Resistance Exponents of Critical Branching Random-Walk Traces

Definitions and mathematical statement

A branching random walk combines a genealogical branching process with spatial displacements along its genealogical edges. In the usual Galton--Watson formulation, criticality means that the offspring random variable $\xi$ satisfies ${\mathbb{E}}[\xi]=1$. This condition alone does not specify the offspring law, the displacement law, or the conditioning of the process. The supplied record refers to dimension $d<6$, but does not specify the underlying spatial state space or the precise construction of the trace.

For reference, on a finite connected electrical network $G=(V,E)$ with edge resistances $r_e>0$, the effective resistance between distinct vertices $a,b\in V$ is \[ R_{\mathrm{eff}}^G(a,b)=\inf_{\theta:\,\operatorname{div}\theta=\delta_a-\delta_b}\sum_{e\in E}r_e\theta(e)^2, \] where each unoriented edge is counted once and $\theta$ is an antisymmetric flow on oriented edges. Applying this definition to a branching-random-walk trace requires specifying how the trace becomes an electrical network, its terminals or boundary sets, and, for an infinite network, the boundary convention.

Problem as supplied. For the trace of critical branching random walk in dimension $d<6$, determine the precise polynomial exponent governing the growth of its effective resistance $R(n)$.

Unresolved formulation. The record does not define $n$, the resistance terminals, the network conductances, or whether $R(n)$ denotes a random resistance or an averaged quantity. Once these choices have been fixed, one possible meaning of a polynomial exponent would be a number $\rho(d)$ satisfying \[ \frac{\log R(n)}{\log n}\longrightarrow\rho(d)\qquad(n\to\infty), \] with a specified mode of convergence when $R(n)$ is random. This is only a possible interpretation, not a formulation established by the supplied record. An exponent for ${\mathbb{E}}[R(n)]$, almost-sure growth, convergence in probability, and two-sided power bounds are distinct targets. The supplied evidence does not determine which target is intended, so a unique precise conjecture cannot yet be stated.

Short English statement

The question asks how effective resistance grows in a critical branching-random-walk trace in dimension $d<6$. The supplied record leaves the trace model, electrical network, meaning of $n$, and probabilistic interpretation of the exponent unspecified. These choices must be recovered from the cited sources before a precise conjecture can be fixed.

Sources

• Local UnsolvedMath registry: permanent ID $30002952$, title Resistance Exponents of Critical Branching Random-Walk Traces, collection Oberwolfach Reports Open Problems, exact source problem code catalogue entry. The supplied proposed year is 2015; no individual proposer is identified.
• Supplied source citation: Scaling Limits in Models of Statistical Mechanics (2016), Oberwolfach Reports, DOI 10.4171/owr/2015/40, pp. 2329--2398. https://doi.org/10.4171/owr/2015/40. The literature-source list labels this same reference Oberwolfach Report 40, 2015; the year discrepancy is unresolved.
• Supplied literature reference: Nachmias--Járai, Electrical Resistance of the Low Dimensional Critical Branching Random Walk. https://arxiv.org/abs/1305.1092.
• Supplied literature reference: Járai--Nachmias, Logarithmic Correction to Resistance. https://arxiv.org/abs/2006.09005.
• Supplied dated literature curation, checked at 2026-08-21: the record labels the problem open and reports sublinear upper bounds and work on critical corrections, while reporting no established precise polynomial growth exponent for $d<6$. This assessment is supplied dated curation derived from the OWR material, not an independently verified literature assessment. The background explicitly requests independent verification of time-sensitive status and cited claims.