15 problems

Parisi formula conjecture. For all $(\kappa,\alpha)\in\mathbb{R}\times\mathbb{R}_+$,

The quenched complexity formula. There exists an open subset $D\subset\mathbb{R}$ such that, for every $f\in D$,

The annealed complexity formula. There exists an open subset $D\subset\mathbb{R}$ containing $[\inf_\zeta\mathcal{P}\mathrm{arisi}(\zeta),+\infty)$ such that, for every $f\in D$,

Let $\xi:\mathbb{R}^D_+\to\mathbb{R}$ be the model's interaction function, and let $S_t\chi$ denote the operator defined according to equation … . Theorem … are the Hopf–Lax-type a…

Let $\mathcal{L}$ be the class of functions on $[0,1]$ with appropriately bounded total variation, let $\mathcal{P}$ be the Parisi variational functional, and let $p=2$. Extended P…

Let $\mu_{p,\rm OPT}$ be the Parisi measure solving the variational problem for the spin-glass ground-state energy, and let its cumulative distribution function be the associated d…

Parisi PDE conjecture. The enriched free energy $F_N$ should converge to a function $f:\mathbb{R}_+\times\mathcal Q^2\to\mathbb{R}$ solving

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 $r_N(\boldsymbol{\xi},\boldsymbol{\xi})$ be the self-overlap, let $\varrho$ denote the Parisi order parameter, and let the Parisi variational formula contain an infimum over Pa…

Replica-symmetry converse conjecture. At least when $\beta_2>0$, the converse implication is also true. This concerns the equivalence between the instability condition…

Let $\beta$ denote the inverse temperature, and let the thermodynamic limit of the free energy be viewed as a function of the squared inverse temperature $\beta^2$. Parisi's concav…

Let $\mathcal P_{\xi,h}$ be the Parisi functional associated with the mixed $p$-spin model parameters $\xi$ and $h$. A minimizer of the Parisi variational formula is called a Paris…

Let $cmu_P$ be a Parisi measure for the mixed even-spin model, defined as a weak limit of probability measures satisfying condition $cmathrm{MIN}(cvarepsilon_n)$ for some sequence…

Convexity conjecture. The functional ${\cal P}(\zeta)$ is convex in $\zeta$.