Lamperti Limits of Multitype-Forest Profiles

Definitions and mathematical statement

A conventional generation profile can be defined as follows. Let $d\geq 1$, and let $F$ be a finite rooted forest with vertex types $\tau:V(F)\to\{1,\ldots,d\}$. For $k\in{\mathbb{Z}}_{\geq0}$, put \[ Z_F^{(i)}(k)=\#\{v\in V(F):\tau(v)=i,\ \operatorname{dist}(v,\text{root of its component})=k\},\qquad 1\leq i\leq d. \] The vector $Z_F(k)=(Z_F^{(1)}(k),\ldots,Z_F^{(d)}(k))$ is the type-resolved generation profile under this convention; in particular, generation zero consists of the roots. This definition supplies terminology, but the record does not establish that this is the precise profile convention intended by the source.

Supplied conjectural target. For uniform multitype forests with prescribed degree sequences, establish convergence of their profiles and characterize the limiting profile as a solution of a Lamperti transform.

Unresolved formulation. The supplied record does not specify a sequence of forest models, the meaning of a multitype degree sequence, the treatment of labels or child ordering, the root data, the number of types along the sequence, the asymptotic hypotheses, the profile normalization, or the mode and topology of convergence. It also supplies no defining Lamperti equation, driving processes, initial conditions, or boundary conventions. Consequently, the target above is a research objective rather than a complete quantified convergence conjecture. No particular scaling, limiting law, or Lamperti equation is asserted here.

The supplied original excerpt reads: "Finally, using our results, we conjecture the following. Population Dynamics and Statistical Physics in Synergy 631". It contains no mathematical formulation of the conjecture, so its omitted hypotheses and conclusion cannot be recovered from that excerpt.

Short English statement

The supplied conjecture asks whether suitably rescaled generation profiles of uniform multitype forests with prescribed degrees converge to a limit described by a Lamperti transform. The necessary model, scaling, convergence hypotheses, and limiting equation are missing, so a precise theorem statement requires further source evidence and human mathematical review.

Sources

• Local registry: data/unsolvedmath/problems.json, supplied record with permanent UnsolvedMath ID $30005045$, source problem code catalogue entry, title Lamperti Limits of Multitype-Forest Profiles. No individual proposer is supplied; the proposed year is recorded as 2022.
• Supplied report citation: Population Dynamics and Statistical Physics in Synergy (2023), Oberwolfach Reports, pp. 577--656, DOI 10.4171/owr/2022/12. https://doi.org/10.4171/owr/2022/12. The original excerpt includes the page marker 631 but omits the conjecture itself.
• Supplied literature source: Multitype random forests with a given degree sequence. https://arxiv.org/abs/2003.03036.
• Supplied literature source: Breadth first search coding of multitype forests. https://arxiv.org/abs/1410.0250.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and states that available work provides coding and partial convergence results, but not the full profile convergence and Lamperti characterization. This is supplied dated curation, not an independently verified literature assessment.
• The registry labels the statement corrected_verified and reports that it was reconstructed and checked against the official Oberwolfach report. These are supplied provenance claims; the cited sources and reconstruction have not been independently checked in this draft.