31 problems

Replica prediction for the single-index global maximum. Under this double limit,

Let $p_e$ be the point where the 1RSB and replica-symmetric (RS) error exponents coincide, with $p_e>p_{\rm 1RSB}$. Let $p_{\rm RS}$ denote the spinodal point below which the RS so…

The Ising $2$-spin Sherrington–Kirkpatrick model is the mean-field spin-glass model with Ising spins and pairwise Gaussian interactions; its overlap distribution describes the law…

Gardner's conjecture. There are two phase transitions: $\mu_P$ is RS, specifically $\mu_P=\delta_0$, for $0<\beta\leq\beta^p_1$; it is 1RSB, specifically

Generalized Parisi conjecture. Replica symmetry breaking, when properly applied, should always yield the correct free energy; in particular,

Consider a spin-glass model with near-optimal configurations and call a near-optimal configuration non-marginal if it does not exhibit marginal stability, meaning that the relevant…

Let $\pi$ be a distribution over $\Lambda_{k,r}$ and let $H_n(\pi,k,m)$ be the associated random uniquely extendable constraint system. Suppose that either $k\geq 4$ and…

Let $N$ be the number of components and $L^d$ the volume of an elastic manifold of internal dimension $0<d<4$. In the full replica symmetry breaking (FRSB) regime, let…

For the Sherrington–Kirkpatrick model, let $\beta$ denote the inverse temperature, and let $\beta>\frac{1}{\sqrt{2}}+\eta$, where…

Consider the superposition-of-plane-waves model described in the paper, with replica-symmetric and replica-symmetry-broken (RSB) solutions, and let the annealed complexity and quen…

An indecomposable metastate is a metastate that cannot be decomposed into a nontrivial mixture of metastates, and it has the relative weak mixing property when the corresponding Gi…

Intrinsic replica-symmetry breaking conjecture. A somewhat similar picture should apply to the low-temperature states of the SK model and to the Edwards–Anderson model under a magn…

Let $\alpha$ denote the constraint density in the symmetric binary perceptron. Let $\Sigma(s)$ be the complexity as a function of the entropy $s$, with one curve obtained from the…

Let $\alpha$ denote the constraint density, let $\Sigma(s)$ be the complexity as a function of the entropy $s$ for clusters in the symmetric binary perceptron, and let $\kappa_0$ b…

Physicists' full-RSB conjecture. When $betarightarrowinfty$, the SK model exhibits the full-RSB property.

Consider a spherical spin glass with a full replica-symmetry breaking phase and covariance exponent $r>2$. Define … Let $P_{\kappa,g}(\delta)$ be the crossover family of fluctuatio…

Let $P_{\kappa,g}(\delta)$ denote a family of typical-fluctuation distributions for spherical spin glasses with a one-step replica-symmetry breaking phase, where $g$ specifies the…

Consider a spherical spin glass with a full replica-symmetry breaking phase and covariance exponent $r>2$. Define $\eta=\frac{3(r-2)}{2r-3}$, and let $e_c$ be the limiting ground-s…

Consider a spherical spin glass with a full replica-symmetry breaking phase, covariance correction exponent $r>2$, and $\eta=\frac{3(r-2)}{2r-3}$. Let $e_c$ be the limiting ground-…

Consider the 1-RSB ansatz for the maximum-average submatrix model. The Type-I stability condition is obtained from the 2-RSB perturbation specified in the paper; Type-II and more g…

Let the 1-RSB solution be tested for stability through the Type-I perturbation obtained from the 2-RSB extremization conditions. The resulting threshold is a bound for the stabilit…

Consider the maximum-average submatrix model in the phase diagram, with $m\geq m_c$. At $a_{\rm stability}(m)$, the replica-symmetric phase undergoes a continuous transition to a f…

Let RSB hold in some dimension, let $\kappa_J$ be the zero-temperature periodic-boundary-condition metastate in that dimension, and let GSP mean ground state pair; two GSP's are in…

In the mixed $p$-spin model with mixture $\xi(t)=\sum_{p\geq 2}\beta_p^2t^p$, inverse temperature $\beta>0$, external field $h\geq 0$, and RS critical-point set … let … The previou…

Consider sparse-recovery cases exhibiting replica-symmetry breaking (RSB), and let $b$ denote the number of replica-symmetry-breaking steps. Full-RSB exact-solution conjecture. At…