14 problems

Bauke–Mertens conjecture. In most circumstances, the local statistics of energies in random spin systems with discrete spin space should be the same as in the random energy model.

Consider discrete disordered systems with an energy function generated by random disorder variables. Local REM universality conjecture. The local REM property should hold for discr…

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…

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…

Let the energies of a number-partitioning model be ordered increasingly, and consider the energy levels near a fixed energy. Local REM conjecture. The local energy spectrum should…

Let $H_n$ be a random Hamiltonian with an exponentially large collection of energy levels, and suppose the levels are properly rescaled. A REM universality conjecture asserts that,…

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 $X_u$ be the CREM field on the binary tree, with $u e0$ denoting a vertex and $|u|$ its depth. Suppose that an algorithm has access only to the leaf values $Xu$ for $|u|=N$. **…

Let $\beta_G$ denote the conjectured sampling hardness threshold for the continuous random energy model (CREM), and let the Gibbs distribution at inverse temperature $\beta$ be the…

Mueller–Munier conjecture. The typical size of $\mathcal{C}([-x,0])$ differs from $\mathrm{e}^{\sqrt{2}x}$ by a stretched-exponentially small factor:

For the quantum $p$-spin model, let the phase diagram be the one described by Figure, and consider its behavior at low temperatures as the interaction order $p$ varies. Smooth…

The $L^1$ convergence conjecture. The convergence above also holds in $L^1$. This concerns the complex BBM energy model with binary branching, for any $\rho\in[-1,1]$ and any…

Let $R_N$ be the random scaling defined in the paper, let $\beta$ and $\alpha$ be the model parameters, and let $h(N)$ be a function growing at most sub-exponentially. Deterministi…

Large deviation-function existence conjecture. In the present case, where $\mu$ is finite only for $-q_c<q<1$, the large deviation function $\psi$ exists; when it exists, it is giv…