{1,2,\ldots\}$ with $m=\mathbf{E}(\eta)>1$, and define the generalized Derrida–Retaux process by … where the…

Let $X_D$ be the class of separable functions defined by $chi(x)=\sum_{d=1}^D\chi_d(x_d)$, let $psi_*$ be the convex dual functional … and let $bar\xi$ possibly be non-convex on…

Let $bxi$ be the interaction function on $S^D_+$, and let the model satisfy the assumptions used for the convex theory except that $bxi$ is not assumed to be convex. The limit free…

Let $\overline\xi$ be the interaction function on $S^D_+$, and let $f$ denote the viscosity solution of the Hamilton–Jacobi equation associated with the model. When $\overline\xi$…

Infinite-order criticality conjecture. The free energy satisfies

Scalar-overlap restriction conjecture. This restriction of the replica formula for the free energy is correct in general, including outside the setting of abelian groups.

Mourat's Hamilton–Jacobi conjecture. While neither $\xi$ nor $\psi$ need be convex nor concave in general, $\overline F_N$ converges to the solution of this Hamilton–Jacobi equatio…

Let $\Gamma_N(\lambda)$ denote the quantity associated with the spin-glass model, and let $h$ be the external field. Assume that $h=0$ and let $\lambda<0$. Dotsenko–Franz–Mézard co…

Let $G_1,G_2,\ldots$ be a sequence of bipartite $Delta$-regular graphs, and let $f(G,k)=\frac{1}{n}\ln|\mathscr{C}_k(G)|$ denote the free energy per variable. Let…

Let $d\in\mathbb{N}$ and let $\gamma>0$. Define the free-energy function $\lambda(\gamma)$ by … Set … and let $\gamma_c$ denote the critical parameter introduced earlier in the pap…

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 $\chi(\rho)$ be the variational free-energy functional over shift-invariant marked point processes $P$ satisfying $P(N_U^{{{\scriptscriptstyle{({\ell}})}}})=\rho$, and let…