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We study models for a directed polymer in a random environment (DPRE) inwhich the polymer traverses a hierarchical diamond graph and the randomenvironment is defined through random variables attached to the vertices. Forthese models, we prove a distributional limit theorem for the partitionfunction in a limiting regime wherein the system grows as the coupling of thepolymer to the random environment is appropriately attenuated. The sequence ofdiamond graphs is determined by a choice of a branching number $b\in\{2,3,\ldots\}$ and segmenting number $s\in \{2,3,\ldots\}$, and our focus ison the critical case of the model where $b=s$. This extends recent work in thecritical case of analogous models with disorder variables placed at the edgesof the graphs rather than the vertices.

Weak-disorder limit for directed polymers on critical hierarchical graphs with vertex disorder