38 problems

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 $\mathcal{D}$ 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 $\mathsf{M}(\bm W)$ denote the Sherrington–Kirkpatrick Hamiltonian instance with random matrix $\bm W$, and let $c(\bm W)$ be the approximation ratio certified by an algorithm.…

Entropy conjecture. The exact location of the multicritical point is given by

Multicritical-point conjecture. The exact location of the multicritical point is determined by $x_0=x_0^{*}$ with $K=K_p$, equivalently

Aspelmeier's conjecture. The variance satisfies

Ultrametric organization conjecture. The TAP critical points at free-energy level $f$ are organized in an ultrametric tree; uniformly sampled overlaps between two such states take…

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 $f\in D$,

Assume the setup of the paper's fl-RDT theorem, including the quantities $\psi_{rp}$, $\psi_{rd}$, $\hat{\mathbf p}$, $\hat{\mathbf q}$, and $\hat{\mathbf c}$. Define the algorithm…

Let $\mu_1,\mu_2,\mu_3$ be probability measures with mean $0$ and variance $1$, and let $\Delta(\mu)$ be the discrepancy of a probability measure. Let $\beta(\mu,\infty)$ denote th…

Let $\mu$ be a probability measure with mean $0$ and variance $1$, and define its discrepancy by … Let $\beta(\mu,\infty)$ denote the scaling exponent of the runtime of $\infty$-re…

Let $\mu$ be a “sufficiently nice” probability measure with mean $0$ and variance $1$, and let $\beta(\mu,0)$ denote the scaling exponent of the runtime of $0$-reluctant, or greedy…

Let $\mu$ be a “sufficiently nice” probability measure on $\mathbb{R}$ with $\mathbb{E}_{X \sim \mu}X=0$ and $\mathbb{E}_{X \sim \mu}X^2=1$, let $\lambda\in[0,+\infty]$, and suppos…

Let $F_N^{SK}(\beta)$ denote the free energy of the Sherrington–Kirkpatrick model with system size $N$ at inverse temperature $\beta$, and let the critical value be $\beta_c=1$. Cr…

Let $Z_{\bm{G}}(\beta)$ denote the partition function of the Sherrington–Kirkpatrick model, let $\varepsilon>0$, and let $\beta_{\mathsf{2nd}}$ be the second-moment threshold. A co…

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

Consider a continuous random energy model (CREM) with concave covariance function $A$, inverse temperature $\beta>\beta_c$, \tilde\text{ and } $\text{?}$, and $\text{?}$. Let…

Let $c$ denote the planted-clique proportion, let $\theta$ and $\theta_1$ be the model parameters, and let $q$, $\mu$, $h$, and $z$ be the quantities appearing in the planted Sherr…

Let $f$ denote the unique viscosity solution of the infinite-dimensional Hamilton–Jacobi equation governing the enriched free energy, and let $\mathbf{G}_N$ and $\sigma^*$ denote t…

Let $H_N$ be a mixed $p$-spin Hamiltonian on the sphere, let $\mathcal{N}(D)$ denote the number of its local maxima on $S_{N-1}$ whose normalized radial derivative lies in $D$, and…

Let $(U,X)$ be a random pair satisfying the consistency equation … where $(U_k,X_k)_{k\geq 1}$ are independent copies of $(U,X)$, $(\xi_k)_{k\geq 1}$ and $(\zeta_k)_{k\geq 1}$ are…

Slow-thermalization conjecture. Starting from the uniform measure, Langevin dynamics takes exponential time to reach equilibrium for every $T<T_{sh}$: there is a constant $c>0$ suc…

Let $T_s$ and $T_{sh}$ denote the static and shattering transition temperatures, respectively. For a temperature $T$, say that the free energy landscape is $(E,q,r)$-shattered when…

Fix a disordered spin system of the type considered in the paper in dimension $d=2$. Let $\beta>0$ be the inverse temperature, $\tau_1,\tau_2\text{?}$ be boundary conditions in…