8 problems

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…

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

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…

Replica-symmetry converse conjecture. At least when $\beta_2>0$, the converse implication is also true. This concerns the equivalence between the instability condition…

For $T\geq 1$, let $\mathcal{P}^T$ be the $T$-fold product of the trial distribution space, let $p_T^{(1)}(\underline{\mathsf{x}})$ be the generalized one-letter functional, and le…

Let $d_{\mathrm{cond}}$ be the condensation threshold, let $\mathrm{corr}(d)$ denote the correlation quantity used to define the reconstruction threshold, and let…

Let $(\beta,h,\gamma)$ be parameters of the independent set problem, and let an MP-structure consist of a hierarchy $(\zeta,\mathcal V_{\boldsymbol m})$ with $K$ levels. For such a…

Equality conjecture. The generalized AT region and the replica-symmetric region are equal: