209 problems

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

Let $\mathbf G_d(n)$ be a random $d$-regular graph on $n$ vertices, and let the max-cut and min-bisection of $G$ be measured by the numbers of edges in the respective cuts. Zdeboro…

Consider the Edwards–Anderson spin-glass model on a sparse random graph drawn from $\mathbold{G}(n,d/n)$, with inverse temperature $\beta>0$ satisfying … where $\mathbold{J}$ is a…

In the classical Sherrington–Kirkpatrick model, set $b=0$, let $q$ be a solution of … where $z$ is a standard Gaussian random variable, and let $R_{1,2}$ denote the overlap of two…

Talagrand's conjecture. The critical overlap satisfies

Let $F_N(\beta)$ denote the free energy of the Sherrington–Kirkpatrick model with $N$ spins at inverse temperature $\beta$, and let the critical inverse temperature be $\beta_c=1$.…

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

Unique-solution conjecture. For the model at small $\alpha$, the TAP equations have a unique solution $(\mathbf{m}^\star,\mathbf{n}^\star)$, and $\mathbf{m}^\star$ approximates the…

Generalized Almeida–Thouless conjecture. At sufficiently high temperature, assuming that this self-consistent equation has a unique fixed point, the condition

Parisi's ultrametricity conjecture. Certain spin glass models have the property that, in the “$n=\infty$ limit”,

The diluted $K$-spin model assigns spin configurations on a random $K$-uniform Erdős–Rényi hypergraph, with objective given by the cut density. A factor of i.i.d. process is a proc…

Let $\mathbb{G}(n,c/n)$ be the Erdős–Rényi random graph sequence and let $\mathbb{G}(n,\Delta)$ be the random regular graph sequence. Right-convergence is convergence of the normal…

Consider disordered systems whose couplings are sampled from a probability distribution, including the Bernoulli distribution as a discrete example. Discrete-distribution conjectur…

The Sherrington–Kirkpatrick model has a Parisi functional whose argument is a functional order parameter. Panchenko's convexity conjecture. The Parisi functional is convex in the f…

Let ${\vec m}$ be the parameter vector in the definition of the functional ${\rm P}_k$ in (Pk). Convexity conjecture. The functional ${\rm P}_k$ is convex in ${\vec m}$. The conjec…

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

A spin glass model is considered with symmetric boundary conditions, and its finite-volume Gibbs measures are studied as the volume tends to infinity. Newman–Stein's conjecture. So…

Stability-threshold divergence conjecture. The critical value $u_{\rm s}(c)$ diverges as the mean connectivity approaches $c_{\rm s}^{(+)}$.

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

Consider unrestricted multi-way partitioning: divide $\{X_1,\ldots,X_n\}$ into $q$ subsets so that their sums are as equal as possible, without requiring every subset to have cardi…

Let the costs of two partitions be represented by spin configurations, and consider configurations associated with nearby fixed typical energies. Typical-energy overlap conjecture.…

Represent partitions by spin configurations and associate overlaps to pairs of configurations corresponding to nearby energy levels. Overlap part of the local REM conjecture. The c…

Conjecture C.1. The point process on $\mathbb R_+^{k-1}$