Critical Fixation Threshold for Zero-Temperature Glauber Dynamics

Definitions and mathematical statement

Let $d\in{\mathbb{N}}$ with $d\geq2$, and let $p\in[0,1]$. Write $\sigma_t(x)\in\{0,1\}$ for the opinion at $x\in{\mathbb{Z}}^d$ at time $t\geq0$, with independent initial opinions satisfying $\mathbb P_p(\sigma_0(x)=1)=p$.

For an explicit conventional interpretation of zero-temperature Glauber dynamics, give each vertex an independent rate-one Poisson clock, independent of the initial opinions. When the clock at $x$ rings, replace its opinion by the strict majority among its $2d$ nearest neighbours; if the neighbours are equally divided, choose either opinion with probability $1/2$, independently at each such update. Here nearest neighbours satisfy $\sum_{i=1}^d|x_i-y_i|=1$. The nearest-neighbour interaction, continuous-time clock convention, and independence of tie-breaking coins are conventional specifications made explicit here; the supplied record does not state them individually, so they require confirmation against the source.

Define the event of sitewise fixation by \[ F_d=\bigcap_{x\in{\mathbb{Z}}^d}\left\{\exists T_x<\infty\ \exists s_x\in\{0,1\}\ \forall t\geq T_x:\ \sigma_t(x)=s_x\right\}. \] The times $T_x$ may depend on the vertex and the realization; this definition does not require a common finite fixation time for the entire lattice. Failure of $F_d$ means that at least one vertex changes opinion infinitely often. Whether the source instead intends infinite changes at a specified vertex, or at every vertex, is not stated.

Source conjecture and unresolved threshold definition. The supplied statement asks whether \[ p_c(d)=\frac12\qquad\text{for every integer }d\geq2, \] particularly whether $p_c(2)=1/2$, and describes $p_c(d)$ as separating eventual fixation from infinitely many opinion changes. It gives no defining event, probability quantifier, or parameter range for this threshold. This omission is material: exchanging the two opinion labels preserves sitewise fixation while replacing $p$ by $1-p$, and both deterministic initial configurations $p=0$ and $p=1$ are fixed. Thus a single monotone threshold for fixation throughout $[0,1]$ cannot be inferred from the wording.

One possible precise interpretation would instead concern fixation to opinion $1$. Define \[ A_d^+=\bigcap_{x\in{\mathbb{Z}}^d}\left\{\exists T_x<\infty\ \forall t\geq T_x:\ \sigma_t(x)=1\right\}, \qquad p_c^+(d)=\inf\{p\in[0,1]:\mathbb P_p(A_d^+)=1\}. \] Under this explicitly conditional interpretation, the proposed target would be $p_c^+(d)=1/2$ for every $d\geq2$. The supplied evidence does not identify $p_c(d)$ with $p_c^+(d)$, so this is a possible clarification, not an attributed replacement statement. An infimum equality alone also does not specify the behaviour at $p=1/2$. The intended critical event and boundary behaviour remain unresolved.

Short English statement

The source conjectures a critical initial density of $1/2$ for zero-temperature Glauber dynamics on every ${\mathbb{Z}}^d$, $d\geq2$. Its threshold is not defined precisely: fixation to either opinion and fixation of every site to opinion $1$ are different events. The draft defines the conventional dynamics and records a conditional interpretation using fixation to opinion $1$, while leaving the intended threshold and critical boundary behaviour for source confirmation.

Sources

• Local UnsolvedMath registry, data/unsolvedmath/problems.json, permanent ID $30004593$, source problem code catalogue entry, title Critical Fixation Threshold for Zero-Temperature Glauber Dynamics. The supplied record lists proposed year 2021 and identifies no individual proposer.
• Supplied report 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 record describes Oberwolfach Reports as documenting workshops and their open-problem discussions. The supplied 2022 report label and 2021 proposed-year and DOI information are retained without resolving their relationship.
• Supplied literature reference: Morris, zero-temperature Glauber dynamics, https://doi.org/10.1007/s00440-009-0259-x). The intended supplied URL, without the closing punctuation, is https://doi.org/10.1007/s00440-009-0259-x.
• Literature assessment dated 2026-08-22: this is supplied dated curation, not independently verified. The supplied status is partially_solved. Its assessment says that the record combines proved finite-time square-lattice properties with unresolved strictness, limiting, hexagonal-lattice, and zero-temperature Glauber questions; it also says that high-dimensional asymptotics are known but the conjecture remains unsettled. No precise statements of those additional results or questions are supplied. The background requests independent verification of time-sensitive status and cited claims.
• The supplied statement status is corrected_verified; its accompanying verification concerns dollar-delimiting explicit formulas and preserving supplied mathematical content. This does not establish mathematical correctness or resolve the threshold definition. The external references have not been independently consulted for this draft.