Sharp Reinforcement Thresholds for Tree and Whisker Components

Definitions and mathematical statement

The supplied record concerns supercritical reinforcement, with parameter $\alpha>1$, on an underlying graph $G$, with vertex firing rates $p(v)$. It does not specify the reinforcement dynamics, admissible graphs, initial conditions, or construction of the surviving graph. Write $H_\alpha$ provisionally for that surviving graph, and $\mathcal C(H_\alpha)$ for its connected components. This notation does not supply the missing construction. A tree is a connected undirected graph containing no cycle. The source does not define a whisker; that term must retain its source-specific meaning pending verification.

The intended problem is to improve the supplied sufficient bounds $\alpha>2$ for the assertion \[ \text{every }C\in\mathcal C(H_\alpha)\text{ is a tree}, \] and $\alpha>25$, under the additional assumption $p(v)=1$ for every vertex $v$, for the assertion \[ \text{every }C\in\mathcal C(H_\alpha)\text{ is a whisker}. \] In particular, determine whether both assertions hold throughout $\alpha>4/3$, under their respective original hypotheses, and whether $4/3$ is the optimal threshold for each. The supplied excerpt states that replacing the bounds by $\alpha>\alpha_0$ with $\alpha_0<4/3$ does not give valid conclusions in general. It does not specify the counterexamples or the behavior at $\alpha=4/3$. The admissible graph and rate quantifiers, the probability qualifier for the conclusions, and the precise meaning of optimality require recovery from the complete original problem statements. Consequently, this is a conditional reconstruction of the target, not a complete specification of the stochastic model.

The supplied original-statement field contains the labels "Open Problem 1" and "Open Problem 2" without their statements, followed by the lower-bound observation and a separate discussion of the critical case $\alpha=1$. That discussion says that a function $L$, not defined in the record, is non-strictly concave and that equilibrium need not be unique. For an even cycle of length $2k$ with $p(v)\equiv1$, it gives equilibria \[ \{(a,b,a,b,\ldots,a,b):a+b=2\}, \] without specifying the admissible domain of $a,b$. It states that odd cycles have a unique equilibrium and that, as the even-cycle length increases, the law of $(a,b)$ converges to the Dirac measure at $(1,1)$; the equilibrium selection and probability law are not defined. Motivated by viewing ${\mathbb{Z}}$ as an infinite cycle, the excerpt also records a separate Conjecture 2: for $G={\mathbb{Z}}$ and $p(v)\equiv1$, almost surely $x_v(t)\to1$ as $t\to\infty$ for every $v$. The quantities $x_v(t)$ and their normalization are not defined here. These critical-case statements are supplied context and are not substituted for the supercritical threshold problem.

Short English statement

Determine whether the surviving components are trees for every $\alpha>4/3$, and whiskers for every $\alpha>4/3$ when all firing rates equal one, and whether these thresholds are optimal. The supplied record omits the stochastic model, graph hypotheses, probability qualifier, and definition of a whisker, so a fully quantified statement requires source review.

Sources

• Local UnsolvedMath registry, supplied record with permanent ID $30005455$, source code catalogue entry, title "Sharp Reinforcement Thresholds for Tree and Whisker Components." The record belongs to the Oberwolfach Reports Open Problems collection and gives the year 2023. No individual proposer is supplied.
• MATRIX-MFO Tandem Workshop: Stochastic Reinforcement Processes and Graphs (2023), Oberwolfach Reports, pp. 639--679, DOI 10.4171/owr/2023/12: https://doi.org/10.4171/owr/2023/12. This is the supplied source report and is also listed as the "original Oberwolfach report" among the literature sources.
• Hirsch--Holmes--Kleptsyn, Infinite WARM graphs III: https://doi.org/10.1088/1361-6544/acc9a0. This reference is supplied in the literature-source list; its contents have not been independently checked here.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem "partially solved" and reports finite surviving components, tree or whisker structure in sufficiently large-$\alpha$ regimes, and phase transitions, while saying that the sharp lower bounds requested by the two problems remain unresolved. This is supplied dated curation, not an independently verified literature assessment; its status and cited claims require independent verification.
• The registry labels the statement corrected_verified and says it was checked against the official report and preserves its hypotheses, notation, and target. This is a supplied verification claim, not verification performed for this draft. The original excerpt's reference label [8] has no matching bibliographic entry in the supplied record.