75 problems

Fix $\b5\in\mathbb{R}$, let $\Sigma_N=\{\pm1\}^N$, and let $M_N=M_N(\b5)$ be the largest integer $M$ for which the Ising perceptron solution set … is nonempty, where…

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

Replica continuity conjecture. The analytic continuation of $f(\cdot)$ from $\mathbb{Z}^{+}$ to $\mathbb{R}^{+}$ equals the same function $f(m)$ for real $m$, and the limits with r…

Let a US-LDPC code be a code from the ensemble under consideration, and let its weight enumerator function (WEF) be calculated using belief propagation (BP) at a fixed point. Asymp…

Symplectic camel conjecture for statistical physics. The principle of the symplectic camel might play a fundamental role in thermodynamics and statistical physics, including Bose–E…

The scaling limits of many two-dimensional systems in statistical physics are expected to exhibit conformally invariant behaviour at criticality. Conformal invariance conjecture. A…

The models under consideration are isotropic short-range percolation models, with details including the underlying lattice and whether percolation is by sites or bonds. Their scali…

Let $P(\alpha)$ be the overlap distribution of a random pair of satisfying assignments of a random $k$-SAT formula, and let $P_{\rm ann}(\alpha)$ denote its annealed approximation,…

Dense-solution-region conjecture. Algorithmic success is associated with the existence of dense regions of solutions which, despite being rare and invisible to the equilibrium meas…

Let $R_\Lambda:\mathcal{K}\to\mathcal{K}$ be the renormalization-group map, let $\beta(k)$ be its beta function, and let $k^*$ be an RG fixed point satisfying $\beta(k^*)=0$. Let…

Let $\mathcal{K}$ be the space of kernels, let $\mathcal{S}^*(k)$ be the optimized path entropy for a kernel held fixed at $k$, and let a self-consistent kernel $k^*$ be a fixed po…

Let $\mathbf G_d(n)$ be a random $d$-regular graph, let $\beta_r$ be the reconstruction threshold, let $Z^{\mathsf{fix}}_{\mathbf G_d(n),\beta}(0)$ denote the fixed-magnetization f…

Let $L=1$ and let $\sigma$ be an activation function with second Hermite coefficient $\mu_2=0$. Exactness conjecture. For this one-hidden-layer perceptron, the replica symmetric th…

Let $\mathsf{P}_{s,\beta}$ denote a one-dimensional non-singular Riesz gas with homogeneity parameter $s$, and let $\mathsf{Leb}_{\mathbb{R}}$ be Lebesgue measure on $\mathbb{R}$.…

Let $\ell$ denote the context length, and let $\beta$ be the inverse temperature in a self-attention model considered in the proportional regime, where the context length grows wit…

Replica-symmetric transition conjecture. The estimate of this threshold obtained by the replica method with a replica-symmetric Ansatz coincides with the actual SAT/UNSAT transitio…

The probe and reference systems are parametrized by inverse temperatures $b$ and $b_{\rm ref}$, an overlap parameter $c$, and solution branches labeled solution 1 and solution 3. T…

Graphical models on locally tree-like ensembles are expected to have thermodynamic behavior similar to that of graphical models on regular lattices, particularly concerning phase t…

Generic statistical-ensemble conjecture. All Attention schemes are variants of a generic, abstract statistical ensemble, with a more complete set of pairwise and/or higher-order in…

For each $d\geq 3$, let $\alpha_d^\star$ denote the limiting independence ratio of a random $d$-regular graph, and let $\alpha_d^{\mathrm{1-RSB}}$ be the value predicted by the one…

Balancedness is a structural property of deep linear networks, represented by the balanced manifold $\balance$. Local equilibrium along a network is described as detailed balance.…

Kraichnan-scale conjecture. This logarithmic divergence in a forced turbulent flow in $d=2$ will depend not upon $L$ but instead upon the Kraichnan dissipation scale that terminate…

Consider a Bayesian inference problem with a noise variance parameter, and let AMP be used for estimation. Define the asymptotically Bayes-optimal estimation regime as the set of v…

Shattering conjecture. Shattering is absent if and only if

Consider a statistical physics game in which player I bets on a possibly coarse-grained partition of a dynamical system's state space and chooses the observation time, while player…