Noise Sensitivity and Exceptional Times in Spin-System Dynamics

Definitions and mathematical statement

The supplied record bundles several questions rather than specifying a single fully quantified conjecture. The following definitions provide a framework for stating those questions; the missing model parameters and quantifiers remain unresolved.

An Ising configuration on a graph $G=(V,E)$ is an element of $\{-1,+1\}^{V}$. Glauber dynamics updates individual spins according to a specified rule preserving a chosen Ising Gibbs measure. The record does not specify the interaction, external field, temperature, boundary conditions, Gibbs measure, or update rates. It specifies transitive graphs for the Ising questions but does not identify a family of such graphs or the intended finite-volume approximation.

For reference, on a finite graph the FK random-cluster measure with parameters $0<p<1$ and $q>0$ has the form \[ \phi_{G,p,q}^{\xi}(\omega) =\frac{1}{Z_{G,p,q}^{\xi}} p^{o(\omega)}(1-p)^{|E|-o(\omega)}q^{k_{\xi}(\omega)}, \qquad \omega\in\{0,1\}^{E}, \] where $o(\omega)$ counts open edges, $\xi$ specifies boundary identifications, and $k_{\xi}(\omega)$ counts the resulting connected components. Heat-bath dynamics resamples an edge from its conditional distribution given the other edges. This is a conventional finite-volume definition, not a selection of the source's intended parameter range, boundary conditions, or update-rate normalization; those are unspecified.

For a sequence of stationary reversible continuous-time chains $(X_t^{(n)})_{t\geq0}$, with stationary laws $\mu_n$, and Boolean observables $f_n$, a standard dynamical noise-sensitivity condition is \[ \forall t>0,\qquad \operatorname{Cov}\left(f_n(X_0^{(n)}),f_n(X_t^{(n)})\right) \longrightarrow 0\quad(n\to\infty). \] The source does not specify this limiting sequence, the clock normalization, whether covariance is normalized by variance, or whether nondegeneracy of the observables is required. Consequently this condition is a reference definition, not a uniquely determined interpretation of the record.

A weighted threshold observable on finitely many spins can be written \[ M_{w,a}(\sigma)=\mathbf{1}\!\left\{\sum_{v}w_v\sigma_v\geq a\right\}. \] A weighted-majority convention must additionally specify admissible weights, the threshold, and treatment of ties. The record does not state the characterization to be extended, the relevant correlation criterion, or its quantifiers.

For a finite irreducible reversible chain with generator $L$ and stationary law $\mu$, put \[ \mathcal E(g,g)=-\langle g,Lg\rangle_{\mu},\qquad \lambda=\inf_{\operatorname{Var}_{\mu}(g)>0} \frac{\mathcal E(g,g)}{\operatorname{Var}_{\mu}(g)},\qquad t_{\mathrm{rel}}=\lambda^{-1}. \] Under one classical log-Sobolev convention, \[ \alpha=\inf_{\operatorname{Ent}_{\mu}(g^2)>0} \frac{\mathcal E(g,g)}{\operatorname{Ent}_{\mu}(g^2)},\qquad \operatorname{Ent}_{\mu}(g^2) =\mu\!\left[g^2\log\frac{g^2}{\mu[g^2]}\right]. \] In this convention, comparison of a log-Sobolev time with relaxation time would mean bounds of the form \[ c\,t_{\mathrm{rel}}\leq\alpha^{-1}\leq C\,t_{\mathrm{rel}}, \] with specified uniformity of $c,C>0$. The source instead says that log-Sobolev constants are comparable to relaxation times, without fixing a reciprocal convention, a classical versus modified inequality, or the family over which comparison is intended. The displayed bounds therefore describe a possible interpretation only.

A crossing event is the existence of a specified spin cluster or open-edge connection between designated boundary portions of a planar domain. The source does not identify the model, cluster type, domains, boundary conditions, critical parameters, or scaling sequence for its crossing question.

For a stationary process with law $\mu$, an exceptional time for a specified property $A$ of probability $\mu(A)=0$ is a time $t$ at which $X_t\in A$. An existence question can be expressed as \[ \mathbb P\left(\exists t>0:\ X_t\in A\right)>0. \] The source does not identify $A$, the intended time interval, or whether existence means positive probability or almost sure existence.

Questions retained from the source. For Ising Glauber dynamics on transitive graphs and heat-bath dynamics of FK random-cluster models, determine: (i) whether a weighted-majority characterization of noise sensitivity persists; (ii) whether the intended log-Sobolev quantity is comparable to relaxation time; (iii) whether critical planar crossing events are noise sensitive; and (iv) whether exceptional times exist. The assignment of these questions to particular models and parameter regimes is unspecified. A complete conjecture requires the qualifications listed above; no universal assertion over all model choices is supplied here.

The supplied dated curation labels the bundle partially solved and reports absence of exceptional times for critical Ising Glauber dynamics on the square lattice, while describing the bundled FK, noise-sensitivity, log-Sobolev, and crossing questions as unresolved. These are supplied literature assessments, not independently verified mathematical conclusions. In particular, the exceptional property and hypotheses of the reported Ising result are not given.

Short English statement

The record asks about weighted-majority criteria for noise sensitivity, comparison of log-Sobolev and relaxation quantities, critical planar crossings, and exceptional times in Ising and FK dynamics. Essential parameters, limiting procedures, and the exceptional property are missing, so a unique fully quantified conjecture cannot be recovered. The supplied partially solved label and square-lattice Ising claim require independent mathematical and source review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004588$, title "Noise Sensitivity and Exceptional Times in Spin-System Dynamics," exact source code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record gives proposed year $2021$ and identifies no proposer.
• Supplied source citation: Spatial Networks and Percolation (2022), Oberwolfach Reports, pp. 149--224, DOI 10.4171/owr/2021/4. Supplied source and literature link: https://doi.org/10.4171/owr/2021/4. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. No passage pinpointing these questions is supplied.
• Supplied literature reference labelled "Schramm and Steif": https://doi.org/10.4007/annals.2010.171.619. The record supplies no title, theorem number, or explicit mapping from this reference to the individual claims.
• Literature assessment date: 2026-08-22. This is supplied dated curation, not independently verified. It labels the bundle partially solved, reports absence of exceptional times for critical Ising Glauber dynamics on the square lattice, and describes the bundled FK, noise-sensitivity, log-Sobolev, and crossing questions as unresolved. The supplied background explicitly calls this dated literature triage derived from OWR curation and requests independent verification of time-sensitive status and cited claims.
• The registry marks the statement status as "exact," while its verification note says that "three conjectural components" were retained with normalized notation and sentence structure. The supplied statement presents four question clauses; their intended grouping and fidelity to the report require review.