Monochromatic Arm Exponents in Critical Percolation

Definitions and mathematical statement

The supplied question concerns arm events in two-dimensional critical percolation. It does not specify the lattice, site or bond percolation, boundary conditions, or the precise arm-event convention. The following gives a conditional annulus formulation; these choices must be fixed before it specifies a unique problem.

Fix a planar percolation model at its critical parameter, with probability law $\mathbb P_c$. For each admissible positive integer $j$, choose a fixed inner radius $r_j>0$ for which the relevant $j$-arm events are possible. For $n>r_j$, let $A_j^{\mathrm{mono}}(r_j,n)$ denote the event that the annular region between radii $r_j$ and $n$ contains $j$ mutually disjoint arms connecting its inner and outer boundaries, all of one specified colour. The interpretation of colour, the underlying paths, and disjointness must follow the chosen model; in particular, closed arms may require a dual or matching-lattice convention. The source does not specify which monochromatic colour is intended.

For $j\geq 2$, let $\sigma=(\sigma_1,\ldots,\sigma_j)\in\{\mathrm{open},\mathrm{closed}\}^j$ contain both colours, and let $A_{j,\sigma}^{\mathrm{poly}}(r_j,n)$ denote the corresponding event with arm colours prescribed by $\sigma$ in cyclic order. When it exists, define its polychromatic exponent by \[ \alpha_{j,\sigma} =-\lim_{n\to\infty} \frac{\log\mathbb P_c\!\left(A_{j,\sigma}^{\mathrm{poly}}(r_j,n)\right)}{\log n}. \] No independence from the colour sequence, the inner radius, or the model is assumed here.

The source asks whether, for each intended admissible $j$, there is a finite exponent $\alpha'_j\geq 0$ such that \[ \mathbb P_c\!\left(A_j^{\mathrm{mono}}(r_j,n)\right) =n^{-\alpha'_j+o(1)} \qquad(n\to\infty), \] where $o(1)$ tends to zero with the model, $j$, colour, and inner radius fixed. Equivalently, it asks whether the finite limit \[ \alpha'_j =-\lim_{n\to\infty} \frac{\log\mathbb P_c\!\left(A_j^{\mathrm{mono}}(r_j,n)\right)}{\log n} \] exists. It further asks for the relation between these monochromatic exponents and the corresponding polychromatic arm exponents. The supplied record gives neither a proposed identity nor a specific comparison to be established, and it does not state the intended range of $j$ or the scope of models covered.

The supplied dated curation labels the problem partially solved: it reports existence and strict bounds for monochromatic arm exponents, a later exact evaluation of the monochromatic two-arm exponent, and remaining questions about exact values for the full higher-arm family and the general relation. This is a supplied literature assessment, not an independently verified theorem statement; its model-specific hypotheses and exact conclusions are not included in the record.

Short English statement

For a specified two-dimensional critical percolation model, ask whether the probability of $j$ disjoint arms of one colour reaching distance $n$ has a well-defined power-law exponent, and how that exponent compares with exponents for arms containing both colours. The model, arm conventions, and intended comparison need clarification. Supplied dated curation describes partial progress; this statement-only draft requires human mathematical review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001319$, source problem code catalogue entry, title Monochromatic Arm Exponents in Critical Percolation, in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied. The registry records proposed year $2009$.
• Supplied report citation: Scaling Limits in Models of Statistical Mechanics (2010), Oberwolfach Reports, pp. 2115--2178, DOI 10.4171/owr/2009/38. The literature-source label is "original Oberwolfach report 4081": https://doi.org/10.4171/owr/2009/38. The record describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. No precise page or passage for this question is supplied.
• Supplied literature reference labelled "Beffara-Nolin theorem": https://arxiv.org/abs/0906.3570. The record associates this reference with existence and strict bounds for monochromatic arm exponents, without supplying the precise theorem or hypotheses.
• Supplied literature reference labelled "Nolin-Qian-Sun-Zhuang backbone exponent": https://arxiv.org/abs/2309.05050. The record associates this reference with an exact evaluation of the monochromatic two-arm exponent, without supplying the precise theorem or hypotheses.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem partially solved, reports existence and strict bounds for monochromatic arm exponents and an exact monochromatic two-arm exponent, and describes exact values for the full higher-arm family and the general relation as remaining open. This is supplied dated curation derived from the OWR record, not independently verified. The source explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified, with the supplied explanation that the arm-exponent questions are complete and that indices, exponent notation, and asymptotic notation were restored. This records the registry's verification claim only; no source was independently visited for this draft.