12 problems

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

Ultrametric organization conjecture. The TAP critical points at free-energy level $f$ are organized in an ultrametric tree; uniformly sampled overlaps between two such states take…

A random overlap structure (ROSt) is a random structure encoding overlaps between states. A $p$-th power cavity field is the cavity field obtained from the $p$-th power interaction…

A random overlap structure (ROSt) is a random structure encoding overlaps between states; its law is stochastically stable when it is invariant under the relevant stochastic-stabil…

In rare-event statistics of complex sparse systems, ultrametricity refers to the emergence of an ultrametric structure in the relevant statistical organization. Ultrametricity conj…

Let $f(z)=\sqrt{-\log|\eta(z)|}$, where $\eta(z)$ is the Dedekind $\eta$-function, and let $z$ tend to rational points on the real axis. Compare this behavior with the limiting spe…

Consider a protein energy landscape decomposed into hierarchically organized basins of local minima. Let the equilibration time within a basin be compared with the time required to…

Ultrametricity is a property of a metric space in which the strong triangle inequality holds. The paper considers complex systems with extremal sparse statistics and rare-event sta…

Let $G_N$ be the Gibbs measures of a mixed $p$-spin glass model on the hypercube with zero external field. The model admits an Orthogonal Structure if there is a sequence…

Let $G$ be the asymptotic Gibbs measure, supported in $\ell_2$. A subset of $\ell_2$ is ultrametric if, for all $x,y,z$ in the subset,…

Let $\ROSt$ be a random overlap structure with law $\mathbb{P}$, and let $\mu$ be its sampling measure. Suppose that $\ROSt$ is stochastically stable for the cavity fields $l_p$ fo…

Let ${\cal S}_N$ be the SK configuration space, let $d_N(\sigma,\tau)=\#\{i:\sigma_i\ne\tau_i\}$ be the Hamming distance, and choose $k$ points $\sigma_1,\dots,\sigma_k$ independen…